Showing papers on "Preconditioner published in 2008"

A Multiplicative Calderon Preconditioner for the Electric Field Integral Equation

Abstract: In this paper, a new technique for preconditioning electric field integral equations (EFIEs) by leveraging Calderon identities is presented. In contrast to all previous Calderon preconditioners, the proposed preconditioner is purely multiplicative in nature, applicable to open and closed structures, straightforward to implement, and easily interfaced with existing method of moments (MoM) code. Numerical results demonstrate that the MoM EFIE system obtained using the proposed preconditioning converges rapidly, independently of the discretization density.

Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier-Stokes Equations

Abstract: We study preconditioners for the iterative solution of the linear systems arising in the implicit time integration of the compressible Navier-Stokes equations. The spatial discretization is carried out using a discontinuous Galerkin method with fourth order polynomial interpolations on triangular elements. The time integration is based on backward difference formulas resulting in a nonlinear system of equations which is solved at each timestep. This is accomplished using Newton's method. The resulting linear systems are solved using a preconditioned GMRES iterative algorithm. We consider several existing preconditioners such as block Jacobi and Gauss-Seidel combined with multilevel schemes which have been developed and tested for specific applications. While our results are consistent with the claims reported, we find that these preconditioners lack robustness when used in more challenging situations involving low Mach numbers, stretched grids, or high Reynolds number turbulent flows. We propose a preconditioner based on a coarse scale correction with postsmoothing based on a block incomplete LU factorization with zero fill-in (ILU0) of the Jacobian matrix. The performance of the ILU0 smoother is found to depend critically on the element numbering. We propose a numbering strategy based on minimizing the discarded fill-in in a greedy fashion. The coarse scale correction scheme is found to be important for diffusion dominated problems, whereas the ILU0 preconditioner with the proposed ordering is effective at handling the convection dominated case. While little can be said in the way of theoretical results, the proposed preconditioner is shown to perform remarkably well for a broad range of representative test problems. These include compressible flows ranging from very low Reynolds numbers to fully turbulent flows using the Reynolds averaged Navier-Stokes equations discretized on highly stretched grids. For low Mach number flows, the proposed preconditioner is more than one order of magnitude more efficient than the other preconditioners considered.

A multiplicative Calderón preconditioner for the electric field integral equation

Abstract: A new technique for preconditioning electric field integral equations (EFIEs) by leveraging Calderon identities is presented. In contrast to all previous Calderon EFIE preconditioners, the proposed preconditioner is purely multiplicative in nature, applicable to open and closed structures, straightforward to implement, and easily interfaced with existing method of moments codes. Numerical results demonstrate that the method of moments (MoM) matrix equations obtained using the proposed preconditioner converge rapidly, independently of the discretization density.

A Newton-CG method for large-scale three-dimensional elastic full-waveform seismic inversion

Abstract: We present a nonlinear optimization method for large-scale 3D elastic full-waveform seismic inversion. The method combines outer Gauss–Newton nonlinear iterations with inner conjugate gradient linear iterations, globalized by an Armijo backtracking line search, solved on a sequence of finer grids and higher frequencies to remain in the vicinity of the global optimum, inexactly terminated to prevent oversolving, preconditioned by L-BFGS/Frankel, regularized by a total variation operator to capture sharp interfaces, finely discretized by finite elements in the Lame parameter space to provide flexibility and avoid bias, implemented in matrix-free fashion with adjoint-based computation of reduced gradient and reduced Hessian-vector products, checkpointed to avoid full spacetime waveform storage, and partitioned spatially across processors to parallelize the solutions of the forward and adjoint wave equations and the evaluation of gradient-like information. Several numerical examples demonstrate the grid independence of linear and nonlinear iterations, the effectiveness of the preconditioner, the ability to solve inverse problems with up to 17 million inversion parameters on up to 2048 processors, the effectiveness of multiscale continuation in keeping iterates in the basin of attraction of the global minimum, and the ability to fit the observational data while reconstructing the model with reasonable resolution and capturing sharp interfaces.

