Randomized exponential integrators for modulated nonlinear Schrödinger equations

Abstract

We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class W α,2 for some α ∈ (0, 1). Due to the loss of smoothness in the problem classical numerical methods face severe order reduction. In this work, we develop and analyze a new randomized exponential integrator based on a stratified Monte Carlo approximation. The new discretization technique averages the high oscillations in the solution allowing for improved convergence rates of order α + 1/2. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods.

A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations

“…We estimate the three terms separately. For J n 1 , we apply estimate (36) with ρ = 2, Assumption 3.4, and the Hölder continuity (15) of the exact solution. This yields…”

“…Compare further with [20,21]. Further randomized numerical methods for partial differential equations are studied in [9,15]. Moreover, in case τ n ≡ 0 for all n ∈ {1, .…”

Application of Randomized Quadrature Formulas to the Finite Element Method for Elliptic Equations

“…We neglect the last term and insert this estimate into the mean-square error. An application of Weitzenböck's inequality (20) then yields…”

“…We refer, for instance, to [8,19,21,25,34,35] for results on randomized one-step methods. Further, these methods have also been applied for the temporal discretization of evolution equations in infinite dimensions, see [10,20], and of stochastic differential equations, see [26,32].…”