Laurence Grammont

Jean Monnet University · Institut Camille JORDAN

This paper establishes error bounds for the convergence of a piecewise linear approximation of the constrained optimal smoothing problem posed in a reproducing kernel Hilbert space (RKHS). This problem can be reformulated as a Bayesian estimation problem involving a Gaussian process related to the kernel of the RKHS. Consequently, error bounds can...

In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) by adding convex constraints to the problem. Through a sequence of approximating Hilbertian subspaces and a discretized model, we prove that the Maximum a posteriori (MAP) of the posterior distribution is exactl...

Consider an integral equation λu-Tu=f, where T is an integral operator, defined on C[0, 1], with a kernel having an algebraic or a logarithmic singularity. Let πm denote an interpolatory projection onto a space of piecewise polynomials of degree ≤r-1 with respect to a graded partition of [0, 1] consisting of m subintervals. In the product integrati...

new product integration scheme is proposed to solve Hammerstein equations which are weakly singular. The standard way of implementing the product integration method to a nonlinear equation is to transform the functional equation to an nonlinear finite dimensional algebraic system by the product integration scheme and then linearize the system to so...

In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) adding a convexe constraints on the solution. Through a sequence of approximating Hilbertian spaces and a discretized model, we prove that the Maximum A Posteriori (MAP) of the posterior distribution is exactly...

In this paper, smoothing curve or surface with both interpolation conditions and inequality constraints is considered as a general convex optimization problem in a Hilbert space. We propose a new approximation method based on a discretized optimization problem in a finite-dimensional Hilbert space under the same set of constraints. We prove that th...

In this paper, we extend the correspondence between Bayes’ estimation and optimal interpolation in a Reproducing Kernel Hilbert Space (RKHS) to the case of convex constraints such as boundedness, monotonicity or convexity. In the unconstrained interpolation case, the mean of the posterior distribution of a Gaussian Process (GP) given data interpola...

We present a new method to solve nonlinear Hammerstein equations with weakly singular kernels. The process to approximate the solution, followed usually, consists in adapting the discretization scheme from the linear case in order to obtain a nonlinear system in a finite dimensional space and solve it by any linearization method. In this paper, we...

Consider a nonlinear operator equation x-K(x)=f, where K is a Urysohn integral operator with a Green's 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 sho...

In this paper, we extend the correspondence between Bayes' estimation and optimal interpolation in a Reproducing Kernel Hilbert Space (RKHS) to the case of linear inequality constraints such as boundedness, monotonicity or convexity. In the unconstrained interpolation case, the mean of the posterior distribution of a Gaussian Process (GP) given dat...

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 ap...

This paper deals with nonlinear Fredholm integral equations of the second kind. We study the case of a weakly singular kernel and we set the problem in the space L 1 ([a, b], C). As numerical method, we extend the product integration scheme from C 0 ([a, b], C) to L 1 ([a, b], C).

To tackle a nonlinear equation in a functional space, two numerical processes are involved: discretization and linearization. In this paper we study the differences between applying them in one or in the other order. Linearize first and discretize the linear problem will be in the sequel called option (A). Discretize first and linearize the discret...

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.

Consider a nonlinear operator equation x-K(x)=f, where K is a Urysohn integral operator with a smooth kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree ≤r, previous authors have established an order r+1 convergence for the Galerkin solution and 2r+2 for the iterated Galerkin solution. Equivalent r...

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 thi...

Mes travaux peuvent se diviser en deux thèmes : L'algèbre linéaire numérique. La théorie des opérateurs intégraux. L'algèbre linéaire numérique fut le cadre de ma thèse de doctorat, dédiée aux propriétés spectrales des opérateurs de Sylvester, endomorphismes d'espaces matriciels. J'ai tout naturellement utilisé mes connaissances, mes compétences et...

Associated with an n×n matrix polynomial of degree , are the eigenvalue problem P(λ)x=0 and the linear system problem P(ω)x=b, where in the latter case x is to be computed for many values of the parameter ω. Both problems can be solved by conversion to an equivalent problem L(λ)z=0 or L(ω)z=c that is linear in the parameter λ or ω. This linearizati...

We propose a fixed point approximation of a compact operator. It is adapted from the method proposed by R.P. Kulkarni for linear operator equations. It is proved to be superconvergent, while the iterated Galerkin method, proposed by K.E. Atkinson and F.A. Potra, needs an additional assumption in order to be superconvergent.

The purpose of this chapter is to compute at a low cost an approximate solution of a Fredholm integral equation at a given accuracy. Vainikko proposed to compute the Nyström approximation of order n with quadrature two-grid iterations. We propose here to compute it with a two-grid method based on a projection method of a new type developed by Kulka...

In [1414. R.P. Kulkarni ( 2003 ). A superconvergence result for solutions of compact operator equations . Bull. Aust. Math. Soc. 68 : 517 – 528 . [CrossRef], [Web of Science ®]View all references], a new method based on projections onto a space of piecewise polynomials of degree ≤r − 1 has been shown to give a convergence of order 4r for second-kin...

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...

We propose a method based on projections for approximating fixed points of a compact nonlinear operator. Under the same assumptions as in the Galerkin method, the proposed solution is shown to converge faster than the Galerkin solution. To cite this article: L. Grammont, R. Kulkarni, C. R. Acad. Sci. Paris, Ser. I 342 (2006).RésuméNous proposons un...

We characterize the stability of discrete-time Lyapunov equations with periodic coefficients. The characterization can be seen as the analog of the classical stability theorem of Lyapunov equations with constant coefficients. It involves quantities readily computable with good accuracy.

In this paper, we study the ε-lemniscate of the characteristic polynomial in relation to the pseudospectrum of the associated matrix. It is natural to investigate this question because these two sets can be seen as gener-alizations of eigenvalues. The question of numerical determination of the ε-lemniscate raises the problem of computing the charac...

Techniques of Krylov subspace iterations play an important role in computing ε-spectra of large matrices. To obtain results about the reliability of this kind of approximations, we propose to compare the position of the ε-spectrum of A with those of its diagonal submatrices. We give theoretical results which are valid for any block decomposition in...

We consider the classical Wicksell problem of estimating an unknown probability density function of the radii of spheres in a medium, based on their observed random cross-sections. This problem is known as Wicksell's corpuscle problem. We first convert this problem to a form suitable for the application of thresholding vaguelet–wavelet methods for...

There exist many definitions of the ε-spectrum. Unlike the spectrum, the ε-spectrum definitions depend on the choice of the norm. We propose to study the conditions on the norm that make all these definitions equivalent. The spectral radius of a related matrix is a useful tool in the eigenvalues perturbation analysis. The study of its connection wi...

The sensivtiity of the solution of the matrix Sylvester equation AX-XB=C is considered in the context of the classical perturbation theory. Our purpose is to find the most influent parameters in the sensitivity of the solution under perturbations in the data, and to compare the theoretical error bounds with numerical evidence.

This paper deals with the equation We propose a new approach to it in the context of Fredholm equations of the second kind.