Abderrahman Bouhamidi

A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications

Abstract In the present paper, we propose a Krylov subspace method for solving large and sparse generalized Sylvester matrix equations. The proposed method is an iterative projection method onto matrix Krylov subspaces. As a particular case, we show how to adapt the ILU and the SSOR preconditioners for solving large Sylvester matrix equations. Numerical examples and applications to some PDE’s will be given.

Multivariate interpolating (m, l, s)-splines

The multivariate interpolating (m, l, s)-splines are a natural generalization of Duchons thin plate splines (TPS). More precisely, we consider the problem of interpolation with respect to some finite number of linear continuous functionals defined on a semi-Hilbert space and minimizing its semi-norm. The (m, l, s)-splines are explicitly given as a linear combination of translates of radial basis functions. We prove the existence and uniqueness of the interpolating (m, l, s)-splines and investigate some of their properties. Finally, we present some practical examples of (m, l, s)-splines for Lagrange and Hermite interpolation.

Approximation of vectors fields by thin plate splines with tension

We study a vectorial approximation problem based on thin plate splines with tension involving two positive parameters: one for the control of the oscillations and the other for the control of the divergence and rotational components of the field. The existence and uniqueness of the solution are proved and the solution is explicitly given. As special cases, we study the limit problems as the parameter controlling the divergence and the rotation converges to zero or infinity. The divergence-free and the rotation-free approximation problems are also considered. The convergence in Sobolev space is studied.

Conditional gradient Tikhonov method for a convex optimization problem in image restoration

In this paper, we consider the problem of image restoration with Tikhonov regularization as a convex constrained minimization problem. Using a Kronecker decomposition of the blurring matrix and the Tikhonov regularization matrix, we reduce the size of the image restoration problem. Therefore, we apply the conditional gradient method combined with the Tikhonov regularization technique and derive a new method. We demonstrate the convergence of this method and perform some numerical examples to illustrate the effectiveness of the proposed method as compared to other existing methods.

Convex constrained optimization for large-scale generalized Sylvester equations

We propose and study the use of convex constrained optimization techniques for solving large-scale Generalized Sylvester Equations (GSE). For that, we adapt recently developed globalized variants of the projected gradient method to a convex constrained least-squares approach for solving GSE. We demonstrate the effectiveness of our approach on two different applications. First, we apply it to solve the GSE that appears after applying left and right preconditioning schemes to the linear problems associated with the discretization of some partial differential equations. Second, we apply the new approach, combined with a Tikhonov regularization term, to restore some blurred and highly noisy images.

Radial basis functions under tension

We discuss multivariate interpolation with some radial basis function, called radial basis function under tension (RBFT). The RBFT depends on a positive parameter which provides a convenient way of controlling the behavior of the interpolating surface. We show that our RBFT is conditionally positive definite of order at least one and give a construction of the native space, namely a semi-Hilbert space with a semi-norm, minimized by such an interpolant. Error estimates are given in terms of this semi-norm and numerical examples illustrate the behavior of interpolating surfaces.

A preconditioned block Arnoldi method for large Sylvester matrix equations

SUMMARY In this paper, we propose a block Arnoldi method for solving the continuous low-rank Sylvester matrix equation AX + XB = EFT. We consider the case where both A and B are large and sparse real matrices, and E and F are real matrices with small rank. We first apply an alternating directional implicit preconditioner to our equation, turning it into a Stein matrix equation. We then apply a block Krylov method to the Stein equation to extract low-rank approximate solutions. We give some theoretical results and report numerical experiments to show the efficiency of this method. Copyright

Pseudo-polyharmonic vectorial approximation for div-curl and elastic semi-norms

Vector field reconstruction is a problem arising in many scientific applications. In this paper, we study a div-curl approximation of vector fields by pseudo-polyharmonic splines. This leads to the variational smoothing and interpolating spline problems with minimization of an energy involving the curl and the divergence of the vector field. The relationship between the div-curl energy and elastic energy is established. Some examples are given to illustrate the effectiveness of our approach for a vector field reconstruction.

WEIGHTED THIN PLATE SPLINES

A widely known method in multivariate interpolation and approximation theory consists of the use of thin plate splines. In this paper, we investigate some results and properties relative to a wide variety of variational splines in some space of functions arising from a nonnegative weight function. This model includes thin plate splines, splines in tension and discusses smoothing and interpolating splines. Pointwise error estimates are given for both problems.

A Kronecker approximation with a convex constrained optimization method for blind image restoration

In many problems of linear image restoration, the point spread function is assumed to be known even if this information is usually not available. In practice, both the blur matrix and the restored image should be performed from the observed noisy and blurred image. In this case, one talks about the blind image restoration. In this paper, we propose a method for blind image restoration by using convex constrained optimization techniques for solving large-scale ill-conditioned generalized Sylvester equations. The blur matrix is approximated by a Kronecker product of two matrices having Toeplitz and Hankel forms. The Kronecker product approximation is obtained from an estimation of the point spread function. Numerical examples are given to show the efficiency of our proposed method.