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 nonuniform time discretizations is crucial.
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Muller-Gronbach, T., Ritter, K. Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations. Found Comput Math 7, 135–181 (2007). https://doi.org/10.1007/s10208-005-0166-6
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DOI: https://doi.org/10.1007/s10208-005-0166-6