Block Symmetric Successive Overrelaxation Research Articles

Conjugate gradient method preconditioned with modified block SSOR iteration for multiplicative half-quadratic image restoration

Image restoration problem is often solved by minimizing a cost function which consists of data-fidelity terms and regularization terms. Half-quadratic regularization has the advantage that it can preserve image details well in the recovered images. In this paper, we consider solving the image restoration model which involves multiplicative half-quadratic regularization term. Newton method is employed to solve the nonlinear system of equations resulted from the optimization problem for image restoration. At each Newton iteration step, a linear system of equations with symmetric positive definite coefficient matrix arises. The preconditioned conjugate gradient method with the proposed modified block SSOR (symmetric successive over-relaxation) preconditioner is applied to solve this linear system of equations. The condition number of the preconditioned matrix is estimated and numerical experiments are also implemented for image restoration. Both theoretical and numerical results show that the modified block SSOR preconditioned PCG methods can greatly improve the computation efficiency when solving the multiplicative half-quadratic regularized image restoration problem.

Robust partitioned block preconditioners for large-scale geotechnical applications with soil-structure interactions

SUMMARY Soil–structure interaction problems are commonly encountered in geotechnical practice and remarkably characterized with significant material stiffness contrast. When solving the soil–structure interaction problems, the employed Krylov subspace iterative method may converge slowly or even fail, indicating that the adopted preconditioning method may not suit for such problems. The inexact block diagonal preconditioners proposed recently have been shown effective for the soil–structure interaction problems; however, they haven't been exploited to full capabilities. By using the same partition strategy according to the structure elements and soil elements, the partitioned block symmetric successive over-relaxation preconditioners or partitioned block constraint preconditioners are proposed. Based on two pile-group foundation problems and a tunnel problem, the proposed preconditioners are evaluated and compared with the available preconditioners for the consolidation analysis and the drained analysis, respectively. In spite of one additional solve associated with the structure block and multiplications with off-diagonal blocks in the preconditioning step, numerical results reveal that the proposed preconditioners obviously possess better performance than the recently developed inexact block preconditioners. Copyright © 2013 John Wiley & Sons, Ltd.

GPU-accelerated iterative solutions for finite element analysis of soil–structure interaction problems

Soil–structure interaction problems are commonly encountered in engineering practice, and the resulting linear systems of equations are difficult to solve due to the significant material stiffness contrast. In this study, a novel partitioned block preconditioner in conjunction with the Krylov subspace iterative method symmetric quasiminimal residual is proposed to solve such linear equations. The performance of these investigated preconditioners is evaluated and compared on both the CPU architecture and the hybrid CPU–graphics processing units (GPU) computing environment. On the hybrid CPU–GPU computing platform, the capability of GPU in parallel implementation and high-intensity floating point operations is exploited to accelerate the iterative solutions, and particular attention is paid to the matrix–vector multiplications involved in the iterative process. Based on a pile-group foundation example and a tunneling example, numerical results show that the partitioned block preconditioners investigated are very efficient for the soil–structure interaction problems. However, their comparative performances may apparently depend on the computer architecture. When the CPU computer architecture is used, the novel partitioned block symmetric successive over-relaxation preconditioner appears to be the most efficient, but when the hybrid CPU–GPU computer architecture is adopted, it is shown that the inexact block diagonal preconditioners embedded with simple diagonal approximation to the soil block outperform the others.

Modified alternating direction-implicit iteration method for linear systems from the incompressible Navier–Stokes equations

In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, Ran and Yuan [On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems, Appl. Math. Comput. 217 (2010), pp. 3050–3068] presented the block symmetric successive over-relaxation (BSSOR) and the modified BSSOR iteration methods based on the special structures of the coefficient matrices. In this study, we present the modified alternating direction-implicit (MADI) iteration method for solving the linear systems. Under suitable conditions, we establish convergence theorems for the MADI iteration method. In addition, the optimal parameter involved in the MADI iteration method is estimated in detail. Numerical experiments show that the MADI iteration method is a feasible and effective iterative solver.

Towards the determination of the optimal p-cyclic SSOR

Suppose that A ∈ C n,n is a block p-cyclic consistently ordered matrix and let B and S ω denote the block Jacobi and the block symmetric successive overrelaxation (SSOR) iteration matrices associated with A, respectively. Extending previous work by Hadjidimos and Neumann, the present authors have determined the exact regions of convergence of the SSOR method in the ( ϱ( B), ω)-plane, for any p ⩾ 3, under the further assumption that the eigenvalues of B p are real of the same sign. In the present work the investigation goes on further, several questions are raised and among others the problem of the determination of the optimal regions of convergence in the spirit of Niethammer and Varga as well as that of the optimal relaxation factor are examined.

On the exact p-cyclic SSOR convergence domains

Suppose that A ∈ C n, n is a block p-cyclic consistently ordered matrix, and let B and S ω denote, respectively, the block Jacobi and the block symmetric successive overrelaxation (SSOR) iteration matrices associated with A. Neumaier and Varga found [in the ( ϱ(| B|), ω) plane] the exact convergence and divergence domains of the SSOR method for the class of H-matrices. Hadjidimos and Neumann applied Rouché's theorem to the functional equation connecting the eigenvalue spectra σ( B) and σ( S ω ) obtained by Varga, Niethammer, and Cai, and derived in the ( ϱ( B), ω) plane the convergence domains for the SSOR method associated with p-cyclic consistently ordered matrices, for any p ⩾ 3. In the present work it is further assumed that the eigenvalues of B p are real of the same sign. Under this assumption the exact convergence domains in the ( ϱ( B), ω) plane are derived in both the nonnegative and the nonpositive cases for any p ⩾ 3.

Superior convergence domains for the p-cyclic SSOR majorizer

Thus far for an n × n complex nonsingular matrix A, the symmetric successive overrelaxation (SSOR) majorizing operator has been used to establish convergence properties of the SSOR method mostly in the case where A is an H-matrix. In this paper we use (actually a similarity transformation of) the SSOR majorizer to investigate convergence properties of the block SSOR method when A is a block p-cyclic matrix. Let J A denote the block Jacobi method and let ν = ϱ(¦J A¦) . We establish regions in the ( ν, ω)-plane where ϱ( S ω A) ⩽ ϱ( Q ω A) < ¦ω − 1¦ [⩽ ϱ( L ω A)] . Here S ω A is the block SSOR iteration operator associated with A, L ω A is the block successive overrelaxation (SOR) iteration operator associated with A, and Q ω A is a convenient similarity transformation of the majorizing operator for S ω A . Of special interest to us are the values of ν for which the above inequality holds for the corresponding values of the relaxation parameter ω(A) = 2 (1 + ν) , the latter being an important quantity in the SOR-SSOR theory for H-matrices.

Convergence domains of the SSOR method for a class of generalized consistently ordered matrices

The functional equation relating the eigenvalues of the block Symmetric Successive Overrelaxation (SSOR) iteration matrix with those of the block Jacobi iteration matrix found by Chong and Cai (1985) is used in order to obtain precise domains of convergence of the block SSOR iteration method associated with a class of generalized consistently ordered (GCO) ( k, p− k)-matrices A( p⩾2, k=1, …, p−1). We show that the domain of convergence depends only on the relaxation parameter ω, the spectral radius of the block Jacobi iteration matrix and the value of l = k p . Unlike the case ( k, p− k) = (1, p−1), p⩾3, which was studied in an earlier paper by the authors (1989), beyond certain critical values of l the domain of convergence does not grow monotonically with l and manifests various sorts of behavior. However, as in the case ( k, p− k) = (1, p−1), it is shown that the intersection of the convergence domains taken over all pairs ( p, k) coincides with the exact domain of convergence of the point SSOR iteration method associated with nonsingular H-matrices A, that is, matrices which can be diagonally scaled to being strictly diagonally dominant.

Precise domains of convergence for the block SSOR method associated with p-Cyclic Matrices

We apply Rouche's theorem to the functional equation relating the eigenvalues of theblock symmetric successive overrelaxation (SSOR) matrix with those of the block Jacobi iteration matrix found by Varga, Niethammer, and Cai, in order to obtain precise domains of convergence for the block SSOR iteration method associated withp-cyclic matricesA, p≥3. The intersection of these domains, taken over all suchp's, is shown to coincide with the exact domain of convergence of thepoint SSOR iteration method associated withH-matricesA. The latter domain was essentially discovered by Neumaier and Varga, but was recently sharpened by Hadjidimos and Neumann.

