In this example, an integral equation is to be solved by representing the discretised operator with an H-matrix, whereby the H-matrix is constructed using a matrix coefficient function for entries of the equation system. Furthermore, for the iterative solver a preconditioner is computed using H-LU factorisation.
Problem Description
Given the integral equation
\begin{displaymath} \int_0^1 \log|x-y| \mathbf{u}(y) dy = \mathbf{f}(x), \quad x \in [0,1] \end{displaymath}
with \(\mathbf{u} : [0,1] \to \mathbf{R}\) being sought for a given right hand side \(\mathbf{f} : [0,1] \to \mathbf{R}\), the Galerkin discretisation with constant ansatz functions \(\phi_i, 0 \le i < n\)
\[ \phi_i(x) = \left\{ \begin{array}{ll} 1, & x \in \left[\frac{i}{n},\frac{i+1}{n} \right] \\ 0, & \mathrm{otherwise} \end{array} \right. \]
leads to a linear equation system \(A u = f\) where \(u\) contains the coefficients of the discretised \(\mathbf{u}\) and \(A\) is defined by
\begin{eqnarray*} a_{ij} & = & \int_0^1 \int_0^1 \phi_i(x) \log|x-y| \phi_j(y) dy dx \\ & = & \int_{\frac{i}{n}}^{\frac{i+1}{n}} \int_{\frac{j}{n}}^{\frac{j+1}{n}} \log|x-y| dy dx \nonumber . \end{eqnarray*}
The right hand side \(f\) is given by
\[ f_i = \int_0^1 \phi_i(x) f(x) dx. \]
Coordinates and Cluster Trees
The code starts with the standard initialisation:
#include <stdio.h>
#include <hlib-c.h>
int main ( int argc, char ** argv )
{
int info;
HLIB_FNDECL void hlib_init(int *info)
HLIB_FNDECL void hlib_set_verbosity(const unsigned int verb)
Together with the initialisation, the verbosity level of HLIBpro was increased to have additional output during execution. The macro CHECK_INFO is defined as
#define CHECK_INFO { if ( info != HLIB_NO_ERROR ) \
{ char buf[1024]; hlib_error_desc( buf, 1024 ); \
printf( "\n%s\n\n", buf ); exit(1); } }
and tests the result of a 𝖧𝖫𝖨𝖡𝗉𝗋𝗈 function call. If an error occured, the description of this error is queried using hlib_error_desc and printed to the screen.
For clustering the unknowns, coordinate information is to be defined. For this 1d example, the indices are placed in the middle of equally sized intervals.
int i;
int n = ...
double h = 1.0 / ((double) n);
double ** vertices;
hlib_coord_t coord;
vertices = (double**) malloc( n * sizeof(double*) );
for ( i = 0; i < n; i++ )
{
vertices[i] = (double*) malloc( sizeof(double) );
vertices[i][0] = h * ((double) i) + (h / 2.0);
}
HLIB_FNDECL hlib_coord_t hlib_coord_import(const size_t n, const unsigned int dim, double **coord, const double *period, int *info)
Coordinates in HLIBpro are vectors of the corresponding dimension, e.g. one in for this problem.
Having coordinates for each index, the cluster tree and block cluster tree can be constructed.
hlib_clustertree_t ct;
hlib_admcond_t adm;
hlib_blockclustertree_t bct;
HLIB_FNDECL hlib_clustertree_t hlib_clt_build_bsp(const hlib_coord_t coord, const hlib_bsp_t bsptype, const unsigned int nmin, int *info)
HLIB_FNDECL hlib_blockclustertree_t hlib_bct_build(const hlib_clustertree_t rowct, const hlib_clustertree_t colct, const hlib_admcond_t ac, int *info)
HLIB_FNDECL hlib_admcond_t hlib_admcond_geom(const hlib_adm_t crit, const double eta, int *info)
@ HLIB_BSP_AUTO
Definition: hlib-c.h:118
@ HLIB_ADM_STD_MIN
Definition: hlib-c.h:139
For the cluster tree, the partitioning strategy for the indices is determined automatically by choosing HLIB_BSP_AUTO. For the block cluster tree, the standard admissibility condition is used as defined by HLIB_ADM_STD_MIN with \(\eta = 2\).
Matrix Construction and Coefficient Function
Using the block cluster tree, finally the H-matrix can be constructed. For this, adaptive cross approximation (ACA, see ???) is applied to compute low rank approximations of admissibly matrix blocks. In HLIBpro, an improved version of ACA, ACA+ is implemented.
