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This paper describes a new numerical method for the solution of large linear discrete ill-posed problems, whose matrix is a Kronecker product. Problems of this kind arise, for instance, from the discretization of Fredholm integral equations of the first kind in two space-dimensions with a separable kernel. The available data (right-hand side) of many linear discrete ill-posed problems that arise in applications is contaminated by measurement errors. Straightforward solution of these problems generally is not meaningful because of severe error propagation. We discuss how to combine the truncated singular value decomposition (TSVD) with reduced rank vector extrapolation to determine computed approximate solutions that are fairly insensitive to the error in the data. Exploitation of the structure of the problem keeps the computational effort quite small.

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 divergencefree and the rotation-free approximation problems are also considered. The convergence in Sobolev space is studied.

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In this paper, we propose a block Arnoldi method for solving the continuous low-rank Sylvester matrix equation AX C XB D EF T . 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.where the unknown matrix X 2 R n s , the coefficient matrices A 2 R n n , B 2 R s s and E 2 R n r , F 2 R s r are full rank with r n, s. Sylvester equations arise in numerous applied areas such as control and communication theory and model reduction problems [1][2][3]. The matrix Equation (1) appears also in the numerical solution of matrix differential Riccati equations, in decoupling techniques for ordinary and partial differential equations, and in filtering and image restoration (see, e.g., [4][5][6] and also the references [7,8]).The matrix Equation (1) can be reformulated as the ns ns linear system .

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. r 2004 Elsevier Inc. All rights reserved.MSC: 65D05; 65D07; 65D15; 41A15