1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
|
<!DOCTYPE html>
<html class="writer-html5" lang="en" >
<head>
<meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<title>Getting Started — GridTools 2.3.0 documentation</title>
<link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
<link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
<link rel="stylesheet" href="../_static/css/cscs.css" type="text/css" />
<!--[if lt IE 9]>
<script src="../_static/js/html5shiv.min.js"></script>
<![endif]-->
<script data-url_root="../" id="documentation_options" src="../_static/documentation_options.js"></script>
<script src="../_static/jquery.js"></script>
<script src="../_static/underscore.js"></script>
<script src="../_static/_sphinx_javascript_frameworks_compat.js"></script>
<script src="../_static/doctools.js"></script>
<script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<script src="../_static/js/theme.js"></script>
<link rel="index" title="Index" href="../genindex.html" />
<link rel="search" title="Search" href="../search.html" />
<link rel="next" title="User Manual" href="../user_manual/user_manual.html" />
<link rel="prev" title="Introduction" href="../introduction/introduction.html" />
</head>
<body class="wy-body-for-nav">
<div class="wy-grid-for-nav">
<nav data-toggle="wy-nav-shift" class="wy-nav-side">
<div class="wy-side-scroll">
<div class="wy-side-nav-search" >
<a href="../index.html" class="icon icon-home"> GridTools
<img src="../_static/logo.svg" class="logo" alt="Logo"/>
</a>
<div class="version">
2.3
</div>
<div role="search">
<form id="rtd-search-form" class="wy-form" action="../search.html" method="get">
<input type="text" name="q" placeholder="Search docs" />
<input type="hidden" name="check_keywords" value="yes" />
<input type="hidden" name="area" value="default" />
</form>
</div>
</div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu">
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="../introduction/introduction.html">Introduction</a></li>
<li class="toctree-l1 current"><a class="current reference internal" href="#">Getting Started</a><ul>
<li class="toctree-l2"><a class="reference internal" href="#coordinate-system">Coordinate System</a></li>
<li class="toctree-l2"><a class="reference internal" href="#storages">Storages</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#storage-traits">Storage Traits</a></li>
<li class="toctree-l3"><a class="reference internal" href="#building-storages">Building Storages</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="#stencils">Stencils</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#example-naive-2d-laplacian">Example: Naive 2D Laplacian</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="#first-gridtools-stencil">First GridTools Stencil</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#update-logic-gridtools-2d-laplacian">Update-logic: GridTools 2D Laplacian</a></li>
<li class="toctree-l3"><a class="reference internal" href="#calling-the-stencil">Calling the Stencil</a></li>
<li class="toctree-l3"><a class="reference internal" href="#full-gridtools-laplacian">Full GridTools Laplacian</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="#assembling-stencils-smoothing-filter">Assembling Stencils: Smoothing Filter</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#apply-method-overload"><cite>apply</cite>-method overload</a></li>
<li class="toctree-l3"><a class="reference internal" href="#gt-temporaries"><cite>GridTools</cite> Temporaries</a></li>
<li class="toctree-l3"><a class="reference internal" href="#functor-calls">Functor Calls</a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../user_manual/user_manual.html">User Manual</a></li>
<li class="toctree-l1"><a class="reference internal" href="../glossary/glossary.html">Glossary</a></li>
<li class="toctree-l1"><a class="reference internal" href="../internal/internal.html">Internal Documentation</a></li>
<li class="toctree-l1"><a class="reference internal" href="../faq/faq.html">Frequently Asked Questions</a></li>
</ul>
</div>
</div>
</nav>
<section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" >
<i data-toggle="wy-nav-top" class="fa fa-bars"></i>
<a href="../index.html">GridTools</a>
</nav>
<div class="wy-nav-content">
<div class="rst-content">
<div role="navigation" aria-label="Page navigation">
<ul class="wy-breadcrumbs">
<li><a href="../index.html" class="icon icon-home"></a> »</li>
<li>Getting Started</li>
<li class="wy-breadcrumbs-aside">
<a href="../_sources/getting_started/getting_started.rst.txt" rel="nofollow"> View page source</a>
</li>
</ul>
<hr/>
</div>
<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
<div itemprop="articleBody">
<section id="getting-started">
<span id="id1"></span><h1>Getting Started<a class="headerlink" href="#getting-started" title="Permalink to this heading"></a></h1>
<p>This chapter describes how to use <cite>GridTools</cite> to solve a (simple) PDE.
We will use a fourth-order horizontal smoothing filter
to explain the necessary steps to assemble a
stencil from scratch. We will not go into details in this chapter but
refer to later chapters for more details.</p>
<p>Our example PDE is given by</p>
<div class="math notranslate nohighlight">
\[\begin{split}\frac{\partial \phi}{\partial t} =\begin{cases}
- \alpha \nabla^4 \phi & z \leq z_\text{0}\\
0 & z > z_0
\end{cases}\end{split}\]</div>
<p>where <span class="math notranslate nohighlight">\(\nabla^4\)</span> is the squared two dimensional horizontal
Laplacian and we apply the filter only up to some maximal <span class="math notranslate nohighlight">\(z_0\)</span> (to make
the example more interesting). The filter is calculated in two steps:
first we calculate the Laplacian of <span class="math notranslate nohighlight">\(\phi\)</span></p>
<div class="math notranslate nohighlight">
\[L = \Delta \phi = \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) \phi\]</div>
<p>then we calculate the Laplacian of <span class="math notranslate nohighlight">\(L\)</span> as <span class="math notranslate nohighlight">\(-\alpha \nabla^4 \phi = -\alpha \Delta L\)</span>.</p>
<p>In the following we will walk through the following steps:</p>
<ul class="simple">
<li><p>The <cite>GridTools</cite> coordinate system and its notation.</p></li>
<li><p>Storages: how does <cite>GridTools</cite> manage the input and output fields.</p></li>
<li><p>The first stencil: calculating <span class="math notranslate nohighlight">\(L\)</span>, the second order Laplacian of <span class="math notranslate nohighlight">\(\phi\)</span>.</p></li>
<li><p>The final stencil: function calls, apply-method overloads and temporaries</p></li>
</ul>
<section id="coordinate-system">
<h2>Coordinate System<a class="headerlink" href="#coordinate-system" title="Permalink to this heading"></a></h2>
<p>For a finite difference discretization we restrict the field
<span class="math notranslate nohighlight">\(\phi \in \mathbb{R}^3\)</span> to a discrete grid. We use the notation
<span class="math notranslate nohighlight">\(i = x_i\)</span> and <span class="math notranslate nohighlight">\(j = y_j\)</span> for the horizontal dimension and <span class="math notranslate nohighlight">\(k = z_k\)</span> for
the vertical dimension, where <span class="math notranslate nohighlight">\(x_i, y_j, z_k\)</span> are the <span class="math notranslate nohighlight">\(x,y,z\)</span>
coordinates restricted on the grid. The <em>computation domain</em> is defined
by all grid points in our domain of interest</p>
<div class="math notranslate nohighlight">
\[\Lambda = (i,j,k) \quad \text{with}\quad i \in \{ 0\dots N_i-1\}, j \in \{0\dots N_j-1\}, k\in\{0 \dots N_k-1\}\]</div>
<p><cite>GridTools</cite> supports any number of dimension, however the iteration is always restricted to three dimensions, and <cite>GridTools</cite> will
treat one dimension, here the <span class="math notranslate nohighlight">\(k\)</span> dimension, differently: the <span class="math notranslate nohighlight">\(ij\)</span>-plane is executed in parallel while the
computation in <span class="math notranslate nohighlight">\(k\)</span> can be sequential. The consequence is that there must not be a dependency in <span class="math notranslate nohighlight">\(ij\)</span> within
