In this paper, we present an abstract framework to obtain convergence rates for the approximation of random evolution equations corresponding to a random family of forms determined by finite-dimensional noise. The full discretization error in space, time, and randomness is considered, where polynomial chaos expansion (PCE) is used for the semi-discretization in randomness. The main result are regularity conditions on the random forms under which convergence of polynomial order in randomness is obtained depending on the smoothness of the coefficients and the Sobolev regularity of the initial value. In space and time, the same convergence rates as in the deterministic setting are achieved. To this end, we derive error estimates for vector-valued PCE as well as a quantified version of the Trotter–Kato theorem for form-induced semigroups. We apply the abstract framework to an anisotropic diffusion model with random diffusion coefficients.

In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong error (Formula presented.) where p∈[2,∞), U is the mild solution, U^j is obtained from a time discretisation scheme, k is the step size, and N_k=T/k for final time T>0. This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong error (Formula presented.) which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

In this paper we prove convergence rates for time discretization schemes for semilinear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator is the generator of a strongly continuous semigroup on a Hilbert space, and the focus is on nonparabolic problems. The main results are optimal bounds for the uniform strong errorwhere, is the mild solution, is obtained from a time discretization scheme, is the step size and. The usual schemes such as the exponential Euler (EE), the implicit Euler (IE), the Crank-Nicolson (CN) method, etc. are included as special cases. Under conditions on the nonlinearity and the noise, we show (linear equation, additive noise, general) (nonlinear equation, multiplicative noise, contractive) (nonlinear wave equation, multiplicative noise), for a large class of time discretization schemes. The logarithmic factor can be removed if the EE method is used with a (quasi)-contractive. The obtained bounds coincide with the optimal bounds for SDEs. Most of the existing literature is concerned with bounds for the simpler pointwise strong errorApplications to Maxwell equations, Schrödinger equations and wave equations are included. For these equations, our results improve and reprove several existing results with a unified method and provide the first results known for the IE and the CN method.