Lower Bounds and Non-Uniform Time Discretization for Approximation of Stochastic Heat Equations
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 L_2-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.
Stochastic heat equation
Non-uniform time discretization
minimal errors
upper and lower bounds
1-37
Regular Paper
Klaus
Ritter
Klaus Ritter
Thomas
Müller-Gronbach
Thomas Müller-Gronbach
10.4230/DagSemProc.04401.6
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