Efficient simulation of Gaussian Markov random fields by Chebyshev polynomial approximation

TL;DR

This paper introduces an efficient algorithm using Chebyshev polynomial approximation to simulate Gaussian Markov random fields with linear complexity, especially useful for discretized stochastic PDEs, ensuring asymptotic accuracy.

Contribution

The paper develops a novel Chebyshev polynomial-based method for simulating Gaussian Markov random fields with polynomial precision matrices, achieving linear complexity and asymptotic exactness.

Findings

Algorithm achieves linear computational complexity.

Provides criteria for minimal approximation order.

Ensures asymptotic accuracy as approximation order increases.

Abstract

This paper presents an algorithm to simulate Gaussian random vectors whose precision matrix can be expressed as a polynomial of a sparse matrix. This situation arises in particular when simulating Gaussian Markov random fields obtained by the discretization by finite elements of the solutions of some stochastic partial derivative equations. The proposed algorithm uses a Chebyshev polynomial approximation to compute simulated vectors with a linear complexity. This method is asymptotically exact as the approximation order grows. Criteria based on tests of the statistical properties of the produced vectors are derived to determine minimal orders of approximation.