A multigrid method for adaptive sparse grids

Benjamin Peherstorfer, Stefan Zimmer, Christoph Zenger, Hans Joachim Bungartz

Research output: Contribution to journalArticle

Abstract

Sparse grids have become an important tool to reduce the number of degrees of freedom of discretizations of moderately high-dimensional partial differential equations; however, the reduction in degrees of freedom comes at the cost of an almost dense and unconventionally structured system of linear equations. To guarantee overall efficiency of the sparse grid approach, special linear solvers are required. We present a multigrid method that exploits the sparse grid structure to achieve an optimal runtime that scales linearly with the number of sparse grid points. Our approach is based on a novel decomposition of the right-hand sides of the coarse grid equations that leads to a reformulation in so-called auxiliary coefficients. With these auxiliary coefficients, the right-hand sides can be represented in a nodal point basis on low-dimensional full grids. Our proposed multigrid method directly operates in this auxiliary coefficient representation, circumventing most of the computationally cumbersome sparse grid structure. Numerical results on nonadaptive and spatially adaptive sparse grids confirm that the runtime of our method scales linearly with the number of sparse grid points and they indicate that the obtained convergence factors are bounded independently of the mesh width.

Original languageEnglish (US)
Pages (from-to)S51-S70
JournalSIAM Journal on Scientific Computing
Volume37
Issue number5
DOIs
StatePublished - Jan 1 2015

Fingerprint

Sparse Grids
Adaptive Grid
Multigrid Method
Linear equations
Partial differential equations
Decomposition
Coefficient
Linearly
Degree of freedom
Grid
System of Linear Equations
Reformulation
High-dimensional
Partial differential equation
Discretization
Mesh
Decompose
Numerical Results

Keywords

  • Adaptive sparse grids
  • ANOVA
  • Multidimensional problems
  • Multigrid
  • Q-cycle

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

Peherstorfer, B., Zimmer, S., Zenger, C., & Bungartz, H. J. (2015). A multigrid method for adaptive sparse grids. SIAM Journal on Scientific Computing, 37(5), S51-S70. https://doi.org/10.1137/140974985

A multigrid method for adaptive sparse grids. / Peherstorfer, Benjamin; Zimmer, Stefan; Zenger, Christoph; Bungartz, Hans Joachim.

In: SIAM Journal on Scientific Computing, Vol. 37, No. 5, 01.01.2015, p. S51-S70.

Research output: Contribution to journalArticle

Peherstorfer, B, Zimmer, S, Zenger, C & Bungartz, HJ 2015, 'A multigrid method for adaptive sparse grids', SIAM Journal on Scientific Computing, vol. 37, no. 5, pp. S51-S70. https://doi.org/10.1137/140974985
Peherstorfer B, Zimmer S, Zenger C, Bungartz HJ. A multigrid method for adaptive sparse grids. SIAM Journal on Scientific Computing. 2015 Jan 1;37(5):S51-S70. https://doi.org/10.1137/140974985
Peherstorfer, Benjamin ; Zimmer, Stefan ; Zenger, Christoph ; Bungartz, Hans Joachim. / A multigrid method for adaptive sparse grids. In: SIAM Journal on Scientific Computing. 2015 ; Vol. 37, No. 5. pp. S51-S70.
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