Alain Largillier

Spectral computations for bounded operators

SPECTRAL DECOMPOSITION Genera Notions Decompositions Spectral Sets of Finite Type Adjoint and Product Spaces SPECTRAL APPROXIMATION Convergence of operators Property U Property L Error Estimates IMPROVEMENT OF ACCURACY Iterative Refinement Acceleration FINITE RANK APPROXIMATIONS Approximations Based on Projection Approximations of Integral Operators A Posteriori Error Estimates MATRIX FORMULATIONS Finite Rank Operators Iterative Refinement Acceleration Numerical Examples MATRIX COMPUTATIONS QR Factorization Convergence of a Sequence of Subspaces QR Methods and Inverse Iteration Error Analysis REFERENCES INDEX Each chapter also includes exercises

An L 1 refined projection approximate solution of the radiation transfer equation in stellar atmospheres

This paper deals with the numerical approximation of the solution of a weakly singular integral equation of the second kind which appears in Astrophysics. The reference space is the complex Banach space of Lebesgue integrable functions on a bounded interval whose amplitude represents the optical thickness of the atmosphere. The kernel of the integral operator is defined through the first exponential-integral function and depends on the albedo of the media. The numerical approximation is based on a sequence of piecewise constant projections along the common annihilator of the corresponding local means. In order to produce high precision solutions without solving large scale linear systems, we develop an iterative refinement technique of a low order approximation. For this scheme, parallelization of matrix computations is suitable.

Superconvergence of Some Projection Approximations for Weakly Singular Integral Equations Using General Grids

This paper deals with superconvergence phenomena in general grids when projection-based approximations are used for solving Fredholm integral equations of the second kind with weakly singular kernels. Four variants of the Galerkin method are considered. They are the classical Galerkin method, the iterated Galerkin method, the Kantorovich method, and the iterated Kantorovich method. It is proved that the iterated Kantorovich approximation exhibits the best superconvergence rate if the right-hand side of the integral equation is nonsmooth. All error estimates are derived for an arbitrary grid without any uniformity or quasi-uniformity condition on it, and are formulated in terms of the data without any additional assumption on the solution. Numerical examples concern the equation governing transfer of photons in stellar atmospheres. The numerical results illustrate the fact that the error estimates proposed in the different theorems are quite sharp, and confirm the superiority of the iterated Kantorovich scheme.

THE ROLES OF A WEAK SINGULARITY AND THE GRID UNIFORMITY IN RELATIVE ERROR BOUNDS

We develop a relative error bound specifying the role of the singularity and the one of the grid uniformity in the case of projection type approximations for the solution of a Fredholm integral equation of the second kind with a weakly singular kernel of convolution type. Numerical experiments with equations having solutions of different nature, namely continuous, integrable bounded discontinuous and integrable unbounded, complete this work. These experiments give an idea on how realistic the theoretical relative error estimates are, depending on the nature of the solution.

Krylov Method Revisited with an Application to the Localization of Eigenvalues

Our aim is to localize matrix eigenvalues in the sense that we build a sufficiently small neighborhood for each of them (or for a cluster), through not prohibitively expensive computations. Our results enter the framework started with Gerschgorin disks and deals at the present time with pseudospectra. The set of theoretical tools we have chosen to use does not avoid the notion of the characteristic polynomial. Certainly, when some computations are performed on it, the well-known ill-conditioning of its coefficients with respect to the matrix entries is properly and carefully handled.

Lp error estimates for projection approximations

We provide an error estimate for the local mean projection approximation in Lp([0,τ∗]) for p∈[1,+∞[, in terms of the regularity of the underlying grid, and we apply it to the corresponding projection approximation of weakly singular Fredholm integral equations of the second kind.

Finite-rank methods and their stability for coupled systems of operator equations

Let

Bounds for relative errors of complex matrix factorizations

\mathcal{K}

A variant of the fixed tangent method for spectral computations on integral operators

be a bounded linear operator of finite rank on a normed linear space X. The solution of a coupled system of linear equations involving

Two numerical approximations for a class of weakly singular integral operators

\mathcal{K}