A random Euler scheme for Carathéodory differential equations

Abstract

a b s t r a c tWe study a random Euler scheme for the approximation of Carathéodory differential equations and give a precise error analysis. In particular, we show that under weak assumptions, this approximation scheme obtains the same rate of convergence as the classical Monte-Carlo method for integration problems.

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

“…Let us stress, that the mapping f is not necessarily continuous with respect to the temporal variable t. In addition, the mappings L and K are not assumed to be bounded, in contrast to other results found in the literature [6,20,30,31]. Moreover, from (17) and (18) we directly deduce the linear growth condition |f (t, x)| ≤ K(t)(1 + |x|) (19) for almost all t ∈ [0, T ] and In the following proposition we collect a few properties of the solution u to (1).…”

“…Compare also with [20,Theorem 2], if the coefficient function f is additionally assumed to be locally bounded. …”

“…For instance, we mention the results in [7,19,20,22,30,31] and the references therein. These randomized methods are usually found to be superior over corresponding deterministic methods in the sense that the resulting discretization error decays already with order O(N −γ− 1 2 ) under the same smoothness assumptions as sketched above.…”

Pathwise Taylor schemes for random ordinary differential equations

“…In [13] Jentzen and Neuenkirch analyzed a numerical scheme for RODEs that converges independent of ϑ with the order 1 2 to the exact solution of the RODE. This scheme is called random Euler scheme there and is given by…”

Approximation of solutions to second order nonlinear Picard problems with Carathéodory right-hand side

“…Regular finite difference schemes should be avoided because we cannot guarantee convergence of such methods in the case of a discontinuous right-hand side. There are many other ideas known starting from integral Euler-type methods (with practical usage limited to specific affine cases  see the discussion in [9]) up to random methods (see in particular [9,15,16]). We will suggest a method which does not entirely escape from the integration, but in many cases leads to classical nonlinear problems in R .…”