a stencil while there can be a dependency in <span class="math notranslate nohighlight">\(k\)</span>. For now (this chapter) it is sufficient to just remember that
the <span class="math notranslate nohighlight">\(ij\)</span>-plane and the <span class="math notranslate nohighlight">\(k\)</span> dimension are treated differently by <cite>GridTools</cite>.</p>
<p>The calculation domain is surrounded by a <em>boundary region</em> as depicted
in <a class="reference internal" href="#fig-getting-started-coordinates"><span class="std std-numref">Fig. 1</span></a>. Computation happens
only within the calculation domain but values may be read from grid
points in the boundary region.</p>
<figure class="align-default" id="id2">
<span id="fig-getting-started-coordinates"></span><a class="reference internal image-reference" href="../_images/coordinates.png"><img alt="../_images/coordinates.png" src="../_images/coordinates.png" style="width: 558.0px; height: 303.59999999999997px;" /></a>
<figcaption>
<p><span class="caption-number">Fig. 1 </span><span class="caption-text">Coordinate system</span><a class="headerlink" href="#id2" title="Permalink to this image"></a></p>
</figcaption>
</figure>
</section>
<section id="storages">
<h2>Storages<a class="headerlink" href="#storages" title="Permalink to this heading"></a></h2>
<p>In this section we will set up the fields for our example: we need a
storage for the <span class="math notranslate nohighlight">\(\phi\)</span>-field (<code class="docutils literal notranslate"><span class="pre">phi_in</span></code>) and a storage for the output
(<code class="docutils literal notranslate"><span class="pre">phi_out</span></code>).</p>
<p>Storages in <cite>GridTools</cite> are n-dimensional array-like objects with the
following capabilities:</p>
<ul class="simple">
<li><p>access an element with <span class="math notranslate nohighlight">\((i,j,k)\)</span> syntax</p></li>
<li><p>synchronization between CPU memory and a device (e.g. a CUDA capable GPU)</p></li>
</ul>
<section id="storage-traits">
<span id="getting-started-storage-traits"></span><h3>Storage Traits<a class="headerlink" href="#storage-traits" title="Permalink to this heading"></a></h3>
<p>Since the storages depend on the architecture (e.g. CPU or GPU) our first step
is to define the <em>storage traits</em> type which typically looks like</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><gridtools/storage/gpu.hpp></span><span class="cp"></span>
<span class="k">using</span><span class="w"> </span><span class="n">traits_t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nn">storage</span><span class="o">::</span><span class="n">gpu</span><span class="p">;</span><span class="w"></span>
</pre></div>
</div>
<p>for the CUDA <a class="reference internal" href="../glossary/glossary.html#term-Storage-Traits"><span class="xref std std-term">Storage Traits</span></a> or</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><gridtools/storage/cpu_ifirst.hpp></span><span class="cp"></span>
<span class="k">using</span><span class="w"> </span><span class="n">traits_t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nn">storage</span><span class="o">::</span><span class="n">cpu_ifirst</span><span class="p">;</span><span class="w"></span>
</pre></div>
</div>
<p>for the CPU <a class="reference internal" href="../glossary/glossary.html#term-Storage-Traits"><span class="xref std std-term">Storage Traits</span></a>.</p>
</section>
<section id="building-storages">
<h3>Building Storages<a class="headerlink" href="#building-storages" title="Permalink to this heading"></a></h3>
<p>For efficient memory accesses the index ordering might depend on the target architecture, therefore the
memory layout will be implicitly decided by storage traits.</p>
<p><cite>GridTools</cite> storage classes don’t have user facing constructors. The builder design pattern is used instead.
The library provides the <code class="docutils literal notranslate"><span class="pre">storage::builder</span></code> object template that is instantiated by storage traits.
To create a storage we need to supply the builder with the desired storage properties by chaining
the setter methods and finally call the <code class="docutils literal notranslate"><span class="pre">build()</span></code> method which returns a <code class="docutils literal notranslate"><span class="pre">std::shared_ptr</span></code>
of a newly created storage. The builder also provides overloaded call operator which is a synonym of <code class="docutils literal notranslate"><span class="pre">build()</span></code>.
There are two required properties that need to be set:
the type of element, eg <code class="docutils literal notranslate"><span class="pre">.type<double>()</span></code>, and the sizes for each dimension, eg <code class="docutils literal notranslate"><span class="pre">.dimensions(10,</span> <span class="pre">12,</span> <span class="pre">20)</span></code>.
Other properties are optional.</p>
<p>If we need several storages that share some properties, we can construct a partially specified builder by
setting the shared properties and reuse it while building concrete storages.</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="n">uint_t</span><span class="w"> </span><span class="n">Ni</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span><span class="w"></span>
<span class="n">uint_t</span><span class="w"> </span><span class="n">Nj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">12</span><span class="p">;</span><span class="w"></span>
<span class="n">uint_t</span><span class="w"> </span><span class="n">Nk</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">20</span><span class="p">;</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">builder</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nn">storage</span><span class="o">::</span><span class="n">builder</span><span class="o"><</span><span class="n">traits_t</span><span class="o">></span><span class="p">.</span><span class="n">dimensions</span><span class="p">(</span><span class="n">Ni</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="p">,</span><span class="w"> </span><span class="n">Nk</span><span class="p">).</span><span class="n">id</span><span class="o"><</span><span class="mi">0</span><span class="o">></span><span class="p">();</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="n">phi</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">builder</span><span class="w"></span>
<span class="w"> </span><span class="p">.</span><span class="n">name</span><span class="p">(</span><span class="s">"phi"</span><span class="p">)</span><span class="w"> </span><span class="c1">//</span>
<span class="w"> </span><span class="p">.</span><span class="n">type</span><span class="o"><</span><span class="kt">float</span><span class="o">></span><span class="p">()</span><span class="w"> </span><span class="c1">//</span>
<span class="w"> </span><span class="p">.</span><span class="n">initializer</span><span class="p">([](</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="p">})</span><span class="w"> </span><span class="c1">//</span>
<span class="w"> </span><span class="p">.</span><span class="n">build</span><span class="p">();</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="n">lap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">builder</span><span class="p">.</span><span class="n">name</span><span class="p">(</span><span class="s">"lap"</span><span class="p">).</span><span class="n">type</span><span class="o"><</span><span class="kt">double</span><span class="o">></span><span class="p">().</span><span class="n">value</span><span class="p">(</span><span class="mi">-1</span><span class="p">).</span><span class="n">build</span><span class="p">();</span><span class="w"></span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>It is recommended to use the <code class="docutils literal notranslate"><span class="pre">id</span></code> property each time the <code class="docutils literal notranslate"><span class="pre">dimensions</span></code> property is set.
<code class="docutils literal notranslate"><span class="pre">id</span></code> should identify the unique set of dimension sizes.
