Edward J. Allen

Finite element and difference approximation of some linear stochastic partial differential equations

Difference and finite element methods are described, analyzed, and tested for numerical solution of linear parabolic and elliptic SPDEs driven by white noise. Weak and integral formulations of the stochastic partial differential equations are approximated, respectively, by finite element and difference methods. The white noise processes are approximated by piecewise constant random processes to facilitate convergence proofs for the finite element method. Error analyses of the two numerical methods yield estimates of convergence rates. Computational experiments indicate that the two numerical methods have similar accuracy but the finite element method is computationally more efficient than the difference method

Construction of Equivalent Stochastic Differential Equation Models

Abstract It is shown how different but equivalent Itô stochastic differential equation models of random dynamical systems can be constructed. Advantages and disadvantages of the different models are described. Stochastic differential equation models are derived for problems in chemistry, textile engineering, and epidemiology. Computational comparisons are made between the different stochastic models.

A comparison of three different stochastic population models with regard to persistence time

Results are summarized from the literature on three commonly used stochastic population models with regard to persistence time. In addition, several new results are introduced to clearly illustrate similarities between the models. Specifically, the relations between the mean persistence time and higher-order moments for discrete-time Markov chain models, continuous-time Markov chain models, and stochastic differential equation models are compared for populations experiencing demographic variability. Similarities between the models are demonstrated analytically, and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models are consistently formulated. As an example, the three stochastic models are applied to a population satisfying logistic growth. Logistic growth is interesting as different birth and death rates can yield the same logistic differential equation. However, the persistence behavior of the population is strongly dependent on the explicit forms for the birth and death rates. Computational results demonstrate how dramatically the mean persistence time can vary for different populations that experience the same logistic growth.

Dispersal and competition models for plants

New models for seed dispersal and competition between plant species are formulated and analyzed. The models are integrodifference equations, discrete in time and continuous in space, and have applications to annual and perennial species. The spread or invasion of a single plant species into a geographic region is investigated by studying the travelling wave solutions of these equations. Travelling wave solutions are shown to exist in the single-species models and are compared numerically. The asymptotic wave speed is calculated for various parameter values. The single-species integrodifference equations are extended to a model for two competing annual plants. Competition in the two-species model is based on a difference equation model developed by Pakes and Maller [26]. The two-species model with competition and dispersal yields a system of integrodifference equations. The effects of competition on the travelling wave solutions of invading plant species is investigated numerically.

Numerical approximation of the product of the square root of a matrix with a vector

Abstract Given an n×n symmetric positive definite matrix A and a vector c → , two numerical methods for approximating A 1/2 c → are developed, analyzed, and computationally tested. The first method applies a Newton iteration to a specific nonlinear system to approximate A 1/2 c → while the second method applies a step-control method to numerically solve a specific initial-value problem to approximate A 1/2 c → . Assuming that A is first reduced to tridiagonal form, the first method requires O (n 2 ) operations per iteration while the second method requires O (n) operations per iteration. In contrast, numerical methods that first approximate A 1/2 and then compute A 1/2 c → generally require O (n 3 ) operations per iteration.

Stochastic neutron transport equations for rod and plane geometries

Abstract In the derivation of the neutron transport equation, it is assumed that neutron populations are large enough so that fluctuations in the neutron population due to random neutron interactions can be ignored. This assumption is removed in this investigation. The result is a system of stochastic differential equations that models the random behavior of neutron transport. Rod and plane geometries are considered in the present investigation. Isotropic scattering is assumed. Numerical procedures are developed and tested for solving these systems. The results are compared with Monte-Carlo calculations which confirm the accuracy of these stochastic neutron transport equations.

Extrapolation of difference methods in option valuation

In the present investigation, the fully implicit and Crank-Nicolson difference schemes for solving option prices are analyzed. It is proved that the error expansions for the difference methods have the correct form for applying Richardson extrapolation to increase the order of accuracy of the approximations. The difference methods are applied to European, American, and down-and-out knock-out call options. Computational results indicate that Richardson extrapolation significantly decreases the amount of computational work (by as much as a factor of 16) in estimation of option prices.

Classical and Modern Numerical Analysis: Theory, Methods and Practice

Classical and Modern Numerical Analysis: Theory, Methods and Practice provides a sound foundation in numerical analysis for more specialized topics, such as finite element theory, advanced numerical linear algebra, and optimization. It prepares graduate students for taking doctoral examinations in numerical analysis. The text covers the main areas of introductory numerical analysis, including the solution of nonlinear equations, numerical linear algebra, ordinary differential equations, approximation theory, numerical integration, and boundary value problems. Focusing on interval computing in numerical analysis, it explains interval arithmetic, interval computation, and interval algorithms. The authors illustrate the concepts with many examples as well as analytical and computational exercises at the end of each chapter. This advanced, graduate-level introduction to the theory and methods of numerical analysis supplies the necessary background in numerical methods so that students can apply the techniques and understand the mathematical literature in this area. Although the book is independent of a specific computer program, MATLAB code will be available on the CRC Press website to illustrate various concepts.

A class of second-order Runge-Kutta methods for numerical solution of stochastic differential equations

A class of explicit Runge-Kutta methods for numerical solution of stochastic differential equations is described, analyzed, and numerically tested. It is shown that this class is of second-order accuracy in the weak sense. Also, a varaince reduction technique is presented that reduces the stochastic error involved when computing expectations. Numerical examples are presented to support the theoretical results.

Derivation of stochastic partial differential equations for size- and age-structured populations

Stochastic partial differential equations (SPDEs) for size-structured and age- and size-structured populations are derived from basic principles, i.e. from the changes that occur in a small time interval. Discrete stochastic models of size-structured and age-structured populations are constructed, carefully taking into account the inherent randomness in births, deaths, and size changes. As the time interval decreases, the discrete stochastic models lead to systems of Itô stochastic differential equations. As the size and age intervals decrease, SPDEs are derived for size-structured and age- and size-structured populations. Comparisons between numerical solutions of the SPDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations.