Related papers: Fully discrete approximation of parametric and sto…
We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space $\mathbb{R}^n$, where $n \ge 2$, assuming that the diffusion matrix depends on the space variable $x$ and has a finite…
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…
We are concerned with the homogenization of second-order linear elliptic equations with random coefficient fields. For symmetric coefficient fields with only short-range correlations, quantified through a logarithmic Sobolev inequality for…
We extend the taming techniques for explicit Euler approximations of stochastic differential equations (SDEs) driven by L\'evy noise with super-linearly growing drift coefficients. Strong convergence results are presented for the case of…
We present a unified theoretical framework for parametric low-rank approximation, a research area devoted to the development of efficient algorithms that act as adaptive alternatives of traditional methods such as Singular Value…
The present work introduces and investigates an explicit time discretization scheme, called the projected Euler method,to numerically approximate random periodic solutions of semi-linear SDEs under non-globally Lipschitz conditions. The…
We derive aposteriori error estimates for fully discrete approximations to solutions of linear parabolic equations on the space-time domain. The space discretization uses finite element spaces, that are allowed to change in time. Our main…
A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations -- such as coefficient identification in partial differential…
In this paper, the convergence of the solutions for a discretized linear state-based static peridynamic system to the corresponding continuous solution is analytically proven. To obtain an implementable model, we further apply…
We establish sparsity and summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions of countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with…
The main result of this article establishes strong convergence rates on the whole probability space for explicit space-time discrete numerical approximations for a class of stochastic evolution equations with possibly non-globally monotone…
We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable…
In this paper, we propose a model reduction method for solving multiscale elliptic PDEs with random coefficients in the multiquery setting using an optimization approach. The optimization approach enables us to construct a set of localized…
In this article we investigate the existence of a solution to a semilinear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one…
We consider the solution of second order elliptic PDEs in $\R^d$ with inhomogeneous Dirichlet data by means of an $h$-adaptive FEM with fixed polynomial order $p\in\N$. As model example serves the Poisson equation with mixed…
We develop a geometric and analytic framework for polynomial partial differential equations posed on thin annuli in the plane. Using renormalized Sobolev inner products, we construct Sobolev orthogonal polynomial bases adapted to the thin…
We prove that for compactly perturbed elliptic problems, where the corresponding bilinear form satisfies a Garding inequality, adaptive mesh-refinement is capable of overcoming the preasymptotic behavior and eventually leads to convergence…
In this paper we consider nonlinear parabolic systems with elliptic part which can be also degenerate. We prove optimal error estimates for smooth enough solutions. The main novelty, with respect to previous results, is that we obtain the…
We develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that…
This paper outlines an approach to the approximation of probability density functions by quadratic forms of weighted orthonormal basis functions with positive semi-definite Hermitian matrices of unit trace. Such matrices are called…