Finite difference preconditioning for solving orthogonal collocation equations of boundary value problems

This is the published version, also available here: http://dx.doi.org/10.1137/S0036142993249760.A technique to construct a low-order finite difference preconditioner for solving orthogonal collocation equations for boundary value problems is presented. It is shown numerically and theoretically that the spectral condition numbers of the preconditioned collocation matrices are bounded by constants independent of the number of mesh nodes when certain exact low-order finite difference preconditionings are used. Preconditioners based on incomplete LU factorization are also discussed. Numerical experiments show the efficiency and robustness of the preconditioning

Almost Block Diagonal Linear Systems: Sequential and Parallel Solution Techniques, and Applications

Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in numerical methods for two-point boundary value problems for ordinary differential equations and in related partial differential equation problems. The stable, efficient sequential solution of ABDs has received much attention over the last fifteen years and the parallel solution more recently. We survey the fields of application with emphasis on how ABDs and bordered ABDs (BABDs) arise. We outline most known direct solution techniques, both sequential and parallel, and discuss the comparative efficiency of the parallel methods. Finally, we examine parallel iterative methods for solving BABD systems. Copyright (C) 2000 John Wiley & Sons, Ltd

Isogeometric preconditioners based on fast solvers for the Sylvester equation

We consider large linear systems arising from the isogeometric discretization of the Poisson problem on a single-patch domain. The numerical solution of such systems is considered a challenging task, particularly when the degree of the splines employed as basis functions is high. We consider a preconditioning strategy which is based on the solution of a Sylvester-like equation at each step of an iterative solver. We show that this strategy, which fully exploits the tensor structure that underlies isogeometric problems, is robust with respect to both mesh size and spline degree, although it may suffer from the presence of complicated geometry or coefficients. We consider two popular solvers for the Sylvester equation, a direct one and an iterative one, and we discuss in detail their implementation and efficiency for 2D and 3D problems on single-patch or conforming multi-patch NURBS geometries. Numerical experiments for problems with different domain geometries are presented, which demonstrate the potential of this approach

Institute for Computer Applications in Science and Engineering (ICASE)

Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science during the period April 1, 1983 through September 30, 1983 is summarized

Mesh Free Methods for Differential Models In Financial Mathematics

Philosophiae Doctor - PhDMany problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston's volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided

Mesh free methods for differential models in financial mathematics

Philosophiae Doctor - PhDMany problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.South Afric