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635
High Resolution Forward and Inverse Earthquake Modeling on Terascale Computers
 In SC2003
, 2003
"... For earthquake simulations to play an important role in the reduction of seismic risk, they must be capable of high resolution and high fidelity. We have developed algorithms and tools for earthquake simulation based on multiresolution hexahedral meshes. We have used this capability to carry out 1 H ..."
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Cited by 80 (27 self)
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For earthquake simulations to play an important role in the reduction of seismic risk, they must be capable of high resolution and high fidelity. We have developed algorithms and tools for earthquake simulation based on multiresolution hexahedral meshes. We have used this capability to carry out 1 Hz simulations of the 1994 Northridge earthquake in the LA Basin using 100 million grid points. Our wave propagation solver sustains 1.21 teraflop/s for 4 hours on 3000 AlphaServer processors at 80% parallel efficiency. Because of uncertainties in characterizing earthquake source and basin material properties, a critical remaining challenge is to invert for source and material parameter fields for complex 3D basins from records of past earthquakes. Towards this end, we present results for material and source inversion of highresolution models of basins undergoing antiplane motion using parallel scalable inversion algorithms that overcome many of the difficulties particular to inverse heterogeneous wave propagation problems.
Experimental Study of ILU Preconditioners for Indefinite Matrices
 J. COMPUT. APPL. MATH
, 1997
"... Incomplete LU factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices. However, their failure rate is still too high for them to be useful as blackbox library software for general matrices. Besides fatal breakdowns due to zer ..."
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Cited by 76 (8 self)
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Incomplete LU factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices. However, their failure rate is still too high for them to be useful as blackbox library software for general matrices. Besides fatal breakdowns due to zero pivots, the major causes of failure are inaccuracy, and instability of the triangular solves. When there are small pivots, both these problems can occur, but these problems can also occur without small pivots. Through examples from actual problems, this paper shows how these problems evince themselves, how these problems can be detected, and how these problems can sometimes be circumvented through pivoting, reordering, scaling, perturbing diagonal elements, and preserving symmetric structure. The goal of this paper is to gain a better practical understanding of ILU preconditioners and help improve their reliability.
ConjugateGradient Preconditioning Methods for ShiftVariant PET Image Reconstruction
 IEEE Tr. Im. Proc
, 2002
"... Gradientbased iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian mat ..."
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Cited by 75 (31 self)
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Gradientbased iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shiftinvariant, i.e. for those with approximately blockToeplitz or blockcirculant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantumlimited optical imaging, the Hessian of the weighted leastsquares objective function is quite shiftvariant, and circulant preconditioners perform poorly. Additional shiftvariance is caused by edgepreserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shiftvariant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugategradient (CG) iteration. We also propose a new efficient method for the linesearch step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.
A Computationally Efficient Superresolution Image Reconstruction Algorithm
, 2000
"... Superresolution reconstruction produces a highresolution image from a set of lowresolution images. Previous iterative methods for superresolution had not adequately addressed the computational and numerical issues for this illconditioned and typically underdetermined large scale problem. We propo ..."
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Cited by 72 (4 self)
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Superresolution reconstruction produces a highresolution image from a set of lowresolution images. Previous iterative methods for superresolution had not adequately addressed the computational and numerical issues for this illconditioned and typically underdetermined large scale problem. We propose efficient block circulant preconditioners for solving the Tikhonovregularized superresolution problem by the conjugate gradient method. We also extend to underdetermined systems the derivation of the generalized crossvalidation method for automatic calculation of regularization parameters. Effectiveness of our preconditioners and regularization techniques is demonstrated with superresolution results for a simulated sequence and a forward looking infrared (FLIR) camera image sequence.
Bayesian waveletbased image deconvolution: A GEM algorithm exploiting a class of heavytailed priors
 IEEE Trans. Image Process
, 2006
"... Abstract—Image deconvolution is formulated in the wavelet domain under the Bayesian framework. The wellknown sparsity of the wavelet coefficients of realworld images is modeled by heavytailed priors belonging to the Gaussian scale mixture (GSM) class; i.e., priors given by a linear (finite of inf ..."
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Cited by 69 (11 self)
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Abstract—Image deconvolution is formulated in the wavelet domain under the Bayesian framework. The wellknown sparsity of the wavelet coefficients of realworld images is modeled by heavytailed priors belonging to the Gaussian scale mixture (GSM) class; i.e., priors given by a linear (finite of infinite) combination of Gaussian densities. This class includes, among others, the generalized Gaussian, the Jeffreys, and the Gaussian mixture priors. Necessary and sufficient conditions are stated under which the prior induced by a thresholding/shrinking denoising rule is a GSM. This result is then used to show that the prior induced by the “nonnegative garrote ” thresholding/shrinking rule, herein termed the garrote prior, is a GSM. To compute the maximum a posteriori estimate, we propose a new generalized expectation maximization (GEM) algorithm, where the missing variables are the scale factors of the GSM densities. The maximization step of the underlying expectation maximization algorithm is replaced with a linear stationary secondorder iterative method. The result is a GEM algorithm of ( log) computational complexity. In a series of benchmark tests, the proposed approach outperforms or performs similarly to stateofthe art methods, demanding comparable (in some cases, much less) computational complexity. Index Terms—Bayesian, deconvolution, expectation maximization (EM), generalized expectation maximization (GEM), Gaussian scale mixtures (GSM), heavytailed priors, wavelet. I.
Adaptively Preconditioned Gmres Algorithms
 SIAM J. Sci. Comput
"... . The restarted GMRES algorithm proposed by Saad and Schultz [22] is one of the most popular iterative methods for the solution of large linear systems of equations Ax = b with a nonsymmetric and sparse matrix. This algorithm is particularly attractive when a good preconditioner is available. The pr ..."
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Cited by 68 (2 self)
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. The restarted GMRES algorithm proposed by Saad and Schultz [22] is one of the most popular iterative methods for the solution of large linear systems of equations Ax = b with a nonsymmetric and sparse matrix. This algorithm is particularly attractive when a good preconditioner is available. The present paper describes two new methods for determining preconditioners from spectral information gathered by the Arnoldi process during iterations by the restarted GMRES algorithm. These methods seek to determine an invariant subspace of the matrix A associated with eigenvalues close to the origin, and move these eigenvalues so that a higher rate of convergence of the iterative methods is achieved. Key words. iterative method, nonsymmetric linear system, Arnoldi process AMS subject classifications. 65F10 1. Introduction. Many problems in Applied Mathematics and Engineering give rise to very large linear systems of equations Ax = b; A 2 R n\Thetan ; x; b 2 R n ; (1.1) with a sparse nons...
Flexible conjugate gradients
 SIAM J. Sci. Comput
, 2000
"... Abstract. We analyze the conjugate gradient (CG) method with preconditioning slightly variable from one iteration to the next. To maintain the optimal convergence properties, we consider a variant proposed by Axelsson that performs an explicit orthogonalization of the search directions vectors. For ..."
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Cited by 64 (8 self)
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Abstract. We analyze the conjugate gradient (CG) method with preconditioning slightly variable from one iteration to the next. To maintain the optimal convergence properties, we consider a variant proposed by Axelsson that performs an explicit orthogonalization of the search directions vectors. For this method, which we refer to as flexible CG, we develop a theoretical analysis that shows that the convergence rate is essentially independent of the variations in the preconditioner as long as the latter are kept sufficiently small. We further discuss the real convergence rate on the basis of some heuristic arguments supported by numerical experiments. Depending on the eigenvalue distribution corresponding to the fixed reference preconditioner, several situations have to be distinguished. In some cases, the convergence is as fast with truncated versions of the algorithm or even with the standard CG method, whereas quite large variations are allowed without too much penalty. In other cases, the flexible variant effectively outperforms the standard method, while the need for truncation limits the size of the variations that can be reasonably allowed.
On the solution of equality constrained quadratic programming problems arising . . .
, 1998
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Sparse Approximate Inverse Preconditioning For Dense Linear Systems Arising In Computational Electromagnetics
 Numerical Algorithms
, 1997
"... . We investigate the use of sparse approximate inverse preconditioners for the iterative solution of linear systems with dense complex coefficient matrices arising from industrial electromagnetic problems. An approximate inverse is computed via a Frobenius norm approach with a prescribed nonzero pat ..."
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Cited by 59 (20 self)
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. We investigate the use of sparse approximate inverse preconditioners for the iterative solution of linear systems with dense complex coefficient matrices arising from industrial electromagnetic problems. An approximate inverse is computed via a Frobenius norm approach with a prescribed nonzero pattern. Some strategies for determining the nonzero pattern of an approximate inverse are described. The results of numerical experiments suggest that sparse approximate inverse preconditioning is a viable approach for the solution of largescale dense linear systems on parallel computers. Key words. Dense linear systems, preconditioning, sparse approximate inverses, complex symmetric matrices, scattering calculations, Krylov subspace methods, parallel computing. AMS(MOS) subject classification. 65F10, 65F50, 65R20, 65N38, 7808, 78A50, 78A55. 1. Introduction. In the last decade, a significant amount of effort has been spent on the simulation of electromagnetic wave propagation phenomena to ad...