Lower Bounds and Non-Uniform Time Discretization for Approximation of Stochastic Heat Equations

2022/01/26 by Ritter, Klaus, Müller-Gronbach, Thomas
#Non-uniform time discretization #Stochastic heat equation #minimal errors #upper and lower bounds

paper · doi:10.4230/dagsemproc.04401.6

Abstract

We study algorithms for approximation of the mild solution of stochastic heat equations on the spatial domain ]0,1[^d. The error of an algorithm is defined in L₂-sense. We derive lower bounds for the error of every algorithm that uses a total of N evaluations of one-dimensional components of the driving Wiener process W. For equations with additive noise we derive matching upper bounds and we construct asymptotically optimal algorithms. The error bounds depend on N and d, and on the decay of eigenvalues of the covariance of W in the case of nuclear noise. In the latter case the use of non-uniform time discretizations is crucial.

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