public static void sparse_grid_hermite_index(int dim_num, int level_max, int point_num,
ref int[] grid_index, ref int[] grid_base)
//****************************************************************************80
//
// Purpose:
//
// SPARSE_GRID_HERMITE_INDEX indexes points in a Gauss-Hermite sparse grid.
//
// Discussion:
//
// The sparse grid is assumed to be formed from 1D Gauss-Hermite rules
// of ODD order, which have the property that only the central abscissa,
// X = 0.0, is "nested".
//
// The necessary dimensions of GRID_INDEX can be determined by
// calling SPARSE_GRID_HERMITE_SIZE first.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 05 July 2008
//
// Author:
//
// John Burkardt
//
// Reference:
//
// Fabio Nobile, Raul Tempone, Clayton Webster,
// A Sparse Grid Stochastic Collocation Method for Partial Differential
// Equations with Random Input Data,
// SIAM Journal on Numerical Analysis,
// Volume 46, Number 5, 2008, pages 2309-2345.
//
// Parameters:
//
// Input, int DIM_NUM, the spatial dimension.
//
// Input, int LEVEL_MAX, the maximum value of LEVEL.
//
// Input, int POINT_NUM, the total number of points in the grids.
//
// Output, int GRID_INDEX[DIM_NUM*POINT_NUM], a list of
// point indices, representing a subset of the product grid of level
// LEVEL_MAX, representing (exactly once) each point that will show up in a
// sparse grid of level LEVEL_MAX.
//
// Output, int GRID_BASE[DIM_NUM*POINT_NUM], a list of
// the orders of the Gauss-Hermite rules associated with each point
// and dimension.
//
{
int level;
//
// The outer loop generates LEVELs from LEVEL_MIN to LEVEL_MAX.
//
int point_num2 = 0;
int level_min = Math.Max(0, level_max + 1 - dim_num);
int[] grid_base2 = new int[dim_num];
int[] level_1d = new int[dim_num];
int[] order_1d = new int[dim_num];
for (level = level_min; level <= level_max; level++)
{
//
// The middle loop generates the next partition LEVEL_1D(1:DIM_NUM)
// that adds up to LEVEL.
//
bool more = false;
int h = 0;
int t = 0;
for (;;)
{
Comp.comp_next(level, dim_num, ref level_1d, ref more, ref h, ref t);
//
// Transform each 1D level to a corresponding 1D order.
//
ClenshawCurtis.level_to_order_open(dim_num, level_1d, ref order_1d);
int dim;
for (dim = 0; dim < dim_num; dim++)
{
grid_base2[dim] = (order_1d[dim] - 1) / 2;
}
//
// The product of the 1D orders gives us the number of points in this grid.
//
int order_nd = typeMethods.i4vec_product(dim_num, order_1d);
//
// The inner (hidden) loop generates all points corresponding to given grid.
//
int[] grid_index2 = Multigrid.multigrid_index_z(dim_num, order_1d, order_nd);
//
// Determine the first level of appearance of each of the points.
// This allows us to flag certain points as being repeats of points
// generated on a grid of lower level.
//
// This is SLIGHTLY tricky.
//
int[] grid_level = HermiteQuadrature.index_level_hermite(level, level_max, dim_num, order_nd,
grid_index2, grid_base2);
//
// Only keep those points which first appear on this level.
//
int point;
for (point = 0; point < order_nd; point++)
{
if (grid_level[point] != level)
{
continue;
}
for (dim = 0; dim < dim_num; dim++)
{
grid_index[dim + point_num2 * dim_num] =
grid_index2[dim + point * dim_num];
grid_base[dim + point_num2 * dim_num] = grid_base2[dim];
}
point_num2 += 1;
}
if (!more)
{
break;
}
}
}
}
public static void sparse_grid_hermite(int dim_num, int level_max, int point_num,
ref double[] grid_weight, ref double[] grid_point)
//****************************************************************************80
//
// Purpose:
//
// SPARSE_GRID_HERMITE computes a sparse grid of Gauss-Hermite points.
//
// Discussion:
//
// The quadrature rule is associated with a sparse grid derived from
// a Smolyak construction using a 1D Gauss-Hermite quadrature rule.
//
// The user specifies:
// * the spatial dimension of the quadrature region,
// * the level that defines the Smolyak grid.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 05 July 2008
//
// Author:
//
// John Burkardt
//
// Reference:
//
// Fabio Nobile, Raul Tempone, Clayton Webster,
// A Sparse Grid Stochastic Collocation Method for Partial Differential
// Equations with Random Input Data,
// SIAM Journal on Numerical Analysis,
// Volume 46, Number 5, 2008, pages 2309-2345.
