An Implicit Euler Scheme with Non-uniform Time Discretization for Heat Equations with Multiplicative Noise

Thoms Mueller-Gronbach; Klaus Ritter

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An Implicit Euler Scheme with Non-uniform Time Discretization for Heat Equations with Multiplicative Noise

Thoms Mueller-Gronbach·Klaus Ritter

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Abstract

We present an algorithm for solving stochastic heat equations, whose key ingredient is a non-uniform time discretization of the driving Brownian motion $W$. For this algorithm we derive an error bound in terms of its number of evaluations of one-dimensional components of $W$. The rate of convergence depends on the spatial dimension of the heat equation and on the decay of the eigenfunctions of the covariance of $W$. According to known lower bounds, our algorithm is optimal, up to a constant, and this optimality cannot be achieved by uniform time discretizations.

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BibTeX RIS

Thoms Mueller-Gronbach, Klaus Ritter. 2006-04-27. An Implicit Euler Scheme with Non-uniform Time Discretization for Heat Equations with Multiplicative Noise. https://arxiv.org/abs/math/0604600

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