ACA only needs access to the matrix coefficients of the dense matrix. Those are provided by a coefficient function kernel with an optional parameter h, which will be discussed below. Furthermore, the block wise accuracy of the H-matrix approximation has to be defined. In this case, an accuracy of \(\epsilon = 10^{-4}\) is used and stored in acc. Finally, a flag indicates, whether the matrix is symmetric or not.
hlib_matrix_t A;
acc, 1, & info ); CHECK_INFO;
HLIB_FNDECL hlib_matrix_t hlib_matrix_build_coeff(const hlib_blockclustertree_t bct, const hlib_coeff_t f, void *arg, const hlib_lrapx_t lrapx, const hlib_acc_t acc, const int sym, int *info)
HLIB_FNDECL hlib_acc_t hlib_acc_fixed_eps(const hlib_real_t eps)
@ HLIB_LRAPX_ACAPLUS
Definition: hlib-c.h:205
The coefficient function used during matrix construction comes in the form of a callback function:
void
kernel ( const size_t n, const int * rowidx,
const size_t m, const int * colidx,
hlib_real_t * matrix, void * arg )
{
int rowi, colj;
double h = *((double*) arg);
for ( colj = 0; colj < m; colj++ )
{
const int idx1 = colidx[colj];
for ( rowi = 0; rowi < n; rowi++ )
{
const int idx0 = rowidx[rowi];
double value;
if ( idx0 == idx1 )
value = -1.5*h*h + h*h*log(h);
else
{
const double dist = h * ( fabs( (double) (idx0-idx1) ) - 1.0 );
const double t1 = dist+1.0*h;
const double t2 = dist+2.0*h;
value = ( - 1.5*h*h + 0.5*t2*t2*log(t2) - t1*t1*log(t1) );
if ( fabs(dist) > 1e-8 )
value += 0.5*dist*dist*log(dist);
}
matrix[(colj*n) + rowi] = -value;
}
}
}
It is called, whenever matrix coefficients are needed. Usually, a complete block of coefficients is requested by the internal matrix constructor, e.g. a full dense block in case of an inadmissible leaf or a full row/column in case of ACA. The row (column) indices for the requested coefficients are stored in rowidx (colidx) and are of size n (m). In the array matrix the coefficients have to be stored in column-wise order, e.g. the coefficient with the i'th row index and the j'th column index at position (i,j) or at matrix[j*n+i]. The final argument arg is the additional parameter to hlib_matrix_build_coeff at can be used to provided necessary data for the coefficient function. Here, it is the stepwidth \( h \) of the problem.
For a complex coefficient functions the callback function is identical except the value type of matrix, which is of type hlib_complex_t.
H-LU Factorisation
Having constructed the H-matrix, the corresponding equation system shall be solved. In most cases, a standard iterative solve will not be sufficient, e.g. either the convergence rate is to bad or there is no convergence at all. Therefore, a (good) preconditioner is needed, with the inverse of \( A \) being the best possible. Since iterative schemes, only need matrix-vector multiplication, the corresponding functionality for the inverse is sufficient. This is provided by an (H-) LU factorisation of \( A \), or for the symmetric case, a LDL factorisation:
hlib_matrix_t B;
hlib_linearoperator_t A_inv;
HLIB_FNDECL hlib_linearoperator_t hlib_matrix_factorise_inv(hlib_matrix_t A, const hlib_acc_t acc, int *info)
HLIB_FNDECL hlib_matrix_t hlib_matrix_copy(const hlib_matrix_t A, int *info)
Since A is modified during the factorisation, a copy B is constructed. For the factorisation the accuracy is chosen to be equal as during matrix construction. The result of the factorisation is linear operator represented by the type hlib_linearoperator_t, which can be used to perform matrix vector multiplication. In this case, the evaluation of the inverse of \( A \) is provided.
Iterative Solvers
Still missing is the right hand side of the equation system. In this example, it was chosen, such that \(\mathbf{u} \equiv \mathbf{1}\):
double * b_arr = (hlib_real_t *) malloc( n * sizeof(hlib_real_t) );
for ( i = 0; i < n; i++ )
b_arr[i] = rhs( i, n );
HLIB_FNDECL hlib_permutation_t hlib_clt_perm_e2i(hlib_clustertree_t ct, int *info)
HLIB_FNDECL void hlib_vector_permute(hlib_vector_t v, const hlib_permutation_t perm, int *info)
HLIB_FNDECL void hlib_vector_import(hlib_vector_t v, const hlib_real_t *arr, int *info)
HLIB_FNDECL hlib_vector_t hlib_vector_build(const size_t size, int *info)
Here, hlib_vector_import copies the coefficients from b_arr to b and rhs is defined by
double
rhs ( int i, int n )
{
const double a = ((double)i) / ((double) n);
const double b = ((double)i+1.0) / ((double) n);
double value = -1.5 * (b - a);
if ( fabs( b ) > 1e-8 ) value += 0.5*b*b*log(b);
if ( fabs( a ) > 1e-8 ) value -= 0.5*a*a*log(a);
if ( fabs( 1.0 - b ) > 1e-8 ) value -= 0.5*(1.0-b)*(1.0-b)*log(1.0-b);
if ( fabs( 1.0 - a ) > 1e-8 ) value += 0.5*(1.0-a)*(1.0-a)*log(1.0-a);
return value;
}
For the iterative solver the special auto solver is used, which automatically chooses a suitable solver for the given combination of matrix and preconditioner. Furthermore, statistics about the solution process is stored in solve_info, e.g. convergence rate, number of steps.
