R. Santos

Bio: R. Santos is an academic researcher. The author has contributed to research in topics: Approximation error & Diffusion (business). The author has an hindex of 1, co-authored 1 publications receiving 265 citations.

Diffusion approximation and computation of the critical size

Abstract: This paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation. To obtain this result, we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size. The paper also contains some simple numerical examples and some new results for the Milne problem. Introduction. The computation of the critical size and the diffusion approximation for the transport equation have been closely related and this is due to the following facts. First, the computation of the critical size is much easier for the diffusion approximation than for the original transport equation. Second, for the critical size one can consider a host media X = nXQ with X0 given and 17 a positive number. The problem of the critical size is then reduced to the computation of the parameter zj. It turns out that when the transport operator is almost conservative, the critical value of the parameter 17 is large and it is exactly for this range of value that the diffusion approximation is accurate. On the other hand the "physical" boundary condition for the diffusion approxi- mation is of the form

Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints

Abstract: A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in $\mbRn$. The nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion, including a notion of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory. In particular, it is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider. The numerous applications elucidate different interpretations of the operator and the associated governing equations. For example, a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process. Sufficie...

A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws

Abstract: A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoint operators. Nonlo...

A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit

Abstract: We propose a new numerical scheme for linear transport equations. It is based on a decomposition of the distribution function into equilibrium and nonequilibrium parts. We also use a projection technique that allows us to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic density and a kinetic equation for the nonequilibrium part. By using a suitable time semi-implicit discretization, our scheme is able to accurately approximate the solution in both kinetic and diffusion regimes. It is asymptotic preserving in the following sense: when the mean free path of the particles is small, our scheme is asymptotically equivalent to a standard numerical scheme for the limit diffusion model. A uniform stability property is proved for the simple telegraph model. Various boundary conditions are studied. Our method is validated in one-dimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes.

Inverse transport theory and applications

Abstract: Inverse transport consists of reconstructing the optical properties of a domain from measurements performed at the domain's boundary. This review concerns several types of measurements: time-dependent, time-independent, angularly resolved and angularly averaged measurements. We review recent results on the reconstruction of the optical parameters from such measurements and the stability of such reconstructions. Inverse transport finds applications e.g. in medical imaging (optical tomography, optical molecular imaging) and in geophysical imaging (remote sensing in the Earth's atmosphere).

The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator

Abstract: We analyze a nonlocal diffusion operator having as special cases the fractional Laplacian and fractional differential operators that arise in several applications. In our analysis, a nonlocal vector calculus is exploited to define a weak formulation of the nonlocal problem. We demonstrate that, when sufficient conditions on certain kernel functions hold, the solution of the nonlocal equation converges to the solution of the fractional Laplacian equation on bounded domains as the nonlocal interactions become infinite. We also introduce a continuous Galerkin finite element discretization of the nonlocal weak formulation and we derive a priori error estimates. Through several numerical examples we illustrate the theoretical results and we show that by solving the nonlocal problem it is possible to obtain accurate approximations of the solutions of fractional differential equations circumventing the problem of treating infinite-volume constraints.