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<a name="Nonlinear-Least_002dSquares-Large-Example"></a>
<div class="header">
<p>
Previous: <a href="Nonlinear-Least_002dSquares-Comparison-Example.html#Nonlinear-Least_002dSquares-Comparison-Example" accesskey="p" rel="previous">Nonlinear Least-Squares Comparison Example</a>, Up: <a href="Nonlinear-Least_002dSquares-Examples.html#Nonlinear-Least_002dSquares-Examples" accesskey="u" rel="up">Nonlinear Least-Squares Examples</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Large-Nonlinear-Least-Squares-Example"></a>
<h4 class="subsection">39.12.4 Large Nonlinear Least Squares Example</h4>
<p>The following program illustrates the large nonlinear least
squares solvers on a system with significant sparse structure
in the Jacobian. The cost function is given by
</p>
<div class="example">
<pre class="example">\Phi(x) &= 1/2 \sum_{i=1}^{p+1} f_i^2
f_i &= \sqrt{\alpha} (x_i - 1), 1 \le i \le p
f_{p+1} &= ||x||^2 - 1/4
</pre></div>
<p>with <em>\alpha = 10^{-5}</em>. The residual <em>f_{p+1}</em> imposes a constraint on the <em>p</em>
parameters <em>x</em>, to ensure that <em>||x||^2 \approx {1 \over 4}</em>.
The <em>(p+1)</em>-by-<em>p</em> Jacobian for this system is given by
</p>
<div class="example">
<pre class="example">J(x) = [ \sqrt{alpha} I_p; 2 x^T ]
</pre></div>
<p>and the normal equations matrix is given by
</p>
<div class="example">
<pre class="example">J^T J = [ \alpha I_p + 4 x x^T ]
</pre></div>
<p>Finally, the second directional derivative of <em>f</em> for the
geodesic acceleration method is given by
</p>
<div class="example">
<pre class="example">fvv = [ 0; 2 ||v||^2 ]
</pre></div>
<p>Since the upper <em>p</em>-by-<em>p</em> block of <em>J</em> is diagonal,
this sparse structure should be exploited in the nonlinear solver.
For comparison, the following program solves the system for <em>p = 2000</em>
using the dense direct Cholesky solver based on the normal equations matrix
<em>J^T J</em>, as well as the iterative Steihaug-Toint solver, based on
sparse matrix-vector products <em>J u</em> and <em>J^T u</em>. The
program output is shown below.
</p>
<div class="smallformat">
<pre class="verbatim">Method NITER NFEV NJUEV NJTJEV NAEV Init Cost Final cost cond(J) Final |x|^2 Time (s)
levenberg-marquardt 25 31 26 26 0 7.1218e+18 1.9555e-02 447.50 2.5044e-01 46.28
levenberg-marquardt+accel 22 23 45 23 22 7.1218e+18 1.9555e-02 447.64 2.5044e-01 33.92
dogleg 37 87 36 36 0 7.1218e+18 1.9555e-02 447.59 2.5044e-01 56.05
double-dogleg 35 88 34 34 0 7.1218e+18 1.9555e-02 447.62 2.5044e-01 52.65
2D-subspace 37 88 36 36 0 7.1218e+18 1.9555e-02 447.71 2.5044e-01 59.75
steihaug-toint 35 88 345 0 0 7.1218e+18 1.9555e-02 inf 2.5044e-01 0.09
</pre></div>
<p>The first five rows use methods based on factoring the dense <em>J^T J</em> matrix
while the last row uses the iterative Steihaug-Toint method. While the number
of Jacobian matrix-vector products (NJUEV) is less for the dense methods, the added time
to construct and factor the <em>J^T J</em> matrix (NJTJEV) results in a much larger runtime than the
iterative method (see last column).
</p>
<p>The program is given below.
