Sparse Grids

SpectralToolbox.SparseGrids.CC(l)

SpectralToolbox.SparseGrids.FEJ(l)

SpectralToolbox.SparseGrids.GQN(): function for generating 1D Gaussian quadrature for integral with Gaussian weight (Gauss-Hermite)

Parameters:l (int) – level of accuracy of the quadrature rule Returns:(tuple [2] of ndarray) – nodes and weights

SpectralToolbox.SparseGrids.GQU(): function for generating 1D Gaussian quadrature rules for unweighted integral over [0, 1] (Gauss-Legendre)

Parameters:l (int) – level of accuracy of the quadrature rule Returns:(tuple [2] of ndarray) – nodes and weights

SpectralToolbox.SparseGrids.KPN(l)

KPN(): function for generating 1D Nested rule for integral with Gaussian weight

Parameters:l (int) – level of accuracy of the quadrature rule Returns:(tuple [2] of ndarray) – nodes and weights

SpectralToolbox.SparseGrids.KPU(l)

KPU(): function for generating 1D Nested rule for unweighted integral over [0,1]

Parameters:l (int) – level of accuracy of the quadrature rule Returns:(tuple [2] of ndarray) – nodes and weights

SpectralToolbox.SparseGrids.NESTEDGAUSS(l)

NESTEDGAUSS(): function for generating 1D Nested rule for integral with Uniform weight with 2**l scaling

Parameters:l (int) – level of accuracy of the quadrature rule Returns:(tuple [2] of ndarray) – nodes and weights

SpectralToolbox.SparseGrids.NESTEDLOBATTO(l)

NESTEDLOBATTO(): function for generating 1D Nested rule for integral with Uniform weight with 2**l scaling

Parameters:l (int) – level of accuracy of the quadrature rule Returns:(tuple [2] of ndarray) – nodes and weights

class SpectralToolbox.SparseGrids.SparseGrid(qrule, dim, k, sym)

Bases: object

Initialization of the Sparse Grid instance.

Syntax:

sg = SparseGrid(qrule, dim, k, sym)

Input:

• qrule = Function of 1D integration rule
• dim = dimension of the integration problem
• k = Accuracy level. The rule will be exact for polynomial up to total order 2k-1
• sym = (optional) only used for own 1D quadrature rule (type not “KPU”,...). If sym is supplied and not=0, the code will run faster but will produce incorrect results if 1D quadrature rule is asymmetric.

Description:

Several 1D integration rules are available to be chosen for the qrule input parameter

• KPU = Nested rule for unweighted integral over [0,1]
• KPN = Nested rule for integral with Gaussian weight
• GQU = Gaussian quadrature for unweighted integral over [0,1] (Gauss-Legendre)
• GQN = Gaussian quadrature for integral with Gaussian weight (Gauss-Hermite)
• CC = Clenshaw-Curtis quadrature for unweighted integral over [-1,1]
• FEJ = Fejer’s quadrature for unweighted integral over [-1,1]
• func = any function provided by the user that accept level l and returns nodes n and weights w for univariate quadrature rule with polynomial exactness 2l-1 as [n w] = feval(func,level)

DIM = 0

K = 0

QRULE = ''

SYM = 1

sparseGrid()

sparseGrid(): main function for generating nodes & weights for sparse grids intergration

Syntax:

(n,w) = sparseGrid()

Output:

• n = matrix of nodes with dim columns
• w = row vector of corresponding weights