Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems

We develop in this paper some efficient algorithms which are essential to implementations of spectral methods on the sparse grid by Smolyak's construction based on a nested quadrature. More precisely, we develop a fast algorithm for the discrete transform between the values at the sparse grid and the coefficients of expansion in a hierarchical basis; and by using the aforementioned fast transform, we construct two very efficient sparse spectral-Galerkin methods for a model elliptic equation. In particular, the Chebyshev-Legendre-Galerkin method leads to a sparse matrix with a much lower number of nonzero elements than that of low-order sparse grid methods based on finite elements or wavelets, and can be efficiently solved by a suitable sparse solver. Ample numerical results are presented to demonstrate the efficiency and accuracy of our algorithms.

[1]  JIANLIN XIA,et al.  ROBUST STRUCTURED MULTIFRONTAL FACTORIZATION AND PRECONDITIONING FOR DISCRETIZED PDES , 2011 .

[2]  Thomas Gerstner,et al.  Numerical integration using sparse grids , 2004, Numerical Algorithms.

[3]  Jianlin Xia,et al.  Superfast Multifrontal Method for Large Structured Linear Systems of Equations , 2009, SIAM J. Matrix Anal. Appl..

[4]  K. Hallatschek.  Fouriertransformation auf dünnen Gittern mit hierarchischen Basen , 1992 .

[5]  Dale B. Haidvogel,et al.  The Accurate Solution of Poisson's Equation by Expansion in Chebyshev Polynomials , 1979 .

[6]  H. Bungartz,et al.  Sparse grids , 2004, Acta Numerica.

[7]  Hans-Joachim Bungartz,et al.  A multigrid algorithm for higher order finite elements on sparse grids. , 1997 .

[8]  Hans-joachim Bungartz,et al.  An adaptive poisson solver using hierarchical bases and sparse grids , 1991, Forschungsberichte, TU Munich.

[9]  Jie Shen,et al.  E-cient Chebyshev-Legendre Galerkin Methods for Elliptic Problems , 1996 .

[10]  Hans-Joachim Bungartz,et al.  A Note on the Complexity of Solving Poisson's Equation for Spaces of Bounded Mixed Derivatives , 1999, J. Complex..

[11]  Y. Notay.  An aggregation-based algebraic multigrid method , 2010 .

[12]  Erich Novak,et al.  High dimensional polynomial interpolation on sparse grids , 2000, Adv. Comput. Math..

[13]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[14]  M. Griebel,et al.  Sparse grids for the Schrödinger equation , 2007 .

[15]  Timothy A. Davis,et al.  A column pre-ordering strategy for the unsymmetric-pattern multifrontal method , 2004, TOMS.

[16]  Jie Shen,et al.  Sparse Spectral Approximations of High-Dimensional Problems Based on Hyperbolic Cross , 2010, SIAM J. Numer. Anal..

[17]  Christoph Zenger,et al.  The Solution of Multidimensional Real Helmholtz Equations on Sparse Grids , 1996, SIAM J. Sci. Comput..

[18]  Rob Stevenson,et al.  Adaptive wavelet algorithms for elliptic PDE's on product domains , 2008, Math. Comput..

[19]  Jie Shen,et al.  Efficient Spectral-Galerkin Method II. Direct Solvers of Second- and Fourth-Order Equations Using Chebyshev Polynomials , 1995, SIAM J. Sci. Comput..

[20]  Vasile Gradinaru,et al.  Fourier transform on sparse grids: Code design and the time dependent Schrödinger equation , 2007, Computing.

[21]  Jie Shen,et al.  Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials , 1994, SIAM J. Sci. Comput..

[22]  ShenJie.  Efficient spectral-Galerkin method I , 1994 .

[23]  David Gottlieb,et al.  The Chebyshev-Legendre method: implementing Legendre methods on Chebyshev points , 1994 .

[24]  Hans-Joachim Bungartz,et al.  Acta Numerica 2004: Sparse grids , 2004 .