</p>
<div class="example">
<pre class="verbatim">#include <stdlib.h>
#include <stdio.h>
#include <sys/time.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_multilarge_nlinear.h>
#include <gsl/gsl_spblas.h>
#include <gsl/gsl_spmatrix.h>
/* parameters for functions */
struct model_params
{
double alpha;
gsl_spmatrix *J;
};
/* penalty function */
int
penalty_f (const gsl_vector * x, void *params, gsl_vector * f)
{
struct model_params *par = (struct model_params *) params;
const double sqrt_alpha = sqrt(par->alpha);
const size_t p = x->size;
size_t i;
double sum = 0.0;
for (i = 0; i < p; ++i)
{
double xi = gsl_vector_get(x, i);
gsl_vector_set(f, i, sqrt_alpha*(xi - 1.0));
sum += xi * xi;
}
gsl_vector_set(f, p, sum - 0.25);
return GSL_SUCCESS;
}
int
penalty_df (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x,
const gsl_vector * u, void * params, gsl_vector * v,
gsl_matrix * JTJ)
{
struct model_params *par = (struct model_params *) params;
const size_t p = x->size;
size_t j;
/* store 2*x in last row of J */
for (j = 0; j < p; ++j)
{
double xj = gsl_vector_get(x, j);
gsl_spmatrix_set(par->J, p, j, 2.0 * xj);
}
/* compute v = op(J) u */
if (v)
gsl_spblas_dgemv(TransJ, 1.0, par->J, u, 0.0, v);
if (JTJ)
{
gsl_vector_view diag = gsl_matrix_diagonal(JTJ);
/* compute J^T J = [ alpha*I_p + 4 x x^T ] */
gsl_matrix_set_zero(JTJ);
/* store 4 x x^T in lower half of JTJ */
gsl_blas_dsyr(CblasLower, 4.0, x, JTJ);
/* add alpha to diag(JTJ) */
gsl_vector_add_constant(&diag.vector, par->alpha);
}
return GSL_SUCCESS;
}
int
penalty_fvv (const gsl_vector * x, const gsl_vector * v,
void *params, gsl_vector * fvv)
{
const size_t p = x->size;
double normv = gsl_blas_dnrm2(v);
gsl_vector_set_zero(fvv);
gsl_vector_set(fvv, p, 2.0 * normv * normv);
(void)params; /* avoid unused parameter warning */
return GSL_SUCCESS;
}
void
solve_system(const gsl_vector *x0, gsl_multilarge_nlinear_fdf *fdf,
gsl_multilarge_nlinear_parameters *params)
{
const gsl_multilarge_nlinear_type *T = gsl_multilarge_nlinear_trust;
const size_t max_iter = 200;
const double xtol = 1.0e-8;
const double gtol = 1.0e-8;
const double ftol = 1.0e-8;
const size_t n = fdf->n;
const size_t p = fdf->p;
gsl_multilarge_nlinear_workspace *work =
gsl_multilarge_nlinear_alloc(T, params, n, p);
gsl_vector * f = gsl_multilarge_nlinear_residual(work);
gsl_vector * x = gsl_multilarge_nlinear_position(work);
int info;
double chisq0, chisq, rcond, xsq;
struct timeval tv0, tv1;
gettimeofday(&tv0, NULL);
/* initialize solver */
gsl_multilarge_nlinear_init(x0, fdf, work);
/* store initial cost */
gsl_blas_ddot(f, f, &chisq0);
/* iterate until convergence */
gsl_multilarge_nlinear_driver(max_iter, xtol, gtol, ftol,
NULL, NULL, &info, work);
gettimeofday(&tv1, NULL);
/* store final cost */
gsl_blas_ddot(f, f, &chisq);
/* compute final ||x||^2 */
gsl_blas_ddot(x, x, &xsq);
/* store cond(J(x)) */
gsl_multilarge_nlinear_rcond(&rcond, work);
/* print summary */
fprintf(stderr, "%-25s %-5zu %-4zu %-5zu %-6zu %-4zu %-10.4e %-10.4e %-7.2f %-11.4e %.2f\n",
gsl_multilarge_nlinear_trs_name(work),
gsl_multilarge_nlinear_niter(work),
fdf->nevalf,
fdf->nevaldfu,
fdf->nevaldf2,
fdf->nevalfvv,
chisq0,
chisq,
1.0 / rcond,
xsq,
(tv1.tv_sec - tv0.tv_sec) + 1.0e-6 * (tv1.tv_usec - tv0.tv_usec));
gsl_multilarge_nlinear_free(work);
}
int
main (void)
{
const size_t p = 2000;
const size_t n = p + 1;
gsl_vector *f = gsl_vector_alloc(n);
gsl_vector *x = gsl_vector_alloc(p);
/* allocate sparse Jacobian matrix with 2*p non-zero elements in triplet format */
gsl_spmatrix *J = gsl_spmatrix_alloc_nzmax(n, p, 2 * p, GSL_SPMATRIX_TRIPLET);
gsl_multilarge_nlinear_fdf fdf;
gsl_multilarge_nlinear_parameters fdf_params =
gsl_multilarge_nlinear_default_parameters();
struct model_params params;
size_t i;
params.alpha = 1.0e-5;
params.J = J;
/* define function to be minimized */
fdf.f = penalty_f;
fdf.df = penalty_df;
fdf.fvv = penalty_fvv;
fdf.n = n;
fdf.p = p;
fdf.params = &params;
for (i = 0; i < p; ++i)
{
/* starting point */
gsl_vector_set(x, i, i + 1.0);
/* store sqrt(alpha)*I_p in upper p-by-p block of J */
gsl_spmatrix_set(J, i, i, sqrt(params.alpha));
}
fprintf(stderr, "%-25s %-4s %-4s %-5s %-6s %-4s %-10s %-10s %-7s %-11s %-10s\n",
"Method", "NITER", "NFEV", "NJUEV", "NJTJEV", "NAEV", "Init Cost",
"Final cost", "cond(J)", "Final |x|^2", "Time (s)");
fdf_params.scale = gsl_multilarge_nlinear_scale_levenberg;
fdf_params.trs = gsl_multilarge_nlinear_trs_lm;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_lmaccel;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_dogleg;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_ddogleg;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_subspace2D;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_cgst;
solve_system(x, &fdf, &fdf_params);
gsl_vector_free(f);
gsl_vector_free(x);
gsl_spmatrix_free(J);
return 0;
}
</pre></div>
<hr>
<div class="header">
<p>
Previous: <a href="Nonlinear-Least_002dSquares-Comparison-Example.html#Nonlinear-Least_002dSquares-Comparison-Example" accesskey="p" rel="previous">Nonlinear Least-Squares Comparison Example</a>, Up: <a href="Nonlinear-Least_002dSquares-Examples.html#Nonlinear-Least_002dSquares-Examples" accesskey="u" rel="up">Nonlinear Least-Squares Examples</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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