SPARSE_GRID_CLOSED_DATASET
Sparse Grid from Closed 1D Quadrature Rule

SPARSE_GRID_CLOSED_DATASET is a C++ program which computes a sparse quadrature rule for an arbitrary spatial dimension, associated with a particular "level" of the Smolyak construction, and based on a closed 1D quadrature rule.

Usage:

sparse_grid_closed_dataset dim_num level_max rule

• dim_num is the spatial dimension, 1 or more;
• level_max is the level of the Smolyak construction; 0 or more.
• rule is a code for the 1D quadrature rule to use:
  • CC = Clenshaw Curtis,
  • NCC = Newton Cotes Closed.

Licensing:

The code described and made available on this web page is distributed under the GNU LGPL license.

Languages:

SPARSE_GRID_CLOSED_DATASET is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CC_DISPLAY is a MATLAB library which can compute and display Clenshaw Curtis grids in two dimensions, as well as sparse grids formed from sums of Clenshaw Curtis grids.

CLENSHAW_CURTIS is a C++ library which can compute Clenshaw Curtis grids in multiple dimensions, as well as sparse grids formed from sums of Clenshaw Curtis grids.

QUADRATURE_RULES is a dataset directory which contains quadrature rules; a number of examples of sparse grid quadrature rules are included.

QUADRULE is a C++ library which defines quadrature rules for various intervals and weight functions.

SGMGA, a C++ library which creates sparse grids based on a mixture of 1D quadrature rules, allowing anisotropic weights for each dimension.

SMOLPACK is a C library which implements Novak and Ritter's method for estimating the integral of a function over a multidimensional hypercube using sparse grids.

SPARSE_GRID_CC is a dataset directory which contains the abscissas of sparse grids based on a Clenshaw Curtis rule.

SPARSE_GRID_CC_DATASET, a C++ program which creates a sparse grid dataset based on Clenshaw-Curtis rules.

SPARSE_GRID_CLOSED is a C++ library which define define sparse grids based on closed nested quadrature rules.

SPARSE_GRID_DISPLAY is a MATLAB library which can display a 2D or 3D sparse grid.

SPARSE_GRID_F2 is a dataset directory which contains the abscissas of sparse grids based on a Fejer Type 2 rule.

SPARSE_GRID_MIXED_DATASET, a C++ program which creates a sparse grid dataset based on a mixture of 1D rules.

SPARSE_GRID_NCC is a dataset directory which contains the abscissas of sparse grids based on a Newton Cotes closed rule.

SPARSE_GRID_NCO is a dataset directory which contains the abscissas of sparse grids based on a Newton Cotes open rule.

SPARSE_GRID_OPEN_DATASET, a C++ program which creates a sparse grid dataset based on open rules (Fejer 2, Gauss-Patterson, Newton-Cotes-Open).

TOMS847 is a MATLAB program which uses sparse grids to carry out multilinear hierarchical interpolation. It is commonly known as SPINTERP, and is by Andreas Klimke.

Reference:

1. Volker Barthelmann, Erich Novak, Klaus Ritter,
   High Dimensional Polynomial Interpolation on Sparse Grids,
   Advances in Computational Mathematics,
   Volume 12, Number 4, 2000, pages 273-288.
2. Thomas Gerstner, Michael Griebel,
   Numerical Integration Using Sparse Grids,
   Numerical Algorithms,
   Volume 18, Number 3-4, 1998, pages 209-232.
3. Albert Nijenhuis, Herbert Wilf,
   Combinatorial Algorithms for Computers and Calculators,
   Second Edition,
   Academic Press, 1978,
   ISBN: 0-12-519260-6,
   LC: QA164.N54.
4. 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.
5. Sergey Smolyak,
   Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions,
   Doklady Akademii Nauk SSSR,
   Volume 4, 1963, pages 240-243.
6. Dennis Stanton, Dennis White,
   Constructive Combinatorics,
   Springer, 1986,
   ISBN: 0387963472,
   LC: QA164.S79.

Source Code:

• sparse_grid_closed_dataset.C, the source code.
• sparse_grid_closed_dataset.sh, commands to compile the source code.

Examples and Tests:

CC_D2_LEVEL3 is a computation based on the Clenshaw Curtis rule in dimension 2 and level 3.

• cc_d2_level3_output.txt, the printed output from a run of the program for DIM_NUM=2 and LEVEL_MAX=3 and RULE=1.
• cc_d2_level3_x.txt, the abscissas of the computed quadrature rule.
• cc_d2_level3_w.txt, the weights of the computed quadrature rule.
• cc_d2_level3_r.txt, the integration region of the computed quadrature rule.

NCC_D2_LEVEL3 is a computation based on the Newton Cotes Closed rule in dimension 2 and level 3.

• ncc_d2_level3_output.txt, the printed output from a run of the program for DIM_NUM=2 and LEVEL_MAX=3 and RULE=2.
• ncc_d2_level3_x.txt, the abscissas of the computed quadrature rule.
• ncc_d2_level3_w.txt, the weights of the computed quadrature rule.
• ncc_d2_level3_r.txt, the integration region of the computed quadrature rule.

List of Routines:

• MAIN is the main program for SPARSE_GRID_CLOSED_DATASET.
• ABSCISSA_LEVEL_CLOSED_ND: first level at which given abscissa is generated.
• CC_ABSCISSA returns the I-th abscissa of the Clenshaw Curtis rule.
• CC_WEIGHTS computes Clenshaw Curtis weights.
• CH_CAP capitalizes a single character.
• CHOOSE computes the binomial coefficient C(N,K).
• COMP_NEXT computes the compositions of the integer N into K parts.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_POWER returns the value of I^J.
• I4_TO_STRING converts an I4 to a C++ string.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
• I4VEC_PRODUCT multiplies the entries of an I4VEC.
• INDEX_TO_LEVEL_CLOSED determines the level of a point given its index.
• LEVEL_TO_ORDER converts a level to an order for closed rules.
• MULTIGRID_INDEX0 returns an indexed multidimensional grid.
• MULTIGRID_SCALE_CLOSED renumbers a grid as a subgrid on a higher level.
• NCC_ABSCISSA returns the I-th abscissa of the Newton-Cotes Closed rule.
• NCC_WEIGHTS computes Newton-Cotes Closed weights.
• PRODUCT_WEIGHTS_CLOSED: closed product rule weights.
• R8_EPSILON returns the R8 roundoff unit.
• R8_HUGE returns a "huge" R8.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8MAT_WRITE writes an R8MAT file with no header.
• R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.
• R8VEC_PRINT_SOME prints "some" of an R8VEC.
• R8VEC_SUM returns the sum of an R8VEC.
• S_EQI reports whether two strings are equal, ignoring case.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• SPARSE_GRID_CFN_SIZE sizes a sparse grid using Closed Fully Nested rules.
• SPGRID_CLOSED_INDEX indexes the points forming a sparse grid.
• SPGRID_CLOSED_WEIGHTS gathers the weights.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• VEC_COLEX_NEXT2 generates vectors in colex order.

You can go up one level to the C++ source codes.