This is because the stencil computation engine assumes that the storages that have the same <code class="docutils literal notranslate"><span class="pre">id</span></code> have the same
sizes. However, if only one set of dimensions is used, the <code class="docutils literal notranslate"><span class="pre">id</span></code> property can be skipped.</p>
</div>
<p>We can now</p>
<ul class="simple">
<li><p>retrieve the name of the field,</p></li>
<li><p>create a view and read and write values in the field using the parenthesis syntax,</p></li>
<li><p>query the lengths of each dimension.</p></li>
</ul>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="n">std</span><span class="o">::</span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">phi</span><span class="o">-></span><span class="n">name</span><span class="p">()</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="s">"</span><span class="se">\n</span><span class="s">"</span><span class="p">;</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="n">phi_view</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">phi</span><span class="o">-></span><span class="n">host_view</span><span class="p">();</span><span class="w"></span>
<span class="n">phi_view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">3.1415</span><span class="p">;</span><span class="w"></span>
<span class="n">std</span><span class="o">::</span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="s">"phi_view(1, 2, 3) = "</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">phi_view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">std</span><span class="o">::</span><span class="n">endl</span><span class="p">;</span><span class="w"></span>
<span class="n">std</span><span class="o">::</span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="s">"j length = "</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">phi</span><span class="o">-></span><span class="n">lengths</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">std</span><span class="o">::</span><span class="n">endl</span><span class="p">;</span><span class="w"></span>
</pre></div>
</div>
</section>
</section>
<section id="stencils">
<h2>Stencils<a class="headerlink" href="#stencils" title="Permalink to this heading"></a></h2>
<p>A <em>stencil</em> is a kernel that updates array elements according to a fixed
access pattern.</p>
<section id="example-naive-2d-laplacian">
<h3>Example: Naive 2D Laplacian<a class="headerlink" href="#example-naive-2d-laplacian" title="Permalink to this heading"></a></h3>
<p>The simplest discretization of the 2D Laplacian is the finite difference
five-point stencil as depicted in <a class="reference internal" href="#fig-getting-started-2dlap"><span class="std std-numref">Fig. 2</span></a>.</p>
<figure class="align-default" id="id3">
<span id="fig-getting-started-2dlap"></span><a class="reference internal image-reference" href="../_images/Laplacian2D.png"><img alt="../_images/Laplacian2D.png" src="../_images/Laplacian2D.png" style="width: 469.79999999999995px; height: 328.8px;" /></a>
<figcaption>
<p><span class="caption-number">Fig. 2 </span><span class="caption-text">Access pattern of a 2D Laplacian</span><a class="headerlink" href="#id3" title="Permalink to this image"></a></p>
</figcaption>
</figure>
<p>For the calculation of
the Laplacian at a given grid point we need the value at the grid point
itself and its four direct neighbors along the Cartesian axis.</p>
<p>A naive C++ implementation of the 2D Laplacian stencil looks as follows:</p>
<div class="highlight-cpp notranslate"><div class="highlight"><pre><span></span><span class="k">auto</span><span class="w"> </span><span class="n">laplacian</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[](</span><span class="k">auto</span><span class="w"> </span><span class="o">&</span><span class="n">lap</span><span class="p">,</span><span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="o">&</span><span class="n">in</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">boundary_size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">lengths</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">in</span><span class="p">.</span><span class="n">lengths</span><span class="p">();</span><span class="w"></span>
<span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">Ni</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">lengths</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">Nj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">lengths</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">Nk</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">lengths</span><span class="p">[</span><span class="mi">2</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">boundary_size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">Ni</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">boundary_size</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">boundary_size</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">Nj</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">boundary_size</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">boundary_size</span><span class="p">;</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">Nk</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">boundary_size</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="n">lap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">-4.0</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="c1">//</span>
<span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="c1">//</span>
<span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="p">};</span><span class="w"></span>
</pre></div>
</div>
<p>Apart from the initialization the stencil implementation
consists of 2 main components:</p>
<ul class="simple">
<li><p>Loop-logic: defines the stencil application domain and loop order</p></li>
<li><p>Update-logic: defines the update formula (here: the 2D Laplacian)</p></li>
</ul>
<p>Special care has to be taken at the boundary of the domain. Since the
Laplacian needs the neighboring points, we cannot calculate the Laplacian
on the boundary layer and have to exclude it from the loop.</p>
</section>
</section>
<section id="first-gridtools-stencil">
<h2>First GridTools Stencil<a class="headerlink" href="#first-gridtools-stencil" title="Permalink to this heading"></a></h2>
<p>In <cite>GridTools</cite> the loop logic and the storage order are implemented (and optimized) by the library while the update function, to
be applied to each gridpoint, is implemented by the user. The loop logic (for a given architecture) is combined with the
user-defined update function at compile-time by template meta-programming.</p>
<section id="update-logic-gridtools-2d-laplacian">
<h3>Update-logic: GridTools 2D Laplacian<a class="headerlink" href="#update-logic-gridtools-2d-laplacian" title="Permalink to this heading"></a></h3>
<p>The update-logic is implemented with state-less functors. A
<cite>GridTools</cite> functor is a <code class="docutils literal notranslate"><span class="pre">struct</span></code> or <code class="docutils literal notranslate"><span class="pre">class</span></code> providing a <em>static</em> method
called <code class="docutils literal notranslate"><span class="pre">apply</span></code>. The update-logic is implemented in these <a class="reference internal" href="../glossary/glossary.html#term-Apply-Method"><span class="xref std std-term">Apply-Methods</span></a>.
As the functors are state-less (no member variables, static methods
only) they can be passed by type, i.e. at compile-time, and therefore
allow for compile-time optimizations.</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="k">struct</span><span class="w"> </span><span class="nc">lap_function</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="n">in</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">in_accessor</span><span class="o"><</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="k">extent</span><span class="o"><</span><span class="mi">-1</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">-1</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="o">>></span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="n">lap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">inout_accessor</span><span class="o"><</span><span class="mi">1</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="k">param_list</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">make_param_list</span><span class="o"><</span><span class="n">in</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">template</span><span class="w"> </span><span class="o"><</span><span class="k">typename</span><span class="w"> </span><span class="nc">Evaluation</span><span class="o">></span><span class="w"></span>
<span class="w"> </span><span class="cp">GT_FUNCTION</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="n">apply</span><span class="p">(</span><span class="n">Evaluation</span><span class="w"> </span><span class="o">&&</span><span class="n">eval</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="n">eval</span><span class="p">(</span><span class="n">lap</span><span class="p">())</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">eval</span><span class="p">(</span><span class="mf">-4.</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">in</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="mi">-1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">-1</span><span class="p">));</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="p">};</span><span class="w"></span>
</pre></div>
</div>
<p>In addition to the <code class="docutils literal notranslate"><span class="pre">apply</span></code>-method, the functor contains <code class="docutils literal notranslate"><span class="pre">accessor</span></code> s. These
two <code class="docutils literal notranslate"><span class="pre">accessor</span></code> s are parameters of the functor, i.e. they are mapped to
fields passed to the functor. They contain compile-time information if
they are only used as input parameters, e.g. the <code class="docutils literal notranslate"><span class="pre">in</span></code> accessor in the
example, or if we want to write into the associated field (<code class="docutils literal notranslate"><span class="pre">inout</span></code>). Additionally,
the <code class="docutils literal notranslate"><span class="pre">extent</span></code> defines which grid points are needed by the stencil relative
to the current point. The format for the extent is</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="k">extent</span><span class="o"><</span><span class="n">i_minus</span><span class="p">,</span><span class="w"> </span><span class="n">i_plus</span><span class="p">,</span><span class="w"> </span><span class="n">j_minus</span><span class="p">,</span><span class="w"> </span><span class="n">j_plus</span><span class="p">,</span><span class="w"> </span><span class="n">k_minus</span><span class="p">,</span><span class="w"> </span><span class="n">k_plus</span><span class="o">></span><span class="w"></span>
</pre></div>
</div>
<p>where <code class="docutils literal notranslate"><span class="pre">i_minus</span></code> and <code class="docutils literal notranslate"><span class="pre">i_plus</span></code> define an interval on the <span class="math notranslate nohighlight">\(i\)</span>-axis relative to
the current position; <code class="docutils literal notranslate"><span class="pre">i_minus</span></code> is the negative offset, i.e. zero or a
negative number, while <code class="docutils literal notranslate"><span class="pre">i_plus</span></code> is the positive offset. Analogously for
<span class="math notranslate nohighlight">\(j\)</span> and <span class="math notranslate nohighlight">\(k\)</span>. In the Laplacian example, the first two numbers
in the extent of the <code class="docutils literal notranslate"><span class="pre">in</span></code> accessor define that we want to access the
field at <span class="math notranslate nohighlight">\(i-1\)</span>, <span class="math notranslate nohighlight">\(i\)</span> and <span class="math notranslate nohighlight">\(i+1\)</span>. The accessor type and the extent is needed for a
dependency analysis in the compile-time optimizations for more complex
stencils. (For example, the computation
domain needs to be extended when we calculate the Laplacian of the Laplacian later. This is done automatically by the
library.)</p>
<p>The first template argument is an index defining the order of the
parameters, i.e. the order in which the fields are passed to the
functor. The <code class="docutils literal notranslate"><span class="pre">param_list</span></code> is a <cite>GridTools</cite> keyword which has to be defined for each stencil,
and should contain the list of accessors.</p>
<p>A <code class="docutils literal notranslate"><span class="pre">apply</span></code>-method needs as first parameter a context
object, usually called <code class="docutils literal notranslate"><span class="pre">eval</span></code>, which is created and passed to the method by the library on
invocation. This object contains, among other things, the index of the
active grid point (<a class="reference internal" href="../glossary/glossary.html#term-Iteration-Point"><span class="xref std std-term">Iteration Point</span></a>) and the mapping of data-pointers to the <code class="docutils literal notranslate"><span class="pre">accessor</span></code> s. The
second argument is optional and specifies the interval on the <span class="math notranslate nohighlight">\(k\)</span>-axis where this implementation
of the <a class="reference internal" href="../glossary/glossary.html#term-Apply-Method"><span class="xref std std-term">Apply-Method</span></a> should be executed. This allows to apply a different update-logic on
<a class="reference internal" href="../glossary/glossary.html#term-Vertical-Interval"><span class="xref std std-term">Vertical Intervals</span></a> by overloading the <a class="reference internal" href="../glossary/glossary.html#term-Apply-Method"><span class="xref std std-term">Apply-Method</span></a>. We will define <a class="reference internal" href="../glossary/glossary.html#term-Vertical-Interval"><span class="xref std std-term">Vertical Intervals</span></a>
later. If the second parameter is not specified, a default interval is assumed.</p>
<p>The body of the <code class="docutils literal notranslate"><span class="pre">apply</span></code>-method looks quite similar to the one in the
naive implementation, except that each
field access has to be wrapped by a call to the context object <code class="docutils literal notranslate"><span class="pre">eval</span></code>.