//
// Parameters:
//
// Input, int DIM_NUM, the spatial dimension.
//
// Input, int LEVEL_MAX, controls the size of the final sparse grid.
//
// Input, int POINT_NUM, the number of points in the grid, as determined
// by SPARSE_GRID_HERM_SIZE.
//
// Output, double GRID_WEIGHT[POINT_NUM], the weights.
//
// Output, double GRID_POINT[DIM_NUM*POINT_NUM], the points.
//
{
int level;
int point;
int point3 = 0;
for (point = 0; point < point_num; point++)
{
grid_weight[point] = 0.0;
}
//
// The outer loop generates LEVELs from LEVEL_MIN to LEVEL_MAX.
//
int point_num2 = 0;
int level_min = Math.Max(0, level_max + 1 - dim_num);
int[] grid_base2 = new int[dim_num];
int[] level_1d = new int[dim_num];
int[] order_1d = new int[dim_num];
for (level = level_min; level <= level_max; level++)
{
//
// The middle loop generates the next partition LEVEL_1D(1:DIM_NUM)
// that adds up to LEVEL.
//
bool more = false;
int h = 0;
int t = 0;
for (;;)
{
Comp.comp_next(level, dim_num, ref level_1d, ref more, ref h, ref t);
//
// Transform each 1D level to a corresponding 1D order.
// The relationship is the same as for other OPEN rules.
// The GL rule differs from the other OPEN rules only in the nesting behavior.
//
ClenshawCurtis.level_to_order_open(dim_num, level_1d, ref order_1d);
int dim;
for (dim = 0; dim < dim_num; dim++)
{
grid_base2[dim] = (order_1d[dim] - 1) / 2;
}
//
// The product of the 1D orders gives us the number of points in this grid.
//
int order_nd = typeMethods.i4vec_product(dim_num, order_1d);
//
// Compute the weights for this product grid.
//
double[] grid_weight2 = HermiteQuadrature.product_weight_hermite(dim_num, order_1d, order_nd);
//
// Now determine the coefficient of the weight.
//
int coeff = (int)(Math.Pow(-1, level_max - level)
* Binomial.choose(dim_num - 1, level_max - level));
//
// The inner (hidden) loop generates all points corresponding to given grid.
// The grid indices will be between -M to +M, where 2*M + 1 = ORDER_1D(DIM).
//
int[] grid_index2 = Multigrid.multigrid_index_z(dim_num, order_1d, order_nd);
//
// Determine the first level of appearance of each of the points.
// This allows us to flag certain points as being repeats of points
// generated on a grid of lower level.
//
// This is SLIGHTLY tricky.
//
int[] grid_level = HermiteQuadrature.index_level_hermite(level, level_max, dim_num, order_nd,
grid_index2, grid_base2);
//
// Only keep those points which first appear on this level.
//
for (point = 0; point < order_nd; point++)
{
//
// Either a "new" point (increase count, create point, create weight)
//
if (grid_level[point] == level)
{
HermiteQuadrature.hermite_abscissa(dim_num, 1, grid_index2,
grid_base2, ref grid_point, gridIndex: +point * dim_num,
gridPointIndex: +point_num2 * dim_num);
grid_weight[point_num2] = coeff * grid_weight2[point];
point_num2 += 1;
}
//
// or an already existing point (create point temporarily, find match,
// add weight to matched point's weight).
//
else
{
double[] grid_point_temp = new double[dim_num];
HermiteQuadrature.hermite_abscissa(dim_num, 1, grid_index2,
grid_base2, ref grid_point_temp, gridIndex: +point * dim_num);
int point2;
for (point2 = 0; point2 < point_num2; point2++)
{
point3 = point2;
for (dim = 0; dim < dim_num; dim++)
{
if (!(Math.Abs(grid_point[dim + point2 * dim_num] - grid_point_temp[dim]) >
double.Epsilon))
{
continue;
}
point3 = -1;
break;
}
if (point3 == point2)
{
break;
}
}
switch (point3)
{
case -1:
Console.WriteLine("");
Console.WriteLine("SPARSE_GRID_HERM - Fatal error!");
Console.WriteLine(" Could not match point.");
return;
default:
grid_weight[point3] += coeff * grid_weight2[point];
break;
}
}
}
if (!more)
{
break;
}
}
}
}