Advances in Iterative Methods and Preconditioners for the Helmholtz Equation

Abstract: In this paper we survey the development of fast iterative solvers aimed at solving 2D/3D Helmholtz problems. In the first half of the paper, a survey on some recently developed methods is given. The second half of the paper focuses on the development of the shifted Laplacian preconditioner used to accelerate the convergence of Krylov subspace methods applied to the Helmholtz equation. Numerical examples are given for some difficult problems, which had not been solved iteratively before.

Block-diagonal preconditioning for spectral stochastic finite-element systems

Abstract: Deterministic models of fluid flow and the transport of chemicals in flows in heterogeneous porous media incorporate partial differential equations (PDEs) whose material parameters are assumed to be known exactly. To tackle more realistic stochastic flow problems, it is fitting to represent the permeability coefficients as random fields with prescribed statistics. Traditionally, large numbers of deterministic problems are solved in a Monte Carlo framework and the solutions are averaged to obtain statistical properties of the solution variables. Alternatively, so-called stochastic finite-element methods (SFEMs) discretize the probabilistic dimension of the PDE directly leading to a single structured linear system. The latter approach is becoming extremely popular but its computational cost is still perceived to be problematic as this system is orders of magnitude larger than for the corresponding deterministic problem. A simple block-diagonal preconditioning strategy incorporating only the mean component of the random field coefficient and based on incomplete factorizations has been employed in the literature and observed to be robust, for problems of moderate variance, but without theoretical analysis. We solve the stochastic Darcy flow problem in primal formulation via the spectral SFEM and focus on its efficient iterative solution. To achieve optimal computational complexity, we base our block-diagonal preconditioner on algebraic multigrid. In addition, we provide new theoretical eigenvalue bounds for the preconditioned system matrix. By highlighting the dependence of these bounds on all the SFEM parameters, we illustrate, in particular, why enriching the stochastic approximation space leads to indefinite system matrices when unbounded random variables are employed.

Splitting Methods Based on Algebraic Factorization for Fluid-Structure Interaction

Abstract: We discuss in this paper the numerical approximation of fluid-structure interaction (FSI) problems dealing with strong added-mass effect. We propose new semi-implicit algorithms based on inexact block-$LU$ factorization of the linear system obtained after the space-time discretization and linearization of the FSI problem. As a result, the fluid velocity is computed separately from the coupled pressure-structure velocity system at each iteration, reducing the computational cost. We investigate explicit-implicit decomposition through algebraic splitting techniques originally designed for the FSI problem. This approach leads to two different families of methods which extend to FSI the algebraic pressure correction method and the Yosida method, two schemes that were previously adopted for pure fluid problems. Furthermore, we have considered the inexact factorization of the fluid-structure system as a preconditioner. The numerical properties of these methods have been tested on a model problem representing a blood-vessel system.

Parallel Newton-Krylov Solver for the Euler Equations Discretized Using Simultaneous-Approximation Terms

Abstract: 0mesh continuity at block interfaces, accommodates arbitrary block topologies, and has low interblock-communication overhead. The resulting discrete equations are solved iteratively using an inexact-Newton method. At each Newton iteration, the linear system is solved inexactly using a Krylov-subspace iterative method, and both additive Schwarz and approximate Schur preconditioners are investigated. The algorithm is tested on the ONERA M6 wing geometry. We conclude that the approximate Schur preconditioner is an efficient alternative to the Schwarz preconditioner. Overall, the results demonstrate that the Newton–Krylov algorithm is very efficient: using 24 processors, a transonic flow on a 96-block, 1-million-node mesh requires 12 minutes for a 10-order reduction of the residual norm.