This is necessary to map the compile-time parameter, the <code class="docutils literal notranslate"><span class="pre">accessor</span></code>, to
the run-time data in the <code class="docutils literal notranslate"><span class="pre">data_store</span></code>.</p>
</section>
<section id="calling-the-stencil">
<h3>Calling the Stencil<a class="headerlink" href="#calling-the-stencil" title="Permalink to this heading"></a></h3>
<p>In the naive implementation, the call to the <code class="docutils literal notranslate"><span class="pre">laplacian</span></code> is as simple as</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="kt">int</span><span class="w"> </span><span class="n">boundary_size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
<span class="n">laplacian</span><span class="p">(</span><span class="w"> </span><span class="n">lap</span><span class="p">,</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">boundary_size</span><span class="w"> </span><span class="p">);</span><span class="w"></span>
</pre></div>
</div>
<p>since it contains already all the information: the update-logic <em>and</em> the loop-logic.</p>
<p>The <cite>GridTools</cite> stencil, does not contain any
information about the loop-logic, i.e. about the domain where we want to apply the stencil operation,
since we need to specify it in a platform-independent syntax, a <em>domain specific embedded language</em>
(DSEL), such that the <a class="reference internal" href="../glossary/glossary.html#term-Backend"><span class="xref std std-term">Backend</span></a> can decide on the specific implementation.</p>
<p>For our example this looks as follows</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="linenos"> 1</span><span class="n">uint_t</span><span class="w"> </span><span class="n">Ni</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span><span class="w"></span>
<span class="linenos"> 2</span><span class="n">uint_t</span><span class="w"> </span><span class="n">Nj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">12</span><span class="p">;</span><span class="w"></span>
<span class="linenos"> 3</span><span class="n">uint_t</span><span class="w"> </span><span class="n">Nk</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">20</span><span class="p">;</span><span class="w"></span>
<span class="linenos"> 4</span><span class="kt">int</span><span class="w"> </span><span class="n">halo</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
<span class="linenos"> 5</span>
<span class="linenos"> 6</span><span class="k">auto</span><span class="w"> </span><span class="n">builder</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nn">storage</span><span class="o">::</span><span class="n">builder</span><span class="o"><</span><span class="n">storage_traits_t</span><span class="o">></span><span class="w"> </span><span class="c1">//</span>
<span class="linenos"> 7</span><span class="w"> </span><span class="p">.</span><span class="n">type</span><span class="o"><</span><span class="kt">double</span><span class="o">></span><span class="p">()</span><span class="w"> </span><span class="c1">//</span>
<span class="linenos"> 8</span><span class="w"> </span><span class="p">.</span><span class="n">dimensions</span><span class="p">(</span><span class="n">Ni</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="p">,</span><span class="w"> </span><span class="n">Nk</span><span class="p">)</span><span class="w"> </span><span class="c1">//</span>
<span class="linenos"> 9</span><span class="w"> </span><span class="p">.</span><span class="n">halos</span><span class="p">(</span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span><span class="w"> </span><span class="c1">//</span>
<span class="linenos">10</span>
<span class="linenos">11</span><span class="k">auto</span><span class="w"> </span><span class="n">phi</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">builder</span><span class="p">();</span><span class="w"></span>
<span class="linenos">12</span><span class="k">auto</span><span class="w"> </span><span class="n">lap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">builder</span><span class="p">();</span><span class="w"></span>
<span class="linenos">13</span>
<span class="linenos">14</span><span class="k">halo_descriptor</span><span class="w"> </span><span class="nf">boundary_i</span><span class="p">(</span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">Ni</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">halo</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">Ni</span><span class="p">);</span><span class="w"></span>
<span class="linenos">15</span><span class="k">halo_descriptor</span><span class="w"> </span><span class="nf">boundary_j</span><span class="p">(</span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">halo</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="p">);</span><span class="w"></span>
<span class="linenos">16</span><span class="k">auto</span><span class="w"> </span><span class="n">grid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nl">make_grid</span><span class="p">(</span><span class="n">boundary_i</span><span class="p">,</span><span class="w"> </span><span class="n">boundary_j</span><span class="p">,</span><span class="w"> </span><span class="n">Nk</span><span class="p">);</span><span class="w"></span>
<span class="linenos">17</span>
<span class="linenos">18</span><span class="nl">run_single_stage</span><span class="p">(</span><span class="n">lap_function</span><span class="p">(),</span><span class="w"> </span><span class="n">stencil_backend_t</span><span class="p">(),</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="p">);</span><span class="w"></span>
</pre></div>
</div>
<p>In lines 14-16 we setup the physical dimension of the problem.
First we define which points on the <span class="math notranslate nohighlight">\(i\)</span> and the <span class="math notranslate nohighlight">\(j\)</span>-axis belong
to the computational domain and which points belong to the boundary (or
a padding region). For now it is enough to know that these lines define
a region with a boundary of size 1 surrounding the <span class="math notranslate nohighlight">\(ij\)</span>-plane. In the
next lines the layers in <span class="math notranslate nohighlight">\(k\)</span> are defined. In this case we have only one
interval. We will discuss the details later.</p>
<p>In line 18 we execute the stencil computation. In our example only one stencil participates.
Hence, we can use the simple <code class="docutils literal notranslate"><span class="pre">run_single_stage</span></code> API. We pass the stencil the backend object, the grid
(the information about the loop bounds) and the storages on which the computation needs to be executed.
