Efficient<i>d</i>-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations

Zubair, Hisham bin; Leentvaar, C.C.W.; Oosterlee, Cornelis W.

doi:10.1080/00207160701356365

Cited by 4 publications

(3 citation statements)

References 15 publications

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“…So, in principle, we would expect the asymptotic theoretical optimal sparse grid convergence of O(h 2 (log h −1 ) d−1 ). However, for the multi-dimensional Poisson equation and a smooth solution, bin Zubair et al (2007) showed that the theoretical sparse grid convergence was only achieved on relatively fine grids. Here we have the same situation.…”

Section: Sparse Grid Computationsmentioning

confidence: 99%

Multi-asset option pricing using a parallel Fourier-based technique

Leentvaar¹,

Oosterlee²

2008

JCF

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Section: Sparse Grid Computationsmentioning

confidence: 99%

Multi-asset option pricing using a parallel Fourier-based technique

Leentvaar¹,

Oosterlee²

2008

JCF

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“…[5,18,20]) have grown in popularity in recent years and are widely used for the numerical solution of PDEs. Other authors have considered different hierarchical and multigrid schemes for a variety of option pricing problems [10,15,22]. The first two of these papers develop sophisticated multilevel stand-alone solvers, while the last also considers using a custom-made multilevel method as a preconditioner.…”

Section: Introductionmentioning

confidence: 99%

A multigrid preconditioner for an adaptive Black-Scholes solver

Ramage

Sydow

2011

Bit Numer Math

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This paper is concerned with the efficient solution of the linear systems of equations that arise from an adaptive space-implicit time discretisation of the Black-Scholes equation. These nonsymmetric systems are very large and sparse, so an iterative method will usually be the method of choice. However, such a method may require a large number of iterations to converge, particularly when the timestep used is large (which is often the case towards the end of a simulation which uses adaptive timestepping). An appropriate preconditioner is therefore desirable. In this paper we show that a very simple multigrid algorithm with standard components works well as a preconditioner for these problems. We analyse the eigenvalue spectrum of the multigrid iteration matrix for uniform grid problems and illustrate the method’s efficiency in practice by considering the results of numerical experiments on both uniform grids and those which use adaptivity in space

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“…They are used either as preconditioners [1,11,12,15,16] or within real multigrid schemes [2,3,6,26,33]. We first note that certain problems allow for sparse grid dis-S53 2.1.…”

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A Multigrid Method for Adaptive Sparse Grids

Peherstorfer¹,

Zimmer

Zenger

et al. 2015

SIAM J. Sci. Comput.

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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. 1. Introduction. Many problems in computational science and engineering, e.g., in financial engineering, quantum mechanics, and data-driven prediction, lead to high-dimensional function approximations [23]. We consider here the case where the function is not given explicitly at selected sample points but implicitly as the solution of a high-dimensional partial differential equation (PDE). Standard discretizations on isotropic uniform grids, so-called full grids, become computationally infeasible because the number of degrees of freedom grows exponentially with the dimension. Sparse grid discretizations are a versatile way to circumvent this exponential growth [4]. If the function satisfies certain smoothness assumptions [4, section 3.1] sparse grids reduce the number of degrees of freedom to depend only logarithmically on the dimension; however, the discretization of a PDE on a sparse grid with the (piecewise linear) hierarchical basis results in an almost dense and unconventionally structured system of linear equations. To be computationally efficient solvers must take this structure into account. We develop a multigrid method that exploits the structure of sparse grid discretizations to achieve an optimal runtime that scales linearly with the number of sparse grids points.Multigrid methods have been extensively studied in the context of sparse grids. They are used either as preconditioners [1,11,12,15,16] or within real multigrid schemes [2,3,6,26,33]. We first note that certain problems allow for sparse grid dis-

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

Cited by 4 publications

References 15 publications

Multi-asset option pricing using a parallel Fourier-based technique

Multi-asset option pricing using a parallel Fourier-based technique

A multigrid preconditioner for an adaptive Black-Scholes solver

A Multigrid Method for Adaptive Sparse Grids

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