Laurence Grammont

Nonlinear Integral Equations of the Second Kind: A New Version of Nyström Method

We consider the numerical approximation of a nonlinear integral operator equation by the Nyström method. We propose a new way of applying this method which leads to a major improvement: We can theoretically attain any desired accuracy while for the classical Nyström, the accuracy is limited by the quadrature error formula. The basic idea behind this new proposition is linearizing the nonlinear equation and discretizing the linearization process after, while the usual approach consists in the opposite.

A New Method For Interpolating In A Convex Subset Of A Hilbert Space

In this paper, interpolating curve or surface with linear inequality constraints is considered as a general convex optimization problem in a Reproducing Kernel Hilbert Space. The aim of the present paper is to propose an approximation method in a very general framework based on a discretized optimization problem in a finite-dimensional Hilbert space under the same set of constraints. We prove that the approximate solution converges uniformly to the optimal constrained interpolating function. Numerical examples are provided to illustrate this result in the case of boundedness and monotonicity constraints in one and two dimensions.

A modified iterated projection method adapted to a nonlinear integral equation

The classical way to tackle a nonlinear Fredholm integral equation of the second kind is to adapt the discretization scheme from the linear case. The Iterated projection method is a popular method since it shows, in most cases, superconvergence and it is easy to implement. The problem is that the accuracy of the approximation is limited by the mesh size discretization. Better approximations can only be achieved for fine discretizations and the size of the linear system to be solved then becomes very large: its dimension grows up with an order proportional to the square of the mesh size. In order to overcome this difficulty, we propose a novel approach to first linearize the nonlinear equation by a Newton-type method and only then to apply the Iterated projection method to each of the linear equations issued from the Newton method. We prove that, for any value (large enough) of the discretization parameter, the approximation tends to the exact solution when the number of Newton iterations tends to infinity, so that we can attain any desired accuracy. Numerical experiments confirm this theoretical result.

Modified projection method for Urysohn integral equations with non-smooth kernels

Consider a nonlinear operator equation x - K ( x ) = f , where K is a Urysohn integral operator with a Greens function type kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials, previous authors have investigated approximate solution of this equation using the Galerkin and the iterated Galerkin methods. They have shown that the iterated Galerkin solution is superconvergent. In this paper, a solution obtained using the iterated modified projection method is shown to converge faster than the iterated Galerkin solution. The improvement in the order of convergence is achieved by retaining the size of the system of equations same as for the Galerkin method. Numerical results are given to illustrate the improvement in the order of convergence.

A new degenerate kernel method for a weakly singular integral equation

In order to compute an approximate solution of a weakly singular integral equation, we first regularize the kernel and then truncate the associated Fourier series. Applications to Green and Abel operators are given.

An Extension of the Product Integration Method to L1 with Applications in Astrophysics

A Fredholm integral equation of the second kind in L1([a, b], C) with a weakly singular kernel is considered. Sufficient conditions are given for the existence and uniqueness of the solution. We adapt the product integration method proposed in C0 ([a, b], C) to apply it in L1 ([a, b], C), and discretize the equation. To improve the accuracy of the approximate solution, we use different iterative refinement schemes which we compare one to each other. Numerical evidence is given with an application in Astrophysics.

On ɛ-spectra and stability radii

Techniques of Krylov subspace iterations play an important role in computing e-spectra of large matrices. To obtain results about the reliability of this kind of approximations, we propose to compare the position of the e-spectrum of A with those of its diagonal submatrices. We give theoretical results which are valid for any block decomposition in four blocks, A11,A12,A21,A22. We then illustrate our results by numerical experiments. The same kind of problem arises when we compute the stability radius of a large matrix. In that context, we propose a new sufficient condition for the stability of a matrix involving quantities readily computable such as stability radius of small submatrices.