The number and the order of storage arguments should match the number of stencil accessors.</p>
</section>
<section id="full-gridtools-laplacian">
<h3>Full GridTools Laplacian<a class="headerlink" href="#full-gridtools-laplacian" title="Permalink to this heading"></a></h3>
<p>The full working example looks as follows:</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="linenos"> 1</span><span class="cm">/*</span>
<span class="linenos"> 2</span><span class="cm"> * GridTools</span>
<span class="linenos"> 3</span><span class="cm"> *</span>
<span class="linenos"> 4</span><span class="cm"> * Copyright (c) 2014-2021, ETH Zurich</span>
<span class="linenos"> 5</span><span class="cm"> * All rights reserved.</span>
<span class="linenos"> 6</span><span class="cm"> *</span>
<span class="linenos"> 7</span><span class="cm"> * Please, refer to the LICENSE file in the root directory.</span>
<span class="linenos"> 8</span><span class="cm"> * SPDX-License-Identifier: BSD-3-Clause</span>
<span class="linenos"> 9</span><span class="cm"> */</span><span class="w"></span>
<span class="linenos">10</span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><gridtools/common/defs.hpp></span><span class="cp"></span>
<span class="linenos">11</span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><gridtools/stencil/cartesian.hpp></span><span class="cp"></span>
<span class="linenos">12</span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><gridtools/storage/builder.hpp></span><span class="cp"></span>
<span class="linenos">13</span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><gridtools/storage/sid.hpp></span><span class="cp"></span>
<span class="linenos">14</span>
<span class="linenos">15</span><span class="k">using</span><span class="w"> </span><span class="k">namespace</span><span class="w"> </span><span class="nn">gridtools</span><span class="p">;</span><span class="w"></span>
<span class="linenos">16</span><span class="k">using</span><span class="w"> </span><span class="k">namespace</span><span class="w"> </span><span class="nn">stencil</span><span class="p">;</span><span class="w"></span>
<span class="linenos">17</span><span class="k">using</span><span class="w"> </span><span class="k">namespace</span><span class="w"> </span><span class="nn">cartesian</span><span class="p">;</span><span class="w"></span>
<span class="linenos">18</span><span class="k">using</span><span class="w"> </span><span class="k">namespace</span><span class="w"> </span><span class="nn">expressions</span><span class="p">;</span><span class="w"></span>
<span class="linenos">19</span>
<span class="linenos">20</span><span class="cp">#ifdef GT_CUDACC</span>
<span class="linenos">21</span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><gridtools/stencil/gpu.hpp></span><span class="cp"></span>
<span class="linenos">22</span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><gridtools/storage/gpu.hpp></span><span class="cp"></span>
<span class="linenos">23</span><span class="k">using</span><span class="w"> </span><span class="n">stencil_backend_t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stencil</span><span class="o">::</span><span class="n">gpu</span><span class="o"><></span><span class="p">;</span><span class="w"></span>
<span class="linenos">24</span><span class="k">using</span><span class="w"> </span><span class="n">storage_traits_t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nn">storage</span><span class="o">::</span><span class="n">gpu</span><span class="p">;</span><span class="w"></span>
<span class="linenos">25</span><span class="cp">#else</span>
<span class="linenos">26</span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><gridtools/stencil/cpu_ifirst.hpp></span><span class="cp"></span>
<span class="linenos">27</span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><gridtools/storage/cpu_ifirst.hpp></span><span class="cp"></span>
<span class="linenos">28</span><span class="k">using</span><span class="w"> </span><span class="n">stencil_backend_t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stencil</span><span class="o">::</span><span class="n">cpu_ifirst</span><span class="o"><></span><span class="p">;</span><span class="w"></span>
<span class="linenos">29</span><span class="k">using</span><span class="w"> </span><span class="n">storage_traits_t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nn">storage</span><span class="o">::</span><span class="n">cpu_ifirst</span><span class="p">;</span><span class="w"></span>
<span class="linenos">30</span><span class="cp">#endif</span>
<span class="linenos">31</span>
<span class="linenos">32</span><span class="k">struct</span><span class="w"> </span><span class="nc">lap_function</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="linenos">33</span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="n">in</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">in_accessor</span><span class="o"><</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="k">extent</span><span class="o"><</span><span class="mi">-1</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">-1</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="o">>></span><span class="p">;</span><span class="w"></span>
<span class="linenos">34</span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="n">lap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">inout_accessor</span><span class="o"><</span><span class="mi">1</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="linenos">35</span>
<span class="linenos">36</span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="k">param_list</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">make_param_list</span><span class="o"><</span><span class="n">in</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="linenos">37</span>
<span class="linenos">38</span><span class="w"> </span><span class="k">template</span><span class="w"> </span><span class="o"><</span><span class="k">typename</span><span class="w"> </span><span class="nc">Evaluation</span><span class="o">></span><span class="w"></span>
<span class="linenos">39</span><span class="w"> </span><span class="cp">GT_FUNCTION</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="n">apply</span><span class="p">(</span><span class="n">Evaluation</span><span class="w"> </span><span class="o">&&</span><span class="n">eval</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="linenos">40</span><span class="w"> </span><span class="n">eval</span><span class="p">(</span><span class="n">lap</span><span class="p">())</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">eval</span><span class="p">(</span><span class="mf">-4.</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">in</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="mi">-1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">-1</span><span class="p">));</span><span class="w"></span>
<span class="linenos">41</span><span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="linenos">42</span><span class="p">};</span><span class="w"></span>
<span class="linenos">43</span>
<span class="linenos">44</span><span class="kt">int</span><span class="w"> </span><span class="nf">main</span><span class="p">()</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="linenos">45</span><span class="w"> </span><span class="n">uint_t</span><span class="w"> </span><span class="n">Ni</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span><span class="w"></span>
<span class="linenos">46</span><span class="w"> </span><span class="n">uint_t</span><span class="w"> </span><span class="n">Nj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">12</span><span class="p">;</span><span class="w"></span>
<span class="linenos">47</span><span class="w"> </span><span class="n">uint_t</span><span class="w"> </span><span class="n">Nk</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">20</span><span class="p">;</span><span class="w"></span>
<span class="linenos">48</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">halo</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
<span class="linenos">49</span>
<span class="linenos">50</span><span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">builder</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">storage</span><span class="o">::</span><span class="n">builder</span><span class="o"><</span><span class="n">storage_traits_t</span><span class="o">></span><span class="w"> </span><span class="c1">//</span>
<span class="linenos">51</span><span class="w"> </span><span class="p">.</span><span class="n">type</span><span class="o"><</span><span class="kt">double</span><span class="o">></span><span class="p">()</span><span class="w"> </span><span class="c1">//</span>
<span class="linenos">52</span><span class="w"> </span><span class="p">.</span><span class="n">dimensions</span><span class="p">(</span><span class="n">Ni</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="p">,</span><span class="w"> </span><span class="n">Nk</span><span class="p">)</span><span class="w"> </span><span class="c1">//</span>
<span class="linenos">53</span><span class="w"> </span><span class="p">.</span><span class="n">halos</span><span class="p">(</span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span><span class="w"> </span><span class="c1">//</span>
<span class="linenos">54</span>
<span class="linenos">55</span><span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">phi</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">builder</span><span class="p">();</span><span class="w"></span>
<span class="linenos">56</span><span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">lap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">builder</span><span class="p">();</span><span class="w"></span>
<span class="linenos">57</span>
<span class="linenos">58</span><span class="w"> </span><span class="n">halo_descriptor</span><span class="w"> </span><span class="n">boundary_i</span><span class="p">(</span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">Ni</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">halo</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">Ni</span><span class="p">);</span><span class="w"></span>
<span class="linenos">59</span><span class="w"> </span><span class="n">halo_descriptor</span><span class="w"> </span><span class="n">boundary_j</span><span class="p">(</span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">halo</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="p">);</span><span class="w"></span>
<span class="linenos">60</span><span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">grid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">make_grid</span><span class="p">(</span><span class="n">boundary_i</span><span class="p">,</span><span class="w"> </span><span class="n">boundary_j</span><span class="p">,</span><span class="w"> </span><span class="n">Nk</span><span class="p">);</span><span class="w"></span>
<span class="linenos">61</span>
<span class="linenos">62</span><span class="w"> </span><span class="n">run_single_stage</span><span class="p">(</span><span class="n">lap_function</span><span class="p">(),</span><span class="w"> </span><span class="n">stencil_backend_t</span><span class="p">(),</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="p">);</span><span class="w"></span>
<span class="linenos">63</span><span class="p">}</span><span class="w"></span>
</pre></div>
</div>
<p>There are some points which we did not discuss so far. For a first look at <cite>GridTools</cite> these can be considered fixed patterns and
we won’t discuss them now in detail.</p>
<p>A common pattern is to use the preprocessor flag <code class="docutils literal notranslate"><span class="pre">__CUDACC__</span></code> to distinguish between CPU and GPU code.