Precise domains of convergence for the block SSOR method associated withp-cyclic matrices

We apply Rouche's theorem to the functional equation relating the eigenvalues of theblock symmetric successive overrelaxation (SSOR) matrix with those of the block Jacobi iteration matrix found by Varga, Niethammer, and Cai, in order to obtain precise domains of convergence for the block SSOR iteration method associated withp-cyclic matricesA, p≥3. The intersection of these domains, taken over all suchp's, is shown to coincide with the exact domain of convergence of thepoint SSOR iteration method associated withH-matricesA. The latter domain was essentially discovered by Neumaier and Varga, but was recently sharpened by Hadjidimos and Neumann.

The Block Symmetric Successive Overrelaxation Method

Previous article Next article The Block Symmetric Successive Overrelaxation MethodLouis W. EhrlichLouis W. Ehrlichhttps://doi.org/10.1137/0112068PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. J. Arms, , L. D. Gates and , B. Zondek, A method of block iteration, J. Soc. Indust. Appl. Math., 4 (1956), 220–229 10.1137/0104012 MR0119405 0077.32502 LinkISIGoogle Scholar[2] E. D'Sylva and , G. A. Miles, The S.S.O.R. iteration scheme for equations with $\sigma \sb{1}$ ordering, Comput. J., 6 (1963/1964), 366–367 MR0158549 0134.32704 CrossrefISIGoogle Scholar[3] L. W. Ehrlich, Masters Thesis, The block symmetric successive overrelaxation method, doctoral thesis, University of Texas, Austin, 1963 Google Scholar[4] M. Engeli, , Th. Ginsburg, , H. Rutishauser and , E. Stiefel, Refined iterative methods for computation of the solution and the eigenvalues of self-adjoint boundary value problems, Mitt. Inst. Angew. Math. Zürich. No., 8 (1959), 107– MR0145689 0089.12103 Google Scholar[5] D. J. Evans and , C. V. D. Forrington, An iterative process for optimizing symmetric successive over-relaxation, Comput. J., 6 (1963), 271–273 0127.08202 CrossrefISIGoogle Scholar[6] George E. Forsythe and , Wolfgang R. Wasow, Finite-difference methods for partial differential equations, Applied Mathematics Series, John Wiley & Sons Inc., New York, 1960x+444 MR0130124 0099.11103 Google Scholar[7] G. J. Habetler and , E. L. Wachspress, Symmetric successive overrelaxation in solving diffusion difference equations, Math. Comp., 15 (1961), 356–362 MR0129139 0102.11403 CrossrefGoogle Scholar[8] Alston S. Householder, Principles of numerical analysis, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953x+274 MR0059056 0051.34602 Google Scholar[9] W. Kahn, Masters Thesis, Gauss-Seidel methods of solving large systems of linear equations, doctoral thesis, University of Toronto, Toronto, 1958 Google Scholar[10] John W. Sheldon, On the numerical solution of elliptic difference equations, Math. Tables Aids Comput., 9 (1955), 101–112 MR0074929 0065.35801 CrossrefGoogle Scholar[11] J. W. Sheldon, A. Ralston and , H. S. Wilf, Iterative methods for the solution of elliptic partial differential equationsMathematical methods for digital computers, Wiley, New York, 1960, 144–156 MR0117919 Google Scholar[12] Richard S. Varga, Matrix iterative analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1962xiii+322 MR0158502 0133.08602 Google Scholar[13] David Young, Iterative methods for solving partial difference equations of elliptic type, Trans. Amer. Math. Soc., 76 (1954), 92–111 MR0059635 0055.35704 CrossrefISIGoogle Scholar[14] David Young, On Richardson's method for solving linear systems with positive definite matrices, J. Math. Physics, 32 (1954), 243–255 MR0060306 0055.11202 CrossrefISIGoogle Scholar[15] David Young, J. Todd, The numerical solution of elliptic and parabolic partial differential equationsSurvey of numerical analysis, McGraw-Hill, New York, 1962, 380–438 MR0136084 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Some History of the Conjugate Gradient and Lanczos Algorithms: 1948–1976SIAM Review, Vol. 31, No. 1 | 2 August 2006AbstractPDF (6558 KB)Block Preconditioning for the Conjugate Gradient MethodSIAM Journal on Scientific and Statistical Computing, Vol. 6, No. 1 | 31 July 2006AbstractPDF (2675 KB)A Survey of Modern Numerical AnalysisSIAM Review, Vol. 15, No. 2 | 2 August 2006AbstractPDF (2464 KB) Volume 12, Issue 4| 1964Journal of the Society for Industrial and Applied Mathematics687-909 History Submitted:24 February 1964Published online:13 July 2006 InformationCopyright © 1964 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0112068Article page range:pp. 807-826ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics