Maria Cristina de Castro Cunha

The probability density function to the random linear transport equation

We present a formula to calculate the probability density function of the solution of the random linear transport equation in terms of the density functions of the velocity and the initial condition. We also present an expression for the joint probability density function of the solution in two different points. Our results have shown good agreement with Monte Carlo simulations.

Extension of GKB-FP algorithm to large-scale general-form Tikhonov regularization

SUMMARY In a recent paper an algorithm for large-scale Tikhonov regularization in standard form called GKB-FP was proposed and numerically illustrated. In this paper, further insight into the convergence properties of this method is provided, and extensions to general-form Tikhonov regularization are introduced. In addition, as alternative to Tikhonov regularization, a preconditioned LSQR method coupled with an automatic stopping rule is proposed. Preconditioning seeks to incorporate smoothing properties of the regularization matrix into the computed solution. Numerical results are reported to illustrate the methods on large-scale problems. Copyright

Original articles: On the linear advection equation subject to random velocity fields

This paper deals with the random linear advection equation for which the time-dependent velocity and the initial condition are independent random functions. Expressions for the density and joint density functions of the solution are given. We also verify that in the Gaussian time-dependent velocity case the probability density function of the solution satisfies a convection-diffusion equation with a time-dependent diffusion coefficient. Some exact examples are presented.

A Numerical Scheme for the Variance of the Solution of the Random Transport Equation

Abstract We present a numerical scheme, based on Godunov’s method (REA algorithm), for the variance of the solution of the 1D random linear transport equation, with homogeneous random velocity and stochastic initial condition. We obtain the stability conditions of the method and we also show its consistency with a deterministic nonhomogeneous advective–diffusive equation, which means convergency. Numerical results are considered to validate our scheme.

Finite elements on dyadic grids with applications

A dyadic grid is a d-dimensional hierarchical mesh where a cell at level k is partitioned into two equal children at level k + 1 by a hyperplane perpendicular to coordinate axis (k mod d). We consider here the finite element approach on adaptive grids, static and dynamic, for various functional approximation problems.We review here the theory of adaptive dyadic grids and splines defined on them. Specifically, we consider the space Pcd[G] of all functions that, within any leaf cell of an arbitrary finite dyadic grid G, coincide with a multivariate polynomial of maximum degree d in each coordinate, and are continuous to order c.We describe algorithms to construct a finite-element basis for such spaces. We illustrate the use of such basis for interpolation, least-squares approximation, and the Galerkin-style integration of partial differential equations, such as the heat diffusion equation and two-phase (oil/water) flow in porous media.Compared to tetrahedral meshes, the simple topology of dyadic grids is expected to compensate for their limitations, especially in problems with moving fronts.

A note on the Riemann problem for the random transport equation

We present an explicit expression to the solution of the random Riemann problem for the 1D random linear transport equation. We show that the random solution is a similarity solution and the statistical moments have very simple expressions. Furthermore, we verify that the mean, the variance, and the 3rd central moment agree quite well with Monte Carlo simulations. We point out that our approach could be useful in designing numerical methods for more general random transport problems.

O Método de Galerkin Estocástico e a Equação Diferencial de Transporte Linear Estocástica.

We use the stochastic Garlekin method to solve a linear transport equation with random data. Following the ideias of the method, the statiscal solution is projected on the space generated by generalized Polynomials Chaos, a basis for the space of random functions. Numerical simulations compare our results with the Monte Carlo simulations.

O método de Galerkin estocástico e a equação ao diferencial de transporte linear com dados de entrada Aleatórios

We use the stochastic Garlekin method to solve a linear transport equation with random data. Following the ideias of the method, the statiscal solution is projected on the space generated by generalized Polynomials Chaos, a basis for the space of random functions. Numerical simulations compare our results with the Monte Carlo simulations.