Here we use the <cite>GridTools</cite> internal <code class="docutils literal notranslate"><span class="pre">GT_CUDA</span></code> macro to do this. The macro is used to set the correct <a class="reference internal" href="../glossary/glossary.html#term-Backend"><span class="xref std std-term">Backend</span></a>
and <a class="reference internal" href="../glossary/glossary.html#term-Storage-Traits"><span class="xref std std-term">Storage Traits</span></a> for the CUDA or CPU architecture, respectively.</p>
<p>The code example can be compiled using the following simple CMake script (requires an installation of <cite>GridTools</cite>, see <a class="reference internal" href="../introduction/introduction.html#installation"><span class="std std-ref">Installing and Validating GridTools</span></a>).</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">cmake_minimum_required</span><span class="p">(</span><span class="n">VERSION</span> <span class="mf">3.18.1</span><span class="p">)</span>
<span class="n">project</span><span class="p">(</span><span class="n">GridTools</span><span class="o">-</span><span class="n">laplacian</span> <span class="n">LANGUAGES</span> <span class="n">CXX</span><span class="p">)</span>
<span class="nb">set</span><span class="p">(</span><span class="n">CMAKE_CXX_EXTENSIONS</span> <span class="n">OFF</span><span class="p">)</span> <span class="c1"># Currently required for HIP</span>
<span class="n">find_package</span><span class="p">(</span><span class="n">GridTools</span> <span class="mf">2.2.0</span> <span class="n">REQUIRED</span> <span class="n">QUIET</span><span class="p">)</span>
<span class="k">if</span><span class="p">(</span><span class="n">TARGET</span> <span class="n">GridTools</span><span class="p">::</span><span class="n">stencil_cpu_ifirst</span><span class="p">)</span>
<span class="n">add_executable</span><span class="p">(</span><span class="n">gt_laplacian</span> <span class="n">test_gt_laplacian</span><span class="o">.</span><span class="n">cpp</span><span class="p">)</span>
<span class="n">target_link_libraries</span><span class="p">(</span><span class="n">gt_laplacian</span> <span class="n">GridTools</span><span class="p">::</span><span class="n">gridtools</span> <span class="n">GridTools</span><span class="p">::</span><span class="n">stencil_cpu_ifirst</span><span class="p">)</span>
<span class="n">endif</span><span class="p">()</span>
</pre></div>
</div>
</section>
</section>
<section id="assembling-stencils-smoothing-filter">
<h2>Assembling Stencils: Smoothing Filter<a class="headerlink" href="#assembling-stencils-smoothing-filter" title="Permalink to this heading"></a></h2>
<p>In the preceding section we saw how a first simple <cite>GridTools</cite> stencil
is defined and executed. In this section we will use this stencil to
compute our example PDE. A naive implementation could look as in</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="k">auto</span><span class="w"> </span><span class="n">naive_smoothing</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[](</span><span class="k">auto</span><span class="w"> </span><span class="o">&</span><span class="n">out</span><span class="p">,</span><span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="o">&</span><span class="n">in</span><span class="p">,</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">alpha</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">kmax</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">lap_boundary</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">full_boundary</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">Ni</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">in</span><span class="p">.</span><span class="n">lengths</span><span class="p">()[</span><span class="mi">0</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">Nj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">in</span><span class="p">.</span><span class="n">lengths</span><span class="p">()[</span><span class="mi">1</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">Nk</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">in</span><span class="p">.</span><span class="n">lengths</span><span class="p">()[</span><span class="mi">2</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="c1">// Instantiate temporary fields</span>
<span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">make_storage</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">storage_builder</span><span class="p">.</span><span class="n">dimensions</span><span class="p">(</span><span class="n">Ni</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="p">,</span><span class="w"> </span><span class="n">Nk</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">lap_storage</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">make_storage</span><span class="p">();</span><span class="w"></span>
<span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">lap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">lap_storage</span><span class="o">-></span><span class="n">target_view</span><span class="p">();</span><span class="w"></span>
<span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">laplap_storage</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">make_storage</span><span class="p">();</span><span class="w"></span>
<span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">laplap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">laplap_storage</span><span class="o">-></span><span class="n">target_view</span><span class="p">();</span><span class="w"></span>
<span class="w"> </span><span class="c1">// laplacian of phi</span>
<span class="w"> </span><span class="n">laplacian</span><span class="p">(</span><span class="n">lap</span><span class="p">,</span><span class="w"> </span><span class="n">in</span><span class="p">,</span><span class="w"> </span><span class="n">lap_boundary</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="c1">// laplacian of lap</span>
<span class="w"> </span><span class="n">laplacian</span><span class="p">(</span><span class="n">laplap</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="p">,</span><span class="w"> </span><span class="n">full_boundary</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">full_boundary</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">Ni</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">full_boundary</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">full_boundary</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">Nj</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">full_boundary</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">full_boundary</span><span class="p">;</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">Nk</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">full_boundary</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">kmax</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="n">out</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">alpha</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">laplap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="k">else</span><span class="w"></span>
<span class="w"> </span><span class="n">out</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">in</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="p">};</span><span class="w"></span>
</pre></div>
</div>
<p>For the <cite>GridTools</cite> implementation we will learn three things in this
section: how to define special regions in the <span class="math notranslate nohighlight">\(k\)</span>-direction; how to use
<cite>GridTools</cite> temporaries and how to call functors from functors.</p>
<section id="apply-method-overload">
<h3><cite>apply</cite>-method overload<a class="headerlink" href="#apply-method-overload" title="Permalink to this heading"></a></h3>
<p>Our first <cite>GridTools</cite> implementation will be very close to the naive
implementation: we will call two times the Laplacian functor from the
previous section and store the result in two extra fields. Then we will
call a third functor to compute the final result. This functor shows how
we can specialize the computation in the <span class="math notranslate nohighlight">\(k\)</span>-direction:</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="k">struct</span><span class="w"> </span><span class="nc">smoothing_function_1</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="n">phi</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">in_accessor</span><span class="o"><</span><span class="mi">0</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="n">laplap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">in_accessor</span><span class="o"><</span><span class="mi">1</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="n">out</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">inout_accessor</span><span class="o"><</span><span class="mi">2</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="k">param_list</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">make_param_list</span><span class="o"><</span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">laplap</span><span class="p">,</span><span class="w"> </span><span class="n">out</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">constexpr</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">alpha</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.5</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">template</span><span class="w"> </span><span class="o"><</span><span class="k">typename</span><span class="w"> </span><span class="nc">Evaluation</span><span class="o">></span><span class="w"></span>
<span class="w"> </span><span class="cp">GT_FUNCTION</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="n">apply</span><span class="p">(</span><span class="n">Evaluation</span><span class="w"> </span><span class="o">&</span><span class="n">eval</span><span class="p">,</span><span class="w"> </span><span class="n">lower_domain</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="n">eval</span><span class="p">(</span><span class="n">out</span><span class="p">())</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">eval</span><span class="p">(</span><span class="n">phi</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">alpha</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">laplap</span><span class="p">());</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="w"> </span><span class="k">template</span><span class="w"> </span><span class="o"><</span><span class="k">typename</span><span class="w"> </span><span class="nc">Evaluation</span><span class="o">></span><span class="w"></span>
<span class="w"> </span><span class="cp">GT_FUNCTION</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="n">apply</span><span class="p">(</span><span class="n">Evaluation</span><span class="w"> </span><span class="o">&</span><span class="n">eval</span><span class="p">,</span><span class="w"> </span><span class="n">upper_domain</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="n">eval</span><span class="p">(</span><span class="n">out</span><span class="p">())</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">eval</span><span class="p">(</span><span class="n">phi</span><span class="p">());</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="p">};</span><span class="w"></span>
</pre></div>
</div>
<p>We use two different
<a class="reference internal" href="../glossary/glossary.html#term-Vertical-Interval"><span class="xref std std-term">Vertical Intervals</span></a>, the <code class="docutils literal notranslate"><span class="pre">lower_domain</span></code> and the <code class="docutils literal notranslate"><span class="pre">upper_domain</span></code>, and provide an overload of the
