Error Analysis of Randomized Time-Stepping Methods for Non-autonomous Evolution Equations with Time-Irregular Coefficients

摘要

In this talk, we consider the numerical approximation of Caratheodory-type differential equations of the form u'(t) =f(t, u(t)), t ∈ (0, T), u(0) = u_0, and of nonlinear and non-autonomous evolution equations of the form u'(t)+A(t)u(t)=f(t), t∈(0,T), u(0) = u_0, where / and A may be discontinuous with respect to the time variable. In this non-smooth situation, it is notoriously 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 if, for instance, A and f only satisfy an L~2-integrability condition with respect to t. 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 more explicit parabolic PDE. This talk is based on joint works with Monika Eisenmann (Technische Universitat Berlin), Mihaly Kovacs and Stig Larsson (both Chalmers University of Technology) as well as Yue Wu (University of Edinburgh).