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The problem of diffusion in a porous medium with a spatially varying porosity is considered. The particular microstructure analyzed comprises a collection of impenetrable spheres, though the methods developed are general. Two different…
We present a novel approach of discretizing variable coefficient diffusion operators in the context of meshfree generalized finite difference methods. Our ansatz uses properties of derived operators and combines the discrete Laplace…
In this paper, we study numerical methods for the homogenization of linear second-order elliptic equations in nondivergence-form with periodic diffusion coefficients and large drift terms. Upon noting that the effective diffusion matrix can…
We construct a novel estimator for the diffusion coefficient of the limiting homogenized equation, when observing the slow dynamics of a multiscale model, in the case when the slow dynamics are of bounded variation. Previous research…
We consider multiscale stochastic systems that are partially observed at discrete points of the slow time scale. We introduce a particle filter that takes advantage of the multiscale structure of the system to efficiently approximate the…
Usually, given a continuous-time nonlinear model, a closed form solution for an exact discretization cannot be found explicitly, originating the need of approximating discrete-time models. This note studies the preservation of the Lipschitz…
A new method is proposed to numerically extract the diffusivity of a (typically nonlinear) diffusion equation from underlying stochastic particle systems. The proposed strategy requires the system to be in local equilibrium and have…
We establish uniform Lipschitz estimates for second-order elliptic systems in divergence form with rapidly oscillating, almost-periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded…
Nonlinear time fractional partial differential equations are widely used in modeling and simulations. In many applications, there are high contrast changes in media properties. For solving these problems, one often uses coarse spatial grid…
Numerical solutions of stationary diffusion equations on the unit sphere with isotropic lognormal diffusion coefficients are considered. H\"older regularity in $L^p$ sense for isotropic Gaussian random fields is obtained and related to the…
We prove existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of…
The behavior of slow-fast diffusions as the separation of scale diverges is a well-studied problem in the literature. In this short paper, we revisit this problem and obtain a new proof of existing strong quantitative convergence estimates…
We investigate well-posedness for martingale solutions of stochastic differential equations, under low regularity assumptions on their coefficients, widely extending some results first obtained by A. Figalli. Our main results are a very…
We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three…
In this paper, we establish $L_p$ estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean…
In this paper, we propose some algorithms for the simulation of the distribution of certain diffusions conditioned on terminal point. We prove that the conditional distribution is absolutely continuous with respect to the distribution of…
We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach via…
A simple Markov process is considered involving a diffusion in one direction and a transport in a transverse direction. Quantitative mixing rate estimates are obtained with limited assumptions about the transport field, which might be…
A numerical method for the two-dimensional, incompressible Navier--Stokes equations in vorticity--streamfunction form is proposed, which employs semi-Lagrangian discretizations for both the advection and diffusion terms, thus achieving…
Based on the notion of a construction process consisting of the stepwise addition of particles to the pure fluid, a discrete model for the apparent viscosity as well as for the maximum packing fraction of polydisperse suspensions of…