The Parallel Iterative Methods (PIM) package for the solution of systems of linear equations on parallel computers

Cunha, Rudnei Dias da; Hopkins, Tim

doi:10.1016/0168-9274(95)00017-o

Cited by 38 publications

(35 citation statements)

References 10 publications

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“…The N d × N ε matrix J is the Jacobian matrix, Ω D * is a N ε -dimensional vector with first order derivatives and Σ D a N ε × N ε matrix with second order derivatives of F D (ε). In this paper, the update system (4) is solved with a BiCGSTAB solver [66] (a subspace preconditioned LSQR method [43] is used when MS regularization is applied).…”

Section: Gauss-newton Minimization Of a Regularized Cost Functionmentioning

confidence: 99%

3d Microwave Tomography With Huber Regularization Applied to Realistic Numerical Breast Phantoms

Bai¹,

Franchois²,

Pižurica³

2016

PIER

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Abstract-Quantitative active microwave imaging for breast cancer screening and therapy monitoring applications requires adequate reconstruction algorithms, in particular with regard to the nonlinearity and ill-posedness of the inverse problem. We employ a fully vectorial three-dimensional nonlinear inversion algorithm for reconstructing complex permittivity profiles from multi-view single-frequency scattered field data, which is based on a Gauss-Newton optimization of a regularized cost function. We tested it before with various types of regularizing functions for piecewise-constant objects from Institut Fresnel and with a quadratic smoothing function for a realistic numerical breast phantom. In the present paper we adopt a cost function that includes a Huber function in its regularization term, relying on a Markov Random Field approach. The Huber function favors spatial smoothing within homogeneous regions while preserving discontinuities between contrasted tissues. We illustrate the technique with 3D reconstructions from synthetic data at 2 GHz for realistic numerical breast phantoms from the University of Wisconsin-Madison UWCEM online repository: we compare Huber regularization with a multiplicative smoothing regularization and show reconstructions for various positions of a tumor, for multiple tumors and for different tumor sizes, from a sparse and from a denser data configuration.

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Section: Gauss-newton Minimization Of a Regularized Cost Functionmentioning

confidence: 99%

3d Microwave Tomography With Huber Regularization Applied to Realistic Numerical Breast Phantoms

Bai¹,

Franchois²,

Pižurica³

2016

PIER

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“…A parallel implementation of the DB-GMRES( m) algorithm was made in Fortran 90 using the strategy employed in the Parallel Iterative Methods (PIM) package by da Cunha and Hopkins [6], suitably extended to cope with a multiple right-hand-sides linear solver. This means that the matrix-matrix product U = AV and the parallel reduction and inner-products are left as user-supplied, external subroutines that are called inside DB-GMRES( m).…”

Section: Parallel Implementation Detailsmentioning

confidence: 99%

Dynamic block GMRES: an iterative method for block linear systems

Cunha

Becker

2006

Adv Comput Math

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We present variants of the block-GMRES(m) algorithms due to Vital and the block-LGMRES( m,k) by Baker, Dennis and Jessup, obtained with replacing the standard QR factorization by a rank-revealing QR factorization in the Arnoldi process. The resulting algorithm allows for dynamic block deflation whenever there is a linear dependency between the Krylov vectors or the convergence of a right-handside occurs. Fortran 90 implementations of the algorithms were tested on a number of test matrices and the results show that in some cases a substantial reduction of the execution time is obtained. Also a parallel implementation of our variant of the block-GMRES( m) algorithm, using Fortran 90 and MPI was tested on SunFire 15K parallel computer, showing good parallel efficiency.

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“…the Householder transformation (although not explicitly computed in this form for efficiency reasons) and the BFGS update procedure (see [1]) of a Hessian matrix, used in the solution of non-linear equations. We have a particular interest in parallelizing this operation because we intend to expand our Parallel Iterative Methods [2] package to solve unconstrained minimization problems, using iterative methods which will have a built-in BFGS update procedure. An overview of the paper is as follows: the basic algorithm and its explicit sharedmemory and message-passing versions are described in Section 2.…”

Section: Introductionmentioning

confidence: 99%

A benchmark study based on the parallel computation of the vector outer-product A = uvT operation

Cunha¹

1997

Concurrency: Pract. Exper.

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In this paper we benchmark the performance of the Cray T3D, IBM 9076 SP/1 and Intel Paragon XP/S parallel computers, using implementations of parallel algorithms for the computation of the vector outer‐product A = uvT operation. The vector outer‐product operation, although very simple in nature, requires the computation of a large number of floating‐point operations and its parallelization induces a great level of communication between the processors. It is thus suited to measure the relative speed of the processor, memory subsystem and network capabilities of a parallel computer. It should not be considered a ‘toy problem’, since it arises in numerical methods in the context of the solution of systems of non‐linear equations – still a difficult problem to solve. We present algorithms for both the explicit shared‐memory and message‐passing programming models together with theoretical computation models for those algorithms. Actual experiments were run on those computers, using Fortran 77 implementations of the algorithms. The results obtained with these experiments show that due to the high degree of communication between the processors one needs a parallel computer with fast communications and carefully implemented data exchange routines. The theoretical computation model allows prediction of the speed‐up to be obtained for some problem size on a given number of processors. © 1997 John Wiley & Sons, Ltd.

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The Parallel Iterative Methods (PIM) package for the solution of systems of linear equations on parallel computers

Abstract: The version in the Kent Academic Repository may differ from the final published version. Users are advised to check http://kar.kent.ac.uk for the status of the paper. Users should always cite the published version of record.

Cited by 38 publications

References 10 publications

3d Microwave Tomography With Huber Regularization Applied to Realistic Numerical Breast Phantoms

3d Microwave Tomography With Huber Regularization Applied to Realistic Numerical Breast Phantoms

Dynamic block GMRES: an iterative method for block linear systems

A benchmark study based on the parallel computation of the vector outer-product A = uvT operation

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