Related papers: A modified tamed scheme for stochastic differentia…
Periodic solutions of delay equations are usually approximated as continuous piecewise polynomials on meshes adapted to the solutions' profile. In practical computations this affects the regularity of the (coefficients of the) linearized…
In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such…
We present an algorithm for the efficient sampling of conditional paths of stochastic differential equations (SDEs). While unconditional path sampling of SDEs is straightforward, albeit expensive for high dimensional systems of SDEs,…
Stochastic evolutional equations with monotone operators are considered in Banach spaces. Explicit and implicit numerical schemes are presented. The convergence of the approximations to the solution of the equations is proved.
The equation with the time fractional substantial derivative and space fractional derivative describes the distribution of the functionals of the L\'evy flights; and the equation is derived as the macroscopic limit of the continuous time…
We introduce an explicit, adaptive time-stepping scheme for the simulation of SPDEs with one-sided Lipschitz drift coefficients. Strong convergence rates are proven for the full space-time discretisation with multiplicative trace-class…
We consider the stochastic Allen--Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension $d\le 3$, and study the semidiscretisation in time of the equation by an Euler type split-step…
In this paper, a weak Local Linearization scheme for Stochastic Differential Equations (SDEs) with multiplicative noise is introduced. First, for a time discretization, the solution of the SDE is locally approximated by the solution of the…
Schemes with the second-order approximation in time are considered for numerical solving the Cauchy problem for an evolutionary equation of first order with a self-adjoint operator. The implicit two-level scheme based on the Pad\'{e}…
This paper proposes an adaptive time-stepping mothods for stochastic diffusion systems whose drift and diffusion coefficients are locally Lipschitz continuous and may exhibit polynomial growth. By controlling the growth of both the drift…
In this paper, we prove convergence rates for time discretisation schemes for semi-linear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly…
In this paper, we combine the operator splitting methodology for abstract evolution equations with that of stochastic methods for large-scale optimization problems. The combination results in a randomized splitting scheme, which in a given…
In this paper, we propose a semi-implicit Euler scheme to discretize the stochastic nonlinear Maxwell equations with multiplicative Ito noise, which is implicit in the drift term and explicit in the diffusion term of the equations, in order…
We present a new multi-symplectic formulation of constrained Hamiltonian partial differential equations, and we study the associated local conservation laws. A multi-symplectic discretisation based on this new formulation is exemplified by…
Dynamical systems are essential to model various phenomena in physics, finance, economics, and are also of current interest in machine learning. A central modeling task is investigating parameter sensitivity, whether tuning atmospheric…
A discretization scheme for variable coefficient elliptic PDEs in the plane is presented. The scheme is based on high-order Gaussian quadratures and is designed for problems with smooth solutions, such as scattering problems involving soft…
Stochastic sampling methods are arguably the most direct and least intrusive means of incorporating parametric uncertainty into numerical simulations of partial differential equations with random inputs. However, to achieve an overall error…
We prove weak convergence of order one for a class of exponential based integrators for SDEs with non-globally Lipschtiz drift. Our analysis covers tamed versions of Geometric Brownian Motion (GBM) based methods as well as the standard…
Many complex fluids can be described by continuum hydrodynamic field equations, to which noise must be added in order to capture thermal fluctuations. In almost all cases, the resulting coarse-grained stochastic partial differential…
This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the…