Abstract—In many image restoration/resolution enhancement applications, the blurring process, i.e., point spread function (PSF) of the imaging system, is not known or is known only to within a set of parameters. We estimate these PSF parameters for this ill-posed class of inverse problem from raw data, along with the regularization parameters required to stabilize the solution, using the generalized cross-validation method (GCV). We propose efficient approximation techniques based on the Lanczos algorithm and Gauss quadrature theory, reducing the computational complexity of the GCV. Data-driven PSF and regularization parameter estimation experiments with synthetic and real image sequences are presented to demonstrate the effectiveness and robustness of our method. Index Terms—Blind restoration, blur identification, generalized cross-validation, quadrature rules, superresolution. I.

Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a well-known result of Demko, Moss and Smith on the decay of the inverse we show that when A is banded, the entries of f(A)are bounded in an exponentially decaying manner away from the main diagonal. Bounds obtained by representing the entries of f(A) in terms of Riemann–Stieltjes integrals and by approximating such integrals by Gaussian quadrature rules are also considered. Applications of these bounds to preconditioning are suggested and illustrated by a few numerical examples.

The central mathematical problem of this report is to bound the quantity u T f(A)v, where A is a given n \Theta n real matrix, u and v are given n-vectors, and f is a given smooth function. Estimating the entries and the trace of the inverse of a matrix and the determinant of a matrix can be classified as such problems. There are a number of interesting applications for such matrix computation problems. The applications in fractal and lattice Quantum Chromodynamics (QCD) are our new motivation for studying such problems. In these applications, the matrices involved are sparse and could be up to the order of millions. It is still a challenging problem to efficiently solve such large matrix computation problems on today's supercomputers. 1 Introduction The central problem studied in this chapter is to estimate a lower bound L and/or an upper bound U , such that L u T f(A)v U; (1) where A is an n \Theta n given real matrix, u and v are given n-vectors, and f is a given smooth fun...

this paper, we focus on deriving lower and upper bounds for the quantities tr(A

The Conjugate Gradient method has always been successfully used in solving the symmetric and positive de nite systems obtained by the nite element approximation of selfadjoint elliptic partial dierential equations. Taking into account recent results by Golub and Meurant (1997), Meurant (1997), Meurant (1999a), and Strakos and Tichy (2002) which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process, we adapt the stopping criterion introduced by Arioli, Noulard and Russo (2001). Moreover, we show that the use of ecient preconditioners does not require to change the energy norm used by the stopping criterion. Finally, we present the results of several numerical tests that experimentally validate the eectiveness of our stopping criterion.

In this paper, we present an algorithm for computing a partial sum of eigenvalues of a large symmetric positive definite matrix pair. We show that this computational task is intimately connected to compute a bilinear form u T f(A)u for a properly defined matrix A, a vector u and a function f(\Delta). Compared to existing techniques which compute individual eigenvalues and then sum them up, the new algorithm is generally less accurate, but requires significantly less memory and CPU time. In the application of electronic structure calculations in molecular dynamics, the new algorithm has achieved a speedup factor of 2 for small size problems to 20 for large size problems. Relative accuracy within 0.1% to 2% is satisfactory. Previously intractable large size problems have been solved. Key words: partial eigenvalue sum, bilinear form, Gauss quadrature, Lanczos method, generalized eigenvalue problem, tight-binding molecular dynamics, Monte Carlo simulation AMS subject classifications: 6...

| Superresolution reconstruction produces a high resolution image from a set of aliased low resolution images. We model the low resolution frames as blurred and down-sampled, shifted versions of the high resolution image. In many applications, the blurring process, i.e., point spread function (PSF) parameters of the imaging system, is not known. In our blind superresolution algorithm, we rst estimate the PSF parameters from the rawdatausing the generalized cross-validation method (GCV). To reduce the computational complexity of GCV, we propose e- cientapproximation techniques based on the Lanczos algorithm and Gauss quadrature theory. Blind superresolution experiments are presented to demonstrate the eectiveness and robustness of our method. I. Introduction Given a sequence of aliased low resolution images # # ## = 1# #### #, superresolution reconstructs a high resolution image # by extracting subpixel information from the given images. We model the low resolution frames as blurre...

The computation of some entries of a matrix inverse arises in several important applications in practice. This paper presents a probing method for determining the diagonal of the inverse of a sparse matrix in the common situation when its inverse exhibits a decay property, i.e., when many of the entries of the inverse are small. A few simple properties of the inverse suggest a way to determine effective probing vectors based on standard graph theory results. An iterative method is then applied to solve the resulting sequence of linear systems, from which the diagonal of the matrix inverse is extracted. Results of numerical experiments are provided to demonstrate the effectiveness of the probing method.

. A simple identity for Krylov sequences is used to study the relationship between spectral decompositions, orthogonal polynomials and the Lanczos algorithm. Key words. Krylov sequences, orthogonal polynomials, Lanczos algorithm, threeterm recurrence relation 1. Introduction. The connection between the Lanczos algorithm and orthogonal polynomials has been investigated in a number of papers both from the theoretical as well as from the computational point of view. It seems that the connection with Krylov sequences has been given less attention than it deserves. In the present note we intend to outline a study of the relationship between the Lanczos algorithm and orthogonal polynomials based on a simple identity for Krylov sequences. In this manner we obtain a simplification of the proofs as well as further insight into some of the classical results. 2. Preliminaries and notation. The elements of C n will be represented by column vectors of length n the indices running from 0 to n \G...

. We survey some unusual eigenvalue problems arising in different applications. We show that all these problems can be cast as problems of estimating quadratic forms. Numerical algorithms based on the well-known Gauss-type quadrature rules and Lanczos process are reviewed for computing these quadratic forms. These algorithms reference the matrix in question only through a matrix-vector product operation. Hence it is well suited for large sparse problems. Some selected numerical examples are presented to illustrate the efficiency of such an approach. 1 Introduction Matrix eigenvalue problems play a significant role in many areas of computational science and engineering. It often happens that many eigenvalue problems arising in applications may not appear in a standard form that we usually learn from a textbook and find in software packages for solving eigenvalue problems. In this paper, we described some unusual eigenvalue problems we have encountered. Some of those problems have been ...