<a class="reference internal" href="../glossary/glossary.html#term-Apply-Method"><span class="xref std std-term">Apply-Method</span></a> for each interval.</p>
<p>The <a class="reference internal" href="../glossary/glossary.html#term-Vertical-Interval"><span class="xref std std-term">Vertical Intervals</span></a> are defined as</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="k">using</span><span class="w"> </span><span class="n">axis_t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">axis</span><span class="o"><</span><span class="mi">2</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="k">using</span><span class="w"> </span><span class="n">lower_domain</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">axis_t</span><span class="o">::</span><span class="n">get_interval</span><span class="o"><</span><span class="mi">0</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="k">using</span><span class="w"> </span><span class="n">upper_domain</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">axis_t</span><span class="o">::</span><span class="n">get_interval</span><span class="o"><</span><span class="mi">1</span><span class="o">></span><span class="p">;</span><span class="w"></span>
</pre></div>
</div>
<p>The first line defines an axis with 2 <a class="reference internal" href="../glossary/glossary.html#term-Vertical-Interval"><span class="xref std std-term">Vertical Intervals</span></a>. From this axis retrieve the <a class="reference internal" href="../glossary/glossary.html#term-Vertical-Interval"><span class="xref std std-term">Vertical Intervals</span></a>
and give them a name.</p>
<p>Then we can assemble the computation</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="k">auto</span><span class="w"> </span><span class="n">phi</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">make_storage</span><span class="p">();</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="n">phi_new</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">make_storage</span><span class="p">();</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="n">lap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">make_storage</span><span class="p">();</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="n">laplap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">make_storage</span><span class="p">();</span><span class="w"></span>
<span class="k">halo_descriptor</span><span class="w"> </span><span class="nf">boundary_i</span><span class="p">(</span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">Ni</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">halo</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">Ni</span><span class="p">);</span><span class="w"></span>
<span class="k">halo_descriptor</span><span class="w"> </span><span class="nf">boundary_j</span><span class="p">(</span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">halo</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="p">);</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="n">grid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nl">make_grid</span><span class="p">(</span><span class="n">boundary_i</span><span class="p">,</span><span class="w"> </span><span class="n">boundary_j</span><span class="p">,</span><span class="w"> </span><span class="n">axis_t</span><span class="p">(</span><span class="n">kmax</span><span class="p">,</span><span class="w"> </span><span class="n">Nk</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">kmax</span><span class="p">));</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">spec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[](</span><span class="k">auto</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">phi_new</span><span class="p">,</span><span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">lap</span><span class="p">,</span><span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">laplap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nl">execute_parallel</span><span class="p">()</span><span class="w"></span>
<span class="w"> </span><span class="p">.</span><span class="n">stage</span><span class="p">(</span><span class="n">lap_function</span><span class="p">(),</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="p">.</span><span class="n">stage</span><span class="p">(</span><span class="n">lap_function</span><span class="p">(),</span><span class="w"> </span><span class="n">lap</span><span class="p">,</span><span class="w"> </span><span class="n">laplap</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="p">.</span><span class="n">stage</span><span class="p">(</span><span class="n">smoothing_function_1</span><span class="p">(),</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">laplap</span><span class="p">,</span><span class="w"> </span><span class="n">phi_new</span><span class="p">);</span><span class="w"></span>
<span class="p">};</span><span class="w"></span>
<span class="nl">run</span><span class="p">(</span><span class="n">spec</span><span class="p">,</span><span class="w"> </span><span class="n">stencil_backend_t</span><span class="p">(),</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">phi_new</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="p">,</span><span class="w"> </span><span class="n">laplap</span><span class="p">);</span><span class="w"></span>
</pre></div>
</div>
<p>We cannot use the <code class="docutils literal notranslate"><span class="pre">run_single_stage</span></code> API because now we need to compose a <a class="reference internal" href="../glossary/glossary.html#term-Computation"><span class="xref std std-term">Computation</span></a> from three stencil calls. To achieve that
we use the full featured <code class="docutils literal notranslate"><span class="pre">run</span></code> API. It requires a stencil composition specification as a first parameter. That specification
is provided in form of a generic lambda. Its arguments represent the storages that are used in the computation.
The expression that is returned describes how the stencils should be composed. It is where the <cite>GridTools</cite> DSL is used.
<code class="docutils literal notranslate"><span class="pre">execute_parallel()</span></code> means that each <span class="math notranslate nohighlight">\(k\)</span>-level can be executed in parallel. The <code class="docutils literal notranslate"><span class="pre">stage</span></code> clause represents
a call to the stencil with the given arguments.</p>
<p>In this version we needed to explicitly allocate the temporary fields
<code class="docutils literal notranslate"><span class="pre">lap</span></code> and <code class="docutils literal notranslate"><span class="pre">laplap</span></code>. In the next section we will learn about
<cite>GridTools</cite> temporaries.</p>
</section>
<section id="gt-temporaries">
<h3><cite>GridTools</cite> Temporaries<a class="headerlink" href="#gt-temporaries" title="Permalink to this heading"></a></h3>
<p><cite>GridTools</cite> <em>temporary storages</em> are storages with the lifetime of the
<code class="docutils literal notranslate"><span class="pre">computation</span></code>. This is exactly what we need for the <code class="docutils literal notranslate"><span class="pre">lap</span></code> and <code class="docutils literal notranslate"><span class="pre">laplap</span></code>
fields.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Note that temporaries are not allocated explicitly and we cannot
access them from outside of the computation. Therefore, sometimes it might be
necessary to replace a temporary by a normal storage for debugging.</p>
</div>
<p>To use temporary storages we exclude the correspondent fields from the arguments of our <code class="docutils literal notranslate"><span class="pre">spec</span></code> and declare them as
temporaries, within the <cite>spec</cite> lambda, using the <code class="docutils literal notranslate"><span class="pre">GT_DECLARE_TMP</span></code> macro. We don’t need the explicit
instantiations any more and don’t have to pass them to the <code class="docutils literal notranslate"><span class="pre">run</span></code>. The new code looks as follows</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="k">auto</span><span class="w"> </span><span class="n">phi</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">make_storage</span><span class="p">();</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="n">phi_new</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">make_storage</span><span class="p">();</span><span class="w"></span>
<span class="k">halo_descriptor</span><span class="w"> </span><span class="nf">boundary_i</span><span class="p">(</span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">Ni</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">halo</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">Ni</span><span class="p">);</span><span class="w"></span>
<span class="k">halo_descriptor</span><span class="w"> </span><span class="nf">boundary_j</span><span class="p">(</span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">halo</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="p">);</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="n">grid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nl">make_grid</span><span class="p">(</span><span class="n">boundary_i</span><span class="p">,</span><span class="w"> </span><span class="n">boundary_j</span><span class="p">,</span><span class="w"> </span><span class="n">axis_t</span><span class="p">{</span><span class="n">kmax</span><span class="p">,</span><span class="w"> </span><span class="n">Nk</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">kmax</span><span class="p">});</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">spec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[](</span><span class="k">auto</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">phi_new</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="cp">GT_DECLARE_TMP</span><span class="p">(</span><span class="kt">double</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="p">,</span><span class="w"> </span><span class="n">laplap</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nl">execute_parallel</span><span class="p">()</span><span class="w"></span>
<span class="w"> </span><span class="p">.</span><span class="n">stage</span><span class="p">(</span><span class="n">lap_function</span><span class="p">(),</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="p">.</span><span class="n">stage</span><span class="p">(</span><span class="n">lap_function</span><span class="p">(),</span><span class="w"> </span><span class="n">lap</span><span class="p">,</span><span class="w"> </span><span class="n">laplap</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="p">.</span><span class="n">stage</span><span class="p">(</span><span class="n">smoothing_function_1</span><span class="p">(),</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">laplap</span><span class="p">,</span><span class="w"> </span><span class="n">phi_new</span><span class="p">);</span><span class="w"></span>
<span class="p">};</span><span class="w"></span>
<span class="nl">run</span><span class="p">(</span><span class="n">spec</span><span class="p">,</span><span class="w"> </span><span class="n">stencil_backend_t</span><span class="p">(),</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">phi_new</span><span class="p">);</span><span class="w"></span>
