SANDIA_SPARSE
Sparse Grids for Sandia

SANDIA_SPARSE is a FORTRAN90 library which can be used to compute the points and weights of a Smolyak sparse grid, based on a variety of 1-dimensional quadrature rules.

The sparse grids that can be created may be based on any one of a variety of 1D quadrature rules. However, only isotropic grids are generated, that is, the same 1D quadrature rule is used in each dimension, and the same maximum order is used in each dimension. This is a limitation of this library, and not an inherent limitation of the sparse grid method.

The 1D quadrature rules that can be used to construct sparse grids include:

• CFN, Closed Fully Nested rules:
  • CC, Clenshaw-Curtis:
    defined on [-1,+1], with w(x)=1.
• OFN, Open Fully Nested rules:
  • F1, Fejer Type 1:
    defined on (-1,+1), with w(x)=1.
  • F2, Fejer Type 2:
    defined on (-1,+1), with w(x)=1.
  • GP, Gauss Patterson:
    defined on (-1,+1), with w(x)=1,
    a family of the midpoint rule, the 3 point Gauss Legendre rule, and then successive Patterson refinements.
• OWN, Open Weakly Nested rules:
  • GL, Gauss Legendre:
    defined on (-1,+1), with w(x)=1.
  • GH, Gauss Hermite:
    defined on (-oo,+oo), with w(x)=exp(-x*x).
• ONN, Open Non-Nested rules:
  • LG, Gauss Laguerre:
    defined on (0,+oo) with w(x)=exp(-x).

Point Growth of 1D Rules

A major advantage of sparse grids is that they can achieve accuracy that is comparable to a corresponding product rule, while using far fewer points, that is, evaluations of the function that is to be integrated. We will leave aside the issue of comparing accuracy for now, and simply focus on the pattern of point growth.

A sparse grid is essentially a linear combination of lower order product grids. One way point growth is controlled is to only use product grids based on a set of factors that are nested. In other words, the underlying 1D rules are selected so that, when we increase the order of such a rule, all the points of the current rule are included in the new one.

The exact details of how this works depend on the particular 1D rule being used and the nesting behavior it satisfies. We classify the cases as follows:

• CFN, "Closed, Fully Nested", based on Clenshaw Curtis.
• OFN, "Open, Fully Nested", based on Fejer Type 1, Fejer Type 2, or Gauss Patterson.
• OWN, "Open, Weakly Nested", based on Gauss Legendre or Gauss Hermite rules.
• ONN, "Open, Non-Nested", based on Gauss Laguerre rules;

For CFN rules we have the following relationship between the level (index of the grid) and the 1D order (the number of points in the 1D rule.)

        order = 2level + 1
      

For OFN, OWN and ONN rules, the relationship between level and 1D order is:

        order = 2level+1 - 1
      

Thus, as we allow level to grow, the order of the 1D closed and open rules behaves as follows:

Level	CFN	OFN/OWN/ONN
0	1	1
1	3	3
2	5	7
3	9	15
4	17	31
5	33	63
6	65	127
7	129	255
8	257	511
9	513	1,023
10	1,025	2,057

When we move to multiple dimensions, the counting becomes more complicated. This is because a multidimensional sparse grid is made up of a logical sum of product grids. A multidimensional sparse grid has a multidimensional level, which is a single number. Each product grid that forms part of this sparse grid has a multidimensional level which is the sum of the 1D levels of its factors. A sparse grid whose multidimensional level is represented by LEVEL includes all product grids whose level ranges LEVEL+1-DIM and LEVEL.

Thus, as one example, if DIM is 2, the sparse grid of level 3, formed from a CFN rule, will be formed from the following product rules.

level	level 1	level 2	order 1	order 2	order
1	0	1	1	3	3
1	1	0	3	1	3
2	0	2	1	5	5
2	1	1	3	3	9
2	2	0	5	1	5

Because of the nesting pattern for CFN rules, instead of 25 points (the sum of the orders), we will actually have just 13 unique points.

For a CFN sparse grid, here is the pattern of growth in the number of points, as a function of spatial dimension and grid level:

DIM	1	2	3	4	5
Level
0	1	1	1	1	1
1	3	5	7	9	11
2	5	13	25	41	61
3	9	29	69	137	241
4	17	65	177	401	801
5	33	145	441	1,105	2,433
6	65	321	1,073	2,929	6,993
7	129	705	2,561	7,537	19,313
8	257	1,537	6,017	18,945	51,713

For an OFN sparse grid, here is the pattern of growth in the number of points, as a function of spatial dimension and grid level:

DIM	1	2	3	4	5
Level
0	1	1	1	1	1
1	3	5	7	9	11
2	7	17	31	49	71
3	15	49	111	209	351
4	31	129	351	769	1,471
5	63	321	1,023	2,561	5,503
6	127	769	2,815	7,937	18,943
7	255	1,793	7,423	23,297	61,183
8	511	4,097	18,943	65,537	187,903

For an OWN sparse grid, here is the pattern of growth in the number of points, as a function of spatial dimension and grid level:

DIM	1	2	3	4	5
Level
0	1	1	1	1	1
1	3	5	7	9	11
2	7	21	37	57	81
3	15	73	159	289	471
4	31	225	597	1,265	2,341
5	63	637	2,031	4,969	10,363
6	127	1,693	6,405	17,945	41,913
7	255	4,289	19,023	60,577	157,583
8	511	10,473	53,829	193,457	557,693

For an ONN sparse grid, here is the pattern of growth in the number of points, as a function of spatial dimension and grid level:

DIM	1	2	3	4	5
Level
0	1	1	1	1	1
1	3	7	10	13	16
2	7	29	58	95	141
3	15	95	255	515	906
4	31	273	945	2,309	4,746
5	63	723	3,120	9,065	21,503
6	127	1,813	9,484	32,259	87,358
7	255	4,375	27,109	106,455	325,943
8	511	10,265	73,915	330,985	1,135,893

Usage:

To integrate a function f(x) over a multidimensional cube [-1,+1]^DIM using a sparse grid based on a Clenshaw Curtis rule, we might use a program something like the following:

      dim = 2
      level = 3
      rule = 1

      call levels_index_size ( dim, level, rule, point_num )

      allocate ( w(1:point_num) )
      allocate ( x(1:dim,1:point_num) )

      call sparse_grid ( dim, level, rule, point_num, w, x )

      quad = 0.0
      do j = 1, point_num
        quad = quad + w(j) * f ( x(1:dim,j) )
      end do
    

Licensing:

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

Languages:

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

Related Data and Programs:

SANDIA_RULES, a FORTRAN90 library which generates Gauss quadrature rules of various orders and types.

SGMGA, a FORTRAN90 library which creates sparse grids based on a mixture of 1D quadrature rules, allowing anisotropic weights for each dimension.

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

SPARSE_GRID_CC, a FORTRAN90 library which can define a multidimensional sparse grid based on a 1D Clenshaw Curtis rule.

SPARSE_GRID_CC_DATASET, a FORTRAN90 program which reads user input, creates a multidimensional sparse grid based on a 1D Clenshaw Curtis rule and writes it to three files that define a quadrature rule.

SPARSE_GRID_DISPLAY, a MATLAB program which can display a 2D or 3D sparse grid.

SPINTERP, a MATLAB library which uses a sparse grid to perform multilinear hierarchical interpolation, 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. Charles Clenshaw, Alan Curtis,
   A Method for Numerical Integration on an Automatic Computer,
   Numerische Mathematik,
   Volume 2, Number 1, December 1960, pages 197-205.
3. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
4. Thomas Gerstner, Michael Griebel,
   Numerical Integration Using Sparse Grids,
   Numerical Algorithms,
   Volume 18, Number 3-4, 1998, pages 209-232.
5. Albert Nijenhuis, Herbert Wilf,
   Combinatorial Algorithms for Computers and Calculators,
   Second Edition,
   Academic Press, 1978,
   ISBN: 0-12-519260-6,
   LC: QA164.N54.
6. 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.
7. Fabio Nobile, Raul Tempone, Clayton Webster,
   An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data,
   SIAM Journal on Numerical Analysis,
   Volume 46, Number 5, 2008, pages 2411-2442.
8. Sergey Smolyak,
   Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions,
   Doklady Akademii Nauk SSSR,
   Volume 4, 1963, pages 240-243.
9. Dennis Stanton, Dennis White,
   Constructive Combinatorics,
   Springer, 1986,
   ISBN: 0387963472,
   LC: QA164.S79.

Source Code:

• sandia_sparse.f90, the source code.

Examples and Tests:

• sandia_sparse_test.f90, a sample calling program.
• sandia_sparse_test.txt, the output from a run of the sample program.

List of Routines:

• ABSCISSA_LEVEL_CLOSED_ND: first level at which given abscissa is generated.
• ABSCISSA_LEVEL_OPEN_ND: first level at which given abscissa is generated.
• CC_ABSCISSA returns the I-th abscissa for the Clenshaw Curtis rule.
• CC_WEIGHTS computes Clenshaw Curtis weights.
• COMP_NEXT computes the compositions of the integer N into K parts.
• F1_ABSCISSA returns the I-th abscissa for the Fejer type 1 rule.
• F1_WEIGHTS computes weights for a Fejer type 1 rule.
• F2_ABSCISSA returns the I-th abscissa for the Fejer type 2 rule.
• F2_WEIGHTS computes weights for a Fejer type 2 rule.
• GH_ABSCISSA sets abscissas for multidimensional Gauss Hermite quadrature.
• GH_WEIGHTS returns weights for certain Gauss Hermite quadrature rules.
• GL_ABSCISSA sets abscissas for multidimensional Gauss Legendre quadrature.
• GL_WEIGHTS returns weights for certain Gauss Legendre quadrature rules.
• GP_ABSCISSA returns the I-th abscissa for a Gauss Patterson rule.
• GP_WEIGHTS sets weights for a Gauss Patterson rule.
• I4_LOG_2 returns the integer part of the logarithm base 2 of an I4.
• I4_MODP returns the nonnegative remainder of I4 division.
• INDEX_LEVEL_OWN: determine first level at which given index is generated.
• INDEX_TO_LEVEL_CLOSED determines the level of a point given its index.
• INDEX_TO_LEVEL_OPEN determines the level of a point given its index.
• LEVEL_TO_ORDER_CLOSED converts a level to an order for closed rules.
• LEVEL_TO_ORDER_OPEN converts a level to an order for open rules.
• LEVELS_INDEX indexes a sparse grid.
• LEVELS_INDEX_CFN indexes a sparse grid made from CFN 1D rules.
• LEVELS_INDEX_OFN indexes a sparse grid made from OFN 1D rules.
• LEVELS_INDEX_ONN indexes a sparse grid made from ONN 1D rules.
• LEVELS_INDEX_OWN indexes a sparse grid made from OWN 1D rules.
• LEVELS_INDEX_SIZE sizes a sparse grid.
• LEVELS_INDEX_SIZE_ONN sizes a sparse grid made from ONN 1D rules.
• LEVELS_INDEX_SIZE_OWN sizes a sparse grid made from OWN 1D rules.
• LG_ABSCISSA sets abscissas for multidimensional Gauss Laguerre quadrature.
• LG_WEIGHTS returns weights for certain Gauss Laguerre quadrature rules.
• MONOMIAL_INTEGRAL_HERMITE integrates a Hermite monomial.
• MONOMIAL_INTEGRAL_LAGUERRE integrates a Laguerre monomial.
• MONOMIAL_INTEGRAL_LEGENDRE integrates a Legendre monomial.
• MONOMIAL_QUADRATURE applies a quadrature rule to a monomial.
• MONOMIAL_VALUE evaluates a monomial.
• MULTIGRID_INDEX_CFN indexes a sparse grid based on CFN 1D rules.
• MULTIGRID_INDEX_OFN indexes a sparse grid based on OFN 1D rules.
• MULTIGRID_INDEX_ONN indexes a sparse grid based on ONN 1D rules.
• MULTIGRID_INDEX_OWN indexes a sparse grid based on OWN 1D rules.
• MULTIGRID_SCALE_CLOSED renumbers a grid as a subgrid on a higher level.
• MULTIGRID_SCALE_OPEN renumbers a grid as a subgrid on a higher level.
• PRODUCT_WEIGHTS computes the weights of a product rule.
• R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.
• R8_FACTORIAL computes the factorial function.
• R8_FACTORIAL2 computes the double factorial function.
• R8_HUGE returns a very large R8.
• R8_MOP returns the I-th power of -1 as an R8 value.
• R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.
• SPARSE_GRID computes a sparse grid.
• SPARSE_GRID_CC_SIZE sizes a sparse grid using Clenshaw Curtis rules.
• SPARSE_GRID_CFN computes a sparse grid based on a CFN 1D rule.
• SPARSE_GRID_OFN computes a sparse grid based on an OFN 1D rule.
• SPARSE_GRID_OFN_SIZE sizes a sparse grid using Open Fully Nested rules.
• SPARSE_GRID_ONN computes a sparse grid based on a ONN 1D rule.
• SPARSE_GRID_OWN computes a sparse grid based on an OWN 1D rule.
• SPARSE_GRID_WEIGHTS_CFN computes sparse grid weights based on a CFN 1D rule.
• SPARSE_GRID_WEIGHTS_OFN computes sparse grid weights based on a OFN 1D rule.
• 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 FORTRAN90 source codes.