Related papers: On Galerkin Approximations for the Zakai Equation …
We consider Galerkin finite element methods for semilinear stochastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. We analyze the strong error of convergence for spatially…
We introduce and analyze a class of Galerkin-collocation discretization schemes in time for the wave equation. Its conceptual basis is the establishment of a direct connection between the Galerkin method for the time discretization and the…
The Bayesian smoothing equations are generally intractable for systems described by nonlinear stochastic differential equations and discrete-time measurements. Gaussian approximations are a computationally efficient way to approximate the…
When solving the Poisson equation by the finite element method, we use one degree of freedom for interpolation by the given Laplacian - the right hand side function in the partial differential equation. The finite element solution is the…
We introduce a filtering technique for Discontinuous Galerkin approximations of hyperbolic problems. Following an approach already proposed for the Hamilton-Jacobi equations by other authors, we aim at reducing the spurious oscillations…
Although for a number of semilinear stochastic wave equations existence and uniqueness results for corresponding solution processes are known from the literature, these solution processes are typically not explicitly known and numerical…
This paper is devoted to filtering, smoothing, and prediction of polynomial processes that are partially observed. These problems are known to allow for an explicit solution in the simpler case of linear Gaussian state space models. The key…
This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point…
We resolve a longstanding open problem in the computational modeling of nonlinear plates by introducing a numerical method that exactly enforces the isometry constraint, namely, that the first fundamental form of the mid-surface coincides…
In this paper, we study efficient approximate sampling for probability distributions known up to normalization constants. We specifically focus on a problem class arising in Bayesian inference for large-scale inverse problems in science and…
We propose a linearized semi-implicit and decoupled finite element method for the incompressible Navier--Stokes equations with variable density. Our method is fully discrete and shown to be unconditionally stable. The velocity equation is…
A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For second order elliptic equation, this framework employs $4$ different…
In this paper, we study the numerical approximation of a coupled system of elliptic-parabolic equations posed on two separated spatial scales. The model equations describe the interplay between macroscopic and microscopic pressures in an…
The aim of this paper is to establish a theory of Galerkin approximations to the space of convex and compact subsets of $\R^d$ with favorable properties, both from a theoretical and from a computational perspective. These Galerkin spaces…
A general framework for the numerical approximation of evolution problems is presented that allows to preserve exactly an underlying Hamiltonian- or gradient structure. The approach relies on rewriting the evolution problem in a particular…
We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise. Owing to the damping term, under appropriate conditions on the nonlinearity, the solution admits a unique invariant distribution. We apply…
Mean-field stochastic differential equations, also called McKean--Vlasov equations, are the limiting equations of interacting particle systems with fully symmetric interaction potential. Such systems play an important role in a variety of…
Consider a Urysohn integral equation $x - \mathcal{K} (x) = f$, where $f$ and the integral operator $\mathcal{K}$ with kernel of the type of Green's function are given. In the computation of approximate solutions of the given integral…
Uncertainty Quantification through stochastic spectral methods is rising in popularity. We derive a modification of the classical stochastic Galerkin method, that ensures the hyperbolicity of the underlying hyperbolic system of partial…
This manuscript is devoted to the study of a class of nonlinear non-instantaneous impulsive first order abstract retarded type functional differential equations in an arbitrary separable Hilbert space H. A new set of sufficient conditions…