Related papers: A study of elliptic partial differential equations…
The Richards' equation is a model for flow of water in unsaturated soils. The coefficients of this (nonlinear) partial differential equation describe the permeability of the medium. Insufficient or uncertain measurements are commonly…
To model subsurface flow in uncertain heterogeneous\ fractured media an elliptic equation with a discontinuous stochastic diffusion coefficient - also called random field - may be used. In case of a one-dimensional parameter space, L\'evy…
Subsurface flows are commonly modeled by advection-diffusion equations. Insufficient measurements or uncertain material procurement may be accounted for by random coefficients. To represent, for example, transitions in heterogeneous media,…
General elliptic equations with spatially discontinuous diffusion coefficients may be used as a simplified model for subsurface flow in heterogeneous or fractured porous media. In such a model, data sparsity and measurement errors are often…
We present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient, the source term, the solution and its flux are discontinuous across an irregular interface. The algorithm produces second-order accurate…
The present work is devoted to approximation of the statistical moments of the unknown solution of a class of elliptic transmission problems in $\mathbb R^3$ with randomly perturbed interfaces. Within this model, the diffusion coefficient…
We discuss the application of multilevel Monte Carlo methods to elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification in subsurface flow modeling. We give a brief…
This paper develops and analyzes an efficient numerical method for solving elliptic partial differential equations, where the diffusion coefficients are random perturbations of deterministic diffusion coefficients. The method is based upon…
In this article, we consider elliptic diffusion problems on random domains with non-smooth diffusion coefficients. We start by illustrating the problems that arise from a non-smooth diffusion coefficient by recapitulating the corresponding…
Randomness is ubiquitous in modern engineering. The uncertainty is often modeled as random coefficients in the differential equations that describe the underlying physics. In this work, we describe a two-step framework for numerically…
We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable…
Almost all materials are anisotropic. In this paper, interface relations of anisotropic elliptic partial differential equations involving discontinuities across interfaces are derived in two and three dimensions. Compared with isotropic…
In this paper we consider one-dimensional diffusions with constant coefficients in a finite interval with jump boundary and a certain deterministic jump distribution. We use coupling methods in order to identify the spectral gap in the case…
The standard diffusion processes are known to be obtained as the limits of appropriate random walks. These prelimiting random walks can be quite different however. The diffusion coefficient can be made responsible for the size of jumps or…
We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $s \in (0,1)$, of symmetric, coercive, linear, elliptic, second-order operators in bounded domains…
Elliptic partial differential equations (PDEs) with discontinuous diffusion coefficients occur in application domains such as diffusions through porous media, electro-magnetic field propagation on heterogeneous media, and diffusion…
Convergence of stochastic processes with jumps to diffusion processes is investigated in the case when the limit process has discontinuous coefficients. An example is given in which the diffusion approximation of a queueing model yields a…
In this paper we study rare events associated to solutions of elliptic partial differential equations with spatially varying random coefficients. The random coefficients follow the lognormal distribution, which is determined by a Gaussian…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
A series of recent articles introduced a method to construct stochastic partial differential equations (SPDEs) which are invariant with respect to the distribution of a given conditioned diffusion. These works are restricted to the case of…