A class of ^-stable advanced multistep methods is derived for the numerical solution of initial value problems in ordinary differential equations. The methods, of all orders of accuracy up to ten, only require values of y' and are self starting. Results for the asymptotic behaviour of the discretization error and for estimating local truncation error are also obtained. The practical implementat...

This paper describes a state space discretization scheme based on statistical experimental designs generated from orthogonal arrays of strength three with index unity. Chen et al. (1999) employed this eecient discretization scheme in a numerical solution method for high-dimensional continuous-state stochastic dynamic programming (SDP). These OAs are instrumental in reducing the dimensionality o...

We examine nonstationary inverse problems in which the time evolution of the unknown quantity is modelled by a stochastic partial differential equation. We consider the problem as a state estimation problem. The time discrete state evolution equation is exact since the solution is given by an analytic semigroup. For the practical reasons the space discretization of the time discrete state estim...

A new method is presented for extending metamodeling techniques to include the effects of finite element model mesh discretization errors. The method employs a rational function representation of the discretization error rather than the power series representation used by methods such as Richardson extrapolation. Examples dealing with simple function estimation and estimation of the vibrational...

The effect on the MRI signal of water diffusion in biological tissues in the presence of applied magnetic field gradient pulses can be modelled by a multiple compartment Bloch-Torrey partial differential equation. We present a method for the numerical solution of this equation by coupling a standard Cartesian spatial discretization with an adaptive time discretization. The time discretization i...

The stationary monochromatic radiative transfer equation is a partial differential transport equation stated on a five-dimensional phase space. To obtain a well-posed problem, inflow boundary conditions have to be prescribed. The sparse tensor product discretization has been successfully applied to finite element methods in radiative transfer with wavelet discretization of the angular domain (W...

This dissertation concerns efficient numerical treatment of the elliptic partial differential equations with high-contrast coefficients. High-contrast means that the ratio between highest and lowest values of the coefficients is very high, or even infinite. A finite-element discretization of such equations yields a linear system with an illconditioned matrix which leads to significant issues in...

Electrical impedance tomography (EIT) of closed conductive media is an ill-posed inverse problem. In the general case, the resolution of the direct problem, derived from a second order partial derivative equation, requires a numerical approximation. The solution brought by the Finite Element Method (FEM) is often used in EIT, because it preserves the nonlinear dependence of the observation set ...

Abstract. We consider a fully discrete approximation of the H gradient flow of an energy integral where the energy density is given by the sum of a nonnegative multi-well potential term and a bending energy term. The spatial discretization is based on a Fourier spectral method, which is combined with an implicit Euler time discretization. The numerical method is shown to be stable and to exhibi...

The main purpose of this paper is to examine the effectiveness of the Quarter-Sweep Gauss-Seidel (QSGS) method in solving the dense linear systems generated from the discretization of the linear Fredholm integral equations of the second kind. In addition, the applications of the various orders of closed Newton-Cotes quadrature discretization schemes will be investigated in order to form linear ...