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
|
// ------------------------------------------------------------------------
//
// SPDX-License-Identifier: LGPL-2.1-or-later
// Copyright (C) 2021 - 2025 by the deal.II authors
//
// This file is part of the deal.II library.
//
// Part of the source code is dual licensed under Apache-2.0 WITH
// LLVM-exception OR LGPL-2.1-or-later. Detailed license information
// governing the source code and code contributions can be found in
// LICENSE.md and CONTRIBUTING.md at the top level directory of deal.II.
//
// ------------------------------------------------------------------------
#ifndef dealii_function_tools_h
#define dealii_function_tools_h
#include <deal.II/base/config.h>
#include <deal.II/base/bounding_box.h>
#include <deal.II/base/function.h>
#include <array>
DEAL_II_NAMESPACE_OPEN
namespace FunctionTools
{
/**
* Estimate bounds on the value and bounds on each gradient component of a
* Function, $f$, over a BoundingBox, by approximating it by a 2nd order
* Taylor polynomial starting from the box center.
*
* Each lower and upper bound is returned as a
* <code>std::pair<double, double></code>, such that the first entry is the
* lower bound, $L$, and the second is the upper bound, $U$, i.e.
* $f(x) \in [L, U]$.
*
* The function value, gradient, and Hessian are computed at the box center.
* The bounds on the value of the function are then estimated as
*
* $f(x) \in [f(x_c) - F, f(x_c) + F]$,
* where
* $F = \sum_i |\partial_i f(x_c)| h_i
* + 1/2 \sum_i \sum_j |\partial_i \partial_j f(x_c)| h_i h_j$.
*
* Here, $h_i$ is half the side length of the box in the $i$th coordinate
* direction, which is the distance we extrapolate. The bounds on the gradient
* components are estimated similarly as
*
* $\partial_i f \in [\partial_i f(x_c) - G_i, \partial_i f(x_c) + G_i]$,
* where
* $G_i = \sum_j |\partial_i \partial_j f(x_c)| h_j$.
*
* If the function has more than 1 component the @p component parameter can
* be used to specify which function component the bounds should be computed
* for.
*/
template <int dim>
void
taylor_estimate_function_bounds(
const Function<dim> &function,
const BoundingBox<dim> &box,
std::pair<double, double> &value_bounds,
std::array<std::pair<double, double>, dim> &gradient_bounds,
const unsigned int component = 0);
} // namespace FunctionTools
DEAL_II_NAMESPACE_CLOSE
#endif /* dealii_function_tools_h */
|