</pre></div>
</div>
<p>Besides the simplifications in the code (no explicit storage needed), the
concept of temporaries allows <cite>GridTools</cite> to apply optimization. While normal storages
have a fixed size, temporaries can have block-private <a class="reference internal" href="../glossary/glossary.html#term-Halo"><span class="xref std std-term">Halos</span></a> which are used for redundant computation.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>It might be semantically incorrect to replace a temporary with a normal storage, as normal storages don’t have the <a class="reference internal" href="../glossary/glossary.html#term-Halo"><span class="xref std std-term">Halo</span></a>
region for redundant computation. In such case several threads (OpenMP or CUDA) will write the same location multiple
times. As long as all threads write the same data (which is a requirement for correctness of <cite>GridTools</cite>), this should be
no problem for correctness on current hardware (might change in the future) but might have side-effects on performance.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This change from normal storages to temporaries did not require any code changes to the functor.</p>
</div>
</section>
<section id="functor-calls">
<h3>Functor Calls<a class="headerlink" href="#functor-calls" title="Permalink to this heading"></a></h3>
<p>The next feature we want to use is the <em>stencil function call</em>. In the first example we computed the Laplacian
and the Laplacian of the Laplacian explicitly and stored the intermediate values in the temporaries. Stencil function
calls will allow us do the computation on the fly and will allow us to get rid of the temporaries.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Note that this is
not necessarily a performance optimization. It might well be that the version with temporaries is actually the
faster one.</p>
</div>
<p>In the following we will remove only one of the temporaries. Instead of calling the Laplacian twice from the
<code class="docutils literal notranslate"><span class="pre">spec</span></code>, we will move one of the calls into the smoothing functor. The new smoothing functor looks as follows</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="k">struct</span><span class="w"> </span><span class="nc">smoothing_function_3</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="n">phi</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">in_accessor</span><span class="o"><</span><span class="mi">0</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="n">lap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">in_accessor</span><span class="o"><</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">extent</span><span class="o"><</span><span class="mi">-1</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">-1</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="o">>></span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="n">out</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">inout_accessor</span><span class="o"><</span><span class="mi">2</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="k">param_list</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">make_param_list</span><span class="o"><</span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="p">,</span><span class="w"> </span><span class="n">out</span><span class="o">></span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">constexpr</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">alpha</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.5</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">template</span><span class="w"> </span><span class="o"><</span><span class="k">typename</span><span class="w"> </span><span class="nc">Evaluation</span><span class="o">></span><span class="w"></span>
<span class="w"> </span><span class="cp">GT_FUNCTION</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="n">apply</span><span class="p">(</span><span class="n">Evaluation</span><span class="w"> </span><span class="o">&</span><span class="n">eval</span><span class="p">,</span><span class="w"> </span><span class="n">lower_domain</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="n">eval</span><span class="p">(</span><span class="n">out</span><span class="p">())</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">eval</span><span class="p">(</span><span class="n">phi</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">alpha</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">call</span><span class="o"><</span><span class="n">lap_function</span><span class="o">>::</span><span class="k">with</span><span class="p">(</span><span class="n">eval</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="p">()));</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="w"> </span><span class="k">template</span><span class="w"> </span><span class="o"><</span><span class="k">typename</span><span class="w"> </span><span class="nc">Evaluation</span><span class="o">></span><span class="w"></span>
<span class="w"> </span><span class="cp">GT_FUNCTION</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="n">apply</span><span class="p">(</span><span class="n">Evaluation</span><span class="w"> </span><span class="o">&</span><span class="n">eval</span><span class="p">,</span><span class="w"> </span><span class="n">upper_domain</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="n">eval</span><span class="p">(</span><span class="n">out</span><span class="p">())</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">eval</span><span class="p">(</span><span class="n">phi</span><span class="p">());</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="p">};</span><span class="w"></span>
</pre></div>
</div>
<p>In <code class="docutils literal notranslate"><span class="pre">call</span></code> we specify the functor which we want to apply. In <code class="docutils literal notranslate"><span class="pre">with</span></code> the <code class="docutils literal notranslate"><span class="pre">eval</span></code> is forwarded, followed by all the
input arguments for the functor. The functor in the call is required to have exactly one <code class="docutils literal notranslate"><span class="pre">inout_accessor</span></code> which will
be the return value of the call. Note that <code class="docutils literal notranslate"><span class="pre">smoothing_function_3</span></code> still needs to specify the extents explicitly;
for functor calls they cannot be inferred automatically.</p>
<p>One of the <code class="docutils literal notranslate"><span class="pre">stage(lap_function(),</span> <span class="pre">...)</span></code> was now moved inside of the functor, therefore the new spec is just:</p>
<div class="highlight-gridtools notranslate"><div class="highlight"><pre><span></span><span class="k">auto</span><span class="w"> </span><span class="n">phi</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">make_storage</span><span class="p">();</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="n">phi_new</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">make_storage</span><span class="p">();</span><span class="w"></span>
<span class="k">halo_descriptor</span><span class="w"> </span><span class="nf">boundary_i</span><span class="p">(</span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">Ni</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">halo</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">Ni</span><span class="p">);</span><span class="w"></span>
<span class="k">halo_descriptor</span><span class="w"> </span><span class="nf">boundary_j</span><span class="p">(</span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">halo</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">halo</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">Nj</span><span class="p">);</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="n">grid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nl">make_grid</span><span class="p">(</span><span class="n">boundary_i</span><span class="p">,</span><span class="w"> </span><span class="n">boundary_j</span><span class="p">,</span><span class="w"> </span><span class="n">axis_t</span><span class="p">{</span><span class="n">kmax</span><span class="p">,</span><span class="w"> </span><span class="n">Nk</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">kmax</span><span class="p">});</span><span class="w"></span>
<span class="k">auto</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">spec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[](</span><span class="k">auto</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="k">auto</span><span class="w"> </span><span class="n">phi_new</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="cp">GT_DECLARE_TMP</span><span class="p">(</span><span class="kt">double</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nl">execute_parallel</span><span class="p">()</span><span class="w"> </span><span class="c1">//</span>
<span class="w"> </span><span class="p">.</span><span class="n">stage</span><span class="p">(</span><span class="n">lap_function</span><span class="p">(),</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="p">)</span><span class="w"> </span><span class="c1">//</span>
<span class="w"> </span><span class="p">.</span><span class="n">stage</span><span class="p">(</span><span class="n">smoothing_function_3</span><span class="p">(),</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">lap</span><span class="p">,</span><span class="w"> </span><span class="n">phi_new</span><span class="p">);</span><span class="w"> </span><span class="c1">//</span>
<span class="p">};</span><span class="w"></span>
<span class="nl">run</span><span class="p">(</span><span class="n">spec</span><span class="p">,</span><span class="w"> </span><span class="n">stencil_backend_t</span><span class="p">(),</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">phi</span><span class="p">,</span><span class="w"> </span><span class="n">phi_new</span><span class="p">);</span><span class="w"></span>
</pre></div>
</div>
<p>The attentive reader may have noticed that our first versions did more
work than needed: we calculated the Laplacian of the Laplacian of phi
(<span class="math notranslate nohighlight">\(\Delta \Delta \phi\)</span>) for all <span class="math notranslate nohighlight">\(k\)</span>-levels, however we used it only for
<span class="math notranslate nohighlight">\(k<k_\text{max}\)</span>. In this version we do a bit better: we still calculate
the Laplacian (<span class="math notranslate nohighlight">\(L = \Delta \phi\)</span>) for all levels but we only calculate
<span class="math notranslate nohighlight">\(\Delta L\)</span> for the levels where we need it.</p>
</section>
</section>
</section>
</div>
</div>
<footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer">
<a href="../introduction/introduction.html" class="btn btn-neutral float-left" title="Introduction" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a>
<a href="../user_manual/user_manual.html" class="btn btn-neutral float-right" title="User Manual" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a>
</div>
<hr/>
<div role="contentinfo">
<p>© Copyright 2019, ETH Zurich.</p>
</div>
Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a
<a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a>
provided by <a href="https://readthedocs.org">Read the Docs</a>.
</footer>
</div>
</div>
</section>
</div>
<script>
jQuery(function () {
SphinxRtdTheme.Navigation.enable(true);
});
</script>
</body>
</html>
|
