On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients
Preprint, 2017

In this paper we introduce a randomized version of the backward Euler method, that is applicable to stiff ordinary differential equations and nonlinear evolution equations with time-irregular coefficients. In the finite-dimensional case, we consider Carath\'eodory type functions satisfying a one-sided Lipschitz condition. After investigating the well-posedness and the stability properties of the randomized scheme, we prove the convergence to the exact solution with a rate of 0.5 in the root-mean-square norm assuming only that the coefficient function is square integrable with respect to the temporal parameter.
These results are then extended to the numerical solution of infinite-dimensional evolution equations under monotonicity and Lipschitz conditions. Here we consider a combination of the randomized backward Euler scheme with a Galerkin finite element method. We obtain error estimates that correspond to the regularity of the exact solution. The practicability of the randomized scheme is also illustrated through several numerical experiments.

stratified sampling

Galerkin finite element method.

evolution equations

ordi- nary differential equations

backward Euler method

Monte Carlo method


Monika Eisenmann

Technische Universität Berlin

Mihaly Kovacs

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Raphael Kruse

Technische Universität Berlin

Stig Larsson

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik



Sannolikhetsteori och statistik


Grundläggande vetenskaper

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