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26
Bounds for the entries of matrix functions with applications to preconditioning
 BIT
, 1999
"... Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a wellknown 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 ..."
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Cited by 32 (14 self)
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Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a wellknown 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.
Efficient generalized crossvalidation with applications to parametric image restoration and resolution enhancement
 IEEE Trans. Image Processing
, 2001
"... 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 illposed class of inverse problem from raw da ..."
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Cited by 31 (6 self)
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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 illposed class of inverse problem from raw data, along with the regularization parameters required to stabilize the solution, using the generalized crossvalidation method (GCV). We propose efficient approximation techniques based on the Lanczos algorithm and Gauss quadrature theory, reducing the computational complexity of the GCV. Datadriven 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 crossvalidation, quadrature rules, superresolution. I.
A Stopping Criterion for the Conjugate Gradient Algorithm in a Finite Element Method Framework
, 2002
"... 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), ..."
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Cited by 22 (5 self)
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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.
Some Large Scale Matrix Computation Problems
 J. Comput. Appl. Math
"... 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 nvectors, 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 clas ..."
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Cited by 22 (6 self)
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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 nvectors, 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 nvectors, and f is a given smooth fun...
Bounds for the Trace of the Inverse and the Determinant of Symmetric Positive Definite Matrices
, 1996
"... this paper, we focus on deriving lower and upper bounds for the quantities tr(A ..."
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Cited by 16 (3 self)
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this paper, we focus on deriving lower and upper bounds for the quantities tr(A
A Probing Method for Computing the Diagonal of the Matrix Inverse ∗
, 2010
"... 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 ent ..."
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Cited by 10 (1 self)
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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.
Computing partial eigenvalue sum in electronic structure calculations
, 1998
"... In this paper, we present an algorithm for computing a partial sum of eigenvalues of a large symmetric positive de nite matrix pair. We show that this computational task is intimately connected to compute a bilinear form u T f(A)u for a properly de ned matrix A, avector u and a function f (). Compar ..."
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Cited by 8 (1 self)
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In this paper, we present an algorithm for computing a partial sum of eigenvalues of a large symmetric positive de nite matrix pair. We show that this computational task is intimately connected to compute a bilinear form u T f(A)u for a properly de ned matrix A, avector u and a function f (). Compared to existing techniques which compute individual eigenvalues and then sum them up, the new algorithm is generally less accurate, but requires signi cantly 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.
Computational Methods for UVSuppressed Fermions
"... Abstract. Lattice fermions with suppressed high momentum modes solve the ultraviolet slowing down problem in lattice QCD. This paper describes a stochastic evaluation of the effective action of such fermions. The method is a based on the Lanczos algorithm and it is shown to have the same complexity ..."
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Cited by 4 (2 self)
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Abstract. Lattice fermions with suppressed high momentum modes solve the ultraviolet slowing down problem in lattice QCD. This paper describes a stochastic evaluation of the effective action of such fermions. The method is a based on the Lanczos algorithm and it is shown to have the same complexity as in the case of standard fermions.
Blind superresolution with generalized crossvalidation using Gausstype quadrature rules
 in 33rd Asilomar Conf. Signals, Syst. Comput
, 1999
"... ..."
Blind Restoration and Superresolution Using Generalized CrossValidation with Gauss Quadrature Rules
, 2000
"... In many image restoration/superresolution applications, the blurring process, i.e., point spread function (PSF) of the imaging system, is not known or known only to within a set of parameters. We estimate these PSF parameters for this illposed class of inverse problem from raw data, along with t ..."
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Cited by 2 (0 self)
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In many image restoration/superresolution applications, the blurring process, i.e., point spread function (PSF) of the imaging system, is not known or known only to within a set of parameters. We estimate these PSF parameters for this illposed class of inverse problem from raw data, along with the regularization parameters required to stabilize the solution, using the generalized crossvalidation method (GCV). To reduce the computational complexity of GCV, we propose efficient approximation techniques based on the Lanczos algorithm and Gauss quadrature theory. Datadriven blind restoration/superresolution experiments with synthetic and Forward Looking Infrared (FLIR) image sequences are presented to demonstrate the effectiveness and robustness of our method.