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

Abstract: The recent advances in microarchitectural bone imaging disclose the possibility to assess both the apparent density and the trabecular microstructure of intact bones in a single measurement. Coupling these imaging possibilities with microstructural finite element (µFE) analysis offers a powerful tool to improve bone stiffness and strength assessment for individual fracture risk prediction. Many elements are needed to accurately represent the intricate microarchitectural structure of bone; hence, the resulting µFE models possess a very large number of degrees of freedom. In order to be solved quickly and reliably on state-of-the-art parallel computers, the µFE analyses require advanced solution techniques. In this paper, we investigate the solution of the resulting systems of linear equations by the conjugate gradient algorithm, preconditioned by aggregation-based multigrid methods. We introduce a variant of the preconditioner that does not need assembling the system matrix but uses element-by-element techniques to build the multilevel hierarchy. The preconditioner exploits the voxel approach that is common in bone structure analysis, and it has modest memory requirements, at the same time robust and scalable. Using the proposed methods, we have solved in 12min a model of trabecular bone composed of 247 734 272 elements, yielding a matrix with 1 178 736 360 rows, using 1024 CRAY XT3 processors. The ability to solve, for the first time, large biomedical problems with over 1 billion degrees of freedom on a routine basis will help us improve our understanding of the influence of densitometric, morphological, and loading factors in the etiology of osteoporotic fractures such as commonly experienced at the hip, spine, and wrist. Copyright © 2007 John Wiley & Sons, Ltd.

Recursive Krylov‐based multigrid cycles

Abstract: We consider multigrid (MG) cycles based on the recursive use of a two-grid method, in which the coarse-grid system is solved by μ>1 steps of a Krylov subspace iterative method. The approach is further extended by allowing such inner iterations only at the levels of given multiplicity, whereas V-cycle formulation is used at all other levels. For symmetric positive definite systems and symmetric MG schemes, we consider a flexible (or generalized) conjugate gradient method as Krylov subspace solver for both inner and outer iterations. Then, based on some algebraic (block matrix) properties of the V-cycle MG viewed as a preconditioner, we show that the method can have optimal convergence properties if μ is chosen to be sufficiently large. We also formulate conditions that guarantee both, optimal complexity and convergence, bounded independently of the number of levels. Our analysis shows that the method is, at least, as effective as the standard W-cycle, whereas numerical results illustrate that it can be much faster than the latter, and actually more robust than predicted by the theory. Copyright © 2007 John Wiley & Sons, Ltd.

Hierarchical Bases for Nonhierarchic 3-D Triangular Meshes

Abstract: We describe a novel basis of hierarchical, multiscale functions that are linear combinations of standard Rao-Wilton-Glisson (RWG) functions. When the basis is used for discretizing the electric field integral equation (EFIE) for PEC objects it gives rise to a linear system immune from low-frequency breakdown, and well conditioned for dense meshes. The proposed scheme can be applied to any mesh with triangular facets, and therefore it can be used as if it were an algebraic preconditioner. The properties of the new system are confirmed by numerical results that show fast convergence rates of iterative solvers, significantly better than those for the loop-tree basis. As a byproduct of the basis generation, a generalization of the RWG functions to nonsimplex cells is introduced.

Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients

Abstract: This paper gives a solution to an open problem concerning the performance of various multilevel preconditioners for the linear finite element approximation of second-order elliptic boundary value problems with strongly discontinuous coefficients. By analyzing the eigenvalue distribution of the BPX preconditioner and multigrid V-cycle preconditioner, we prove that only a small number of eigenvalues may deteriorate with respect to the discontinuous jump or meshsize, and we prove that all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and meshsize. As a result, we prove that the convergence rate of the preconditioned conjugate gradient methods is uniform with respect to the large jump and meshsize. We also present some numerical experiments to demonstrate the theoretical results.

Resistivity imaging with controlled-source electromagnetic data: depth and data weighting

Abstract: We discuss some computational aspects of resistivity imaging by inversion of offshore controlled-source electromagnetic data. We adopt the classic approach to imaging by formulating it as an inverse problem. A weighted least-squares functional measures the misfit between synthetic and observed data. Its minimization by a quasi-Newton algorithm requires the gradient