hlib_solver_t solver;
hlib_solve_info_t solve_info;
A_inv, & solve_info, & info ); CHECK_INFO;
HLIB_FNDECL hlib_solver_t hlib_solver_auto(int *info)
HLIB_FNDECL void hlib_solver_solve(const hlib_solver_t solver, const hlib_linearoperator_t A, hlib_vector_t x, const hlib_vector_t b, const hlib_linearoperator_t W, hlib_solve_info_t *solve_info, int *info)
To check the accuracy of the computed solution, we compare it with the known exact solution:
double error;
printf( " error of solution = %.4e\n", error );
HLIB_FNDECL hlib_permutation_t hlib_clt_perm_i2e(hlib_clustertree_t ct, int *info)
HLIB_FNDECL void hlib_vector_fill(hlib_vector_t x, const hlib_real_t f, int *info)
HLIB_FNDECL void hlib_vector_axpy(const hlib_real_t alpha, const hlib_vector_t x, hlib_vector_t y, int *info)
HLIB_FNDECL hlib_real_t hlib_vector_norm_inf(const hlib_vector_t x, int *info)
Finally, all resources have to be freed and 𝖧𝖫𝖨𝖡𝗉𝗋𝗈 be finished:
free( b_arr );
for ( i = 0; i < n; i++ )
free( vertices[i] );
free( vertices );
return 0;
}
HLIB_FNDECL void hlib_matrix_free(hlib_matrix_t A, int *info)
HLIB_FNDECL void hlib_vector_free(hlib_vector_t v, int *info)
HLIB_FNDECL void hlib_clt_free(hlib_clustertree_t ct, int *info)
HLIB_FNDECL void hlib_bct_free(hlib_blockclustertree_t bct, int *info)
HLIB_FNDECL void hlib_done(int *info)
HLIB_FNDECL void hlib_coord_free(hlib_coord_t coord, int *info)
HLIB_FNDECL void hlib_solver_free(hlib_solver_t solver, int *info)
HLIB_FNDECL void hlib_admcond_free(const hlib_admcond_t ac, int *info)
HLIB_FNDECL void hlib_linearoperator_free(hlib_linearoperator_t A, int *info)
The Plain Program
#include <stdio.h>
#include <hlib-c.h>
#define CHECK_INFO { if ( info != HLIB_NO_ERROR ) \
{ char buf[1024]; hlib_error_desc( buf, 1024 ); \
printf( "\n%s\n\n", buf ); exit(1); } }
void
kernel ( const size_t n, const int * rowidx,
const size_t m, const int * colidx,
hlib_real_t * matrix, void * arg )
{
int rowi, colj;
double h = *((double*) arg);
for ( colj = 0; colj < m; colj++ )
{
const int idx1 = colidx[colj];
for ( rowi = 0; rowi < n; rowi++ )
{
const int idx0 = rowidx[rowi];
double value;
if ( idx0 == idx1 )
value = -1.5*h*h + h*h*log(h);
else
{
const double dist = h * ( fabs( (double) (idx0-idx1) ) - 1.0 );
const double t1 = dist+1.0*h;
const double t2 = dist+2.0*h;
value = ( - 1.5*h*h + 0.5*t2*t2*log(t2) - t1*t1*log(t1) );
if ( fabs(dist) > 1e-8 )
value += 0.5*dist*dist*log(dist);
}
matrix[(colj*n) + rowi] = -value;
}
}
}
double
rhs ( int i, int n )
{
const double a = ((double)i) / ((double) n);
const double b = ((double)i+1.0) / ((double) n);
double value = -1.5 * (b - a);
if ( fabs( b ) > 1e-8 ) value += 0.5*b*b*log(b);
if ( fabs( a ) > 1e-8 ) value -= 0.5*a*a*log(a);
if ( fabs( 1.0 - b ) > 1e-8 ) value -= 0.5*(1.0-b)*(1.0-b)*log(1.0-b);
if ( fabs( 1.0 - a ) > 1e-8 ) value += 0.5*(1.0-a)*(1.0-a)*log(1.0-a);
return value;
}
int
main ( int argc, char ** argv )
{
int info;
int i;
int n = ...
double h = 1.0 / ((double) n);
double ** vertices;
hlib_coord_t coord;
hlib_clustertree_t ct;
hlib_admcond_t adm;
hlib_blockclustertree_t bct;
hlib_matrix_t A, B;
hlib_linearoperator_t A_inv;
hlib_vector_t b;
double * b_arr = NULL;
hlib_vector_t x;
hlib_solver_t solver;
hlib_solve_info_t solve_info;
hlib_vector_t one;
double error;
vertices = (double**) malloc( n * sizeof(double*) );
for ( i = 0; i < n; i++ )
{
vertices[i] = (double*) malloc( sizeof(double) );
vertices[i][0] = h * ((double) i) + (h / 2.0);
}
acc, 1, & info ); CHECK_INFO;
b_arr = (hlib_real_t *) malloc( n * sizeof(hlib_real_t) );
for ( i = 0; i < n; i++ )
b_arr[i] = rhs( i, n );
A_inv, & solve_info, & info ); CHECK_INFO;
printf( " error of solution = %.4e\n", error );
free( b_arr );
for ( i = 0; i < n; i++ )
free( vertices[i] );
free( vertices );
return 0;
}
1.9.1