# SPARSE_GRID_CC Sparse Grids Based on the Clenshaw Curtis Rule

SPARSE_GRID_CC is a FORTRAN90 library which can be used to compute the points and weights of a Smolyak sparse grid, based on a 1-dimensional Clenshaw-Curtis quadrature rule, to be used for efficient and accurate quadrature in multiple dimensions.

One way of looking at the construction of sparse grids is to assume that we start out by constructing a (very dense) product grid. We will assume for now that the order, that is, the number of points used in each component grid, is the same for all dimensions. Moreover, we will assume that the order is a power of 2 plus one, so that we have a natural relationship between the order and the logarithm base 2 of the order minus 1:

```
order = 2level + 1
```
except that for the special case of level=0 we assign order=1. (If we used our formula, then this case would give us order=2 instead.

Thus, if we allow level to grow, the order roughly doubles, as follows:
LevelOrder
0 1
1 3
2 5
3 9
4 17
5 33
6 65
7 129
8 257
9 513
10 1025

To keep things simple, let us begin by supposing we are selecting points for a grid to be used in an interpolation or quadrature rule. If you successively compute the locations of the points of each level, you will probably see that the points of a level are all included in the grid associated with the next level. (This is not guaranteed for all rules; it's simply a property of the way most such grids are defined!).

This nesting property is very useful. For one thing, it means that when if we've computed a grid of one level, and now proceed to the next, then all the information associated with the current level (point location, the value of functions at those points) is still useful for the next level, and will save us some computation time as well. This also means that, when we have reached a particular level, all the previous levels are still available to us, with no extra storage. These considerations make it possible, for instance, to do efficient and convenient error estimation.

When we move to a problem whose geometry is two-dimensional or more, we can still take the same approach. However, when working in multidimensional geometry, it is usually not a good idea to form a grid using the product of 1D grids, especially when we are determining the order using the idea of levels. Especially in this case, if we go to the next level in each dimension, the total number of points would increase by a factor of roughly 2 to the spatial dimension. Just a few such steps in, say, 6 dimensions, and we would be far beyond our computational capacity.

Instead, in multidimensions, the idea is to construct a sparse grid, which can be thought of in one of two ways:

• the sparse gird is a logical sum of low order product grids; each product grid has a total level (sum of the levels of the 1d rules) that is less than or equal to level_max;
• the sparse grid is a very sparse selection of points from the very high order product grid formed by using rules of level level_max in each dimension.

(There is still a lot of explaining to do to get from the one-dimensional levels to the N-dimensional levels and the selection of the low-level product grids that sum up to the sparse grid...)

Once the grid indices of the sparse grid points have been selected, there are a variety of schemes for distributing the points. We consider closed quadrature rules, in which the endpoints of the interval are included. The uniform scheme, known as the Newton Cotes Closed rule, is easy to understand. However, it has been observed that greater accuracy and stability can be achieved by arranging the points in a nonuniform way that tends to move points towards the boundary and away from the center. A common scheme for doing this starts with the uniform points on [0,1] and applies the cosine function to arrive at nonuniformly spaced points in [-1,1]. This scheme is known as the Clenshaw Curtis rule.

• Newton Cotes Closed:
Points = [ 0, 1, 2, ..., N-1 ] / (N-1);
Uniformly spaced points on [0,1], including endpoints.
Point spacing is 1/(N-1).
• Clenshaw Curtis:
Theta = pi * [ 0, 1, 2, ..., N-1 ] / (N-1);
Points = cos ( Theta );
Nonuniformly spaced points on [-1,1], including endpoints.

The library of routines presented here will only construct grids based on the Clenshaw Curtis rule.

A version of the sparse grid library is available in http://tasmanian.ornl.gov, the TASMANIAN library, available from Oak Ridge National Laboratory.

### Languages:

SPARSE_GRID_CC is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

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.

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

SPARSE_GRID_CC_DATASET, a FORTRAN90 program which creates a sparse grid dataset based on Clenshaw-Curtis rules.

SPARSE_GRID_GL, a FORTRAN90 library which creates sparse grids based on Gauss-Legendre rules.

SPARSE_GRID_HERMITE, a FORTRAN90 library which creates sparse grids based on Gauss-Hermite rules.

SPARSE_GRID_HW, a FORTRAN90 library which creates sparse grids based on Gauss-Legendre, Gauss-Hermite, Gauss-Patterson, or a nested variation of Gauss-Hermite rules, by Florian Heiss and Viktor Winschel.

SPARSE_GRID_LAGUERRE, a FORTRAN90 library which creates sparse grids based on Gauss-Laguerre rules.

SPARSE_GRID_MIXED, a FORTRAN90 library which constructs a sparse grid using different rules in each spatial dimension.

SPARSE_GRID_OPEN, a FORTRAN90 library which creates sparse grids based on open rules (Fejer 2, Gauss-Patterson, Newton-Cotes-Open).

### Reference:

1. Volker Barthelmann, Erich Novak, Klaus Ritter,
High Dimensional Polynomial Interpolation on Sparse Grids,
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,
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,
Volume 4, 1963, pages 240-243.
6. Dennis Stanton, Dennis White,
Constructive Combinatorics,
Springer, 1986,
ISBN: 0387963472,
LC: QA164.S79.

### List of Routines:

• ABSCISSA_LEVEL_CLOSED_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.
• CHOOSE computes the binomial coefficient C(N,K).
• COMP_NEXT computes the compositions of the integer N into K parts.
• GET_UNIT returns a free FORTRAN unit number.
• I4_MODP returns the nonnegative remainder of I4 division.
• INDEX_TO_LEVEL_CLOSED determines the level of a point given its index.
• LEVEL_TO_ORDER_CLOSED converts a level to an order for closed rules.
• LEVELS_CLOSED_INDEX computes closed grids with 0 <= LEVEL <= LEVEL_MAX.
• MONOMIAL_INT01 returns the integral of a monomial over the [0,1] hypercube.
• MONOMIAL_VALUE evaluates a monomial.
• MULTIGRID_INDEX0 returns an indexed multidimensional grid.
• MULTIGRID_SCALE_CLOSED renumbers a grid as a subgrid on a higher level.
• PRODUCT_WEIGHTS_CC: Clenshaw Curtis product rule weights.
• R8_HUGE returns a very large R8.
• R8MAT_WRITE writes an R8MAT file.
• R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.
• S_BLANK_DELETE removes blanks from a string, left justifying the remainder.
• SPARSE_GRID_CC computes a sparse grid of Clenshaw Curtis points.
• SPARSE_GRID_CC_INDEX indexes the points forming a sparse grid.
• SPARSE_GRID_CFN_SIZE sizes a sparse grid using Closed Fully Nested rules.
• SPARSE_GRID_CC_SIZE_OLD sizes a sparse grid based on nested closed rules.
• SPARSE_GRID_CC_WEIGHTS gathers the weights.
• SPARSE_GRID_CCS_SIZE sizes a sparse grid using Clenshaw Curtis Slow rules.
• 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.

Last revised on 22 December 2009.