Pascal Joly

Complex conjugate gradient methods

Linear systems with complex coefficients arise from various physical problems. Examples are the Helmholtz equation and Maxwell equations approximated by finite difference or finite element methods, that lead to large sparse linear systems. When the continuous problem is reduced to integral equations, after discretization, one obtains a dense linear system. The resulting matrices are generally non-Hermitian but, most of the time, symmetric and consequently the classical conjugate gradient method cannot be directly applied. Usually, these linear systems have to be solved with a large number of unknowns because, for instance, in electromagnetic scattering problems the mesh size must be related to the wave length of the incoming wave. The higher the frequency of the incoming wave, the smaller the mesh size must be. When one wants to solve 3D-problems, it is no longer practical to use direct method solvers, because of the huge memory they need. So iterative methods are attractive for this kind of problems, even though their convergence cannot be always guaranteed with theoretical results. In this paper we derive several methods from a unified framework and we numerically compare these algorithms on some test problems.

Preconditioned biconjugate gradient methods for numerical reservoir simulation

Abstract This paper describes numerical experiments on solving linear systems of equations that arise in reservoir simulations. The well-known conjugate-gradient methods Orthomin and Gmres are compared to the biconjugate-gradient method and to an accelerated version called the conjugate-gradient squared method. An incomplete factorization technique based on the level of fill-in idea is used, with investigations to find the appropriate level. Finally, the influence of a reordering method on the convergence rate is tested.

Towards a method for solving partial differential equations by using wavelet packet bases

Abstract In this paper we present a new methodology based on the wavelet packet concept, in order to define an adaptive method for the approximation of partial differential equations. The wavelet packet framework allows us to define the notion of a minimal basis that has proven to be an efficient procedure for data compression. The purpose here is to take benefit of this compression to represent accurately and economically the solution of a time dependent PDE. The time discretization is a standard multistep scheme. The spacial discretization is defined by inferring a reduced basis for the solution at the new time step, from the knowledge of the previous ones.

An Introduction to Scientific Computing

This book provides twelve computational projects aimed at numerically solving problems from a broad range of applications including Fluid Mechanics, Chemistry, Elasticity, Thermal Science, Computer Aided Design, Signal and Image Processing. For each project the reader is guided through the typical steps of scientific computing from physical and mathematical description of the problem, to numerical formulation and programming and finally to critical discussion of numerical results. Considerable emphasis is placed on practical issues of computational methods. The last section of each project contains the solutions to all proposed exercises and guides the reader in using the MATLAB scripts. The mathematical framework provides a basic foundation in the subject of numerical analysis of partial differential equations and main discretization techniques, such as finite differences, finite elements, spectral methods and wavelets). The book is primarily intended as a graduate-level text in applied mathematics, but it may also be used by students in engineering or physical sciences. It will also be a useful reference for researchers and practicing engineers

A Dynamical Adaptive Concept Based on Wavelet Packet Best Bases: Application to Convection Diffusion Partial Differential Equations

Abstract We exploit in this paper a methodology based on the wavelet packet concept. It allows for solving partial differential equations employing very few degrees of freedom. The main application is the Burgers equation with a small viscosity. The wavelet packet framework allows one to define the notion of a minimal basis that has proven to be an efficient procedure for data compression. The purpose here is to take benefit of this compression to represent accurately and economically the solution of a time-dependent PDE. The time discretization is a standard multistep scheme. The spatial discretization is defined by inferring a reduced basis for the solution at the new time step, from the knowledge of the previous ones. The wavelet packet method is a better approach for adaptivity in the case where the solution to be approximated has many singularities.

Wavelet preconditioning of the Stokes problem in ψ–ω formulation

The diagonal preconditioning in wavelet basis enables one to obtain an optimal preconditioner for Galerkin discretizations of elliptic operators in Sobolev norms of both positive and negative smoothness. We develop these techniques in order to solve efficiently the bi-Laplacian or the bidimensional Stokes problem in ψ–ω formulation using a diagonal preconditioning in wavelet basis for the H−1/2(∂Ω) boundary operator that relates the trace of ∂nψ to the trace of ω.