On randomized time-stepping methods for non-autonomous evolution equations with time-irregular coefficients
A talk in the Oberseminar Numerik series by
Raphael Kruse
| Abstract: | In this talk, we consider the numerical approximation of Carathéodory-type
ODEs and of nonlinear and non-autonomous evolution equations
whose coefficients may be irregular or discontinuous with respect to the
time variable. In this non-smooth situation, it is difficult to construct
numerical algorithms with a positive convergence rate. In fact, it can be
shown that any deterministic algorithm depending only on point evaluations
may fail to converge.
Instead, we propose to apply randomized Runge–Kutta methods to such
time-irregular evolution equations as, for instance, a randomized version of the backward Euler method.
We obtain positive convergence rates with respect
to the mean-square norm under considerably relaxed temporal regularity
conditions. An important ingredient in the error analysis
consists of a well-known variance reduction technique for Monte Carlo
methods, the stratified sampling.
We demonstrate the practicability of the new algorithm in the case of a
fully discrete approximation of a parabolic PDE.
This talk is based on joint work with Monika Eisenmann (TU Berlin),
Mihály Kovács and Stig Larsson (both Chalmers University
of Technology) as well as Yue Wu (U Edinburgh). |