A proof of convergence for the combination technique for the Laplace equation using tools of symbolic computation

Bungartz, H.-J.; Griebel, Michael; Röschke, Dierk; Zenger, Christoph

doi:10.1016/s0378-4754(96)00036-5

Cited by 30 publications

(50 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The table and figures show the dependence of the number of dimensions in the convergence according to the theoretical convergence ratio in equation (16). Although the theoretical convergence of the sparse grid method is low when d is high at small numbers of n max (the largest number of cells in one direction), the convergence in this test experiment is reasonable.…”

Section: D-multigrid As a Preconditioner In The Time-independent Casementioning

confidence: 75%

“…It is known that the error of the discrete solution from a second order finite difference discretization of the 2D Laplacian can be split [15] as With the combination technique as in Definition 3.2 and the splitting in (15), the dimensiondependent absolute error (for the Laplacian), reads, [16]:…”

Section: The Sparse Grid Methodsmentioning

confidence: 99%

“…This reduces the coarsening strategy (explained above) to the more simple case illustrated here for a five-dimensional problem discretized on the non-equidistant grid given by N = 128 4 16 …”

Section: Coarsening Strategies To Handle Anisotropiesmentioning

confidence: 99%

“…The number of problems #probl is as in equation (12) and the theoretical convergence Th. Conv from equation (16).…”

Section: D-multigrid As a Preconditioner In The Time-independent Casementioning

confidence: 99%

“…For the time-dependent case, we choose as the test problem solution, (12) and the theoretical convergence from equation (16).…”

Section: D-multigrid As a Preconditioner In The Time-dependent Casementioning

confidence: 99%

See 4 more Smart Citations

Efficientd-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations

Zubair¹,

Leentvaar²,

Oosterlee³

2007

International Journal of Computer Mathematics

View full text Add to dashboard Cite

Fast and efficient solution techniques are developed for high-dimensional parabolic partial differential equations (PDEs). In this paper we present a robust solver based on the Krylov subspace method Bi-CGSTAB combined with a powerful, and efficient, multigrid preconditioner. Instead of developing the perfect multigrid method, as a stand-alone solver for a single problem discretized on a certain grid, we aim for a method that converges well for a wide class of discrete problems arising from discretization on various anisotropic grids. This is exactly what we encounter during a sparse grid computation of a high-dimensional problem. Different multigrid components are discussed and presented with operator construction formulae. An option-pricing application is focused and presented with results computed with this method.

show abstract

Section: D-multigrid As a Preconditioner In The Time-independent Casementioning

confidence: 75%

Section: The Sparse Grid Methodsmentioning

confidence: 99%

Section: Coarsening Strategies To Handle Anisotropiesmentioning

confidence: 99%

“…The number of problems #probl is as in equation (12) and the theoretical convergence Th. Conv from equation (16).…”

Section: D-multigrid As a Preconditioner In The Time-independent Casementioning

confidence: 99%

“…For the time-dependent case, we choose as the test problem solution, (12) and the theoretical convergence from equation (16).…”

Section: D-multigrid As a Preconditioner In The Time-dependent Casementioning

confidence: 99%

See 3 more Smart Citations

Efficientd-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations

Zubair¹,

Leentvaar²,

Oosterlee³

2007

International Journal of Computer Mathematics

View full text Add to dashboard Cite

show abstract

Optimal scaling parameters for sparse grid discretizations

Griebel

Hullmann

Oswald

2014

Numer. Linear Algebra Appl.

View full text Add to dashboard Cite

We apply iterative subspace correction methods to elliptic PDE problems discretized by generalized sparse grid systems. The involved subspace solvers are based on the combination of all anisotropic full grid spaces that are contained in the sparse grid space. Their relative scaling is at our disposal and has significant influence on the performance of the iterative solver. In this paper, we follow three approaches to obtain close-to-optimal or even optimal scaling parameters of the subspace solvers and thus of the overall subspace correction method. We employ a Linear Program that we derive from the theory of additive subspace splittings, an algebraic transformation that produces partially negative scaling parameters that result in improved asymptotic convergence properties, and finally, we use the OptiCom method as a variable nonlinear preconditioner

show abstract

On the Computation of the Eigenproblems of Hydrogen and Helium in Strong Magnetic and Electric Fields with the Sparse Grid Combination Technique

Garcke¹,

Griebel²

2000

Journal of Computational Physics

View full text Add to dashboard Cite

A proof of convergence for the combination technique for the Laplace equation using tools of symbolic computation

Cited by 30 publications

References 3 publications

Efficientd-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations

Efficientd-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations

Optimal scaling parameters for sparse grid discretizations

On the Computation of the Eigenproblems of Hydrogen and Helium in Strong Magnetic and Electric Fields with the Sparse Grid Combination Technique

Contact Info

Product

Resources

About