An implicit Euler scheme with non-uniform time discretization for heat equations with multiplicative noise

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.