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A higher order sparse grid combination technique

Julia Muñoz-Echániz, Christoph Reisinger

TL;DR

The paper tackles high-dimensional PDEs by elevating the sparse grid combination technique to fourth-order accuracy via a multivariate extrapolation carried out before the combination step. It introduces a novel extrapolated solver $ ilde{u}_{oldsymbol{h}}$ with extrapolation weights $oldsymbol{eta}$ that cancels leading second-order mixed terms, and then combines these higher-order tensor-grid solutions in the sparse-grid framework. A rigorous error-expansion theory is developed under generic stability and regularity assumptions, and is validated for the Poisson problem with smooth data, with numerical experiments confirming fourth-order convergence up to seven dimensions. The approach yields a scalable, high-precision method that maintains sparse-grid efficiency while delivering substantially improved accuracy, evidenced by favorable cost-accuracy comparisons against standard sparse grids and full-grid methods. These results are particularly relevant for high-dimensional PDEs where traditional full-grid methods are prohibitive.

Abstract

We show that a generalised sparse grid combination technique which combines multi-variate extrapolation of finite difference solutions with the standard combination formula lifts a second order accurate scheme on regular meshes to a fourth order combined sparse grid solution. In the analysis, working in a general dimension, we characterise all terms in a multivariate error expansion of the scheme as solutions of a sequence of semi-discrete problems. This is first carried out formally under suitable assumptions on the truncation error of the scheme, stability and regularity of solutions. We then verify the assumptions on the example of the Poisson problem with smooth data, and illustrate the practical convergence in up to seven dimensions.

A higher order sparse grid combination technique

TL;DR

The paper tackles high-dimensional PDEs by elevating the sparse grid combination technique to fourth-order accuracy via a multivariate extrapolation carried out before the combination step. It introduces a novel extrapolated solver with extrapolation weights that cancels leading second-order mixed terms, and then combines these higher-order tensor-grid solutions in the sparse-grid framework. A rigorous error-expansion theory is developed under generic stability and regularity assumptions, and is validated for the Poisson problem with smooth data, with numerical experiments confirming fourth-order convergence up to seven dimensions. The approach yields a scalable, high-precision method that maintains sparse-grid efficiency while delivering substantially improved accuracy, evidenced by favorable cost-accuracy comparisons against standard sparse grids and full-grid methods. These results are particularly relevant for high-dimensional PDEs where traditional full-grid methods are prohibitive.

Abstract

We show that a generalised sparse grid combination technique which combines multi-variate extrapolation of finite difference solutions with the standard combination formula lifts a second order accurate scheme on regular meshes to a fourth order combined sparse grid solution. In the analysis, working in a general dimension, we characterise all terms in a multivariate error expansion of the scheme as solutions of a sequence of semi-discrete problems. This is first carried out formally under suitable assumptions on the truncation error of the scheme, stability and regularity of solutions. We then verify the assumptions on the example of the Poisson problem with smooth data, and illustrate the practical convergence in up to seven dimensions.
Paper Structure (12 sections, 11 theorems, 102 equations, 2 figures)

This paper contains 12 sections, 11 theorems, 102 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and Main Results
  3. Combination Technique
  4. Pre-existing Error Analysis
  5. Main results of this paper
  6. Multivariate Extrapolation and Higher Order Combination
  7. Error Expansion
  8. Application to the Poisson Problem
  9. Theoretical Error Analysis for the Poisson Equation
  10. Numerical Results
  11. Two Dimensional Error Expansion
  12. Proofs from Section \ref{['sec:Chapter 5']}

Key Result

Theorem 2.1

Let $u$ be a solution to the Poisson problem on $I^d$ such that for all $\alpha \in \{0, \ldots, 4\}^d$, $D^{\alpha} u$ is continuous, $\|D^{\alpha} u \|_{\infty} \leq K$ and $D^{\alpha} u\vert_{\partial I^d}=0$. Let $\mathbf{u}_h$ denote the standard central finite difference approximation of order

Figures (2)

  • Figure 1: Surplus vs. level $n$ for HO‐SG, SG, HO‐FG, and FG for $1\le d\le 7$.
  • Figure 2: Surplus vs. number of grid points $N$ for HO‐SG, SG, HO‐FG, and FG for $1\le d\le 7$.

Theorems & Definitions (27)

  • Theorem 2.1: Error Expansion of Finite Differences for Poisson problem
  • Theorem 2.2: Sparse Grid Error Bounds; Theorem 5.4 in reisinger2013analysis
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6: Error bounds for the Higher Order Combination Solution
  • Theorem 2.7
  • proof
  • Remark 2.8
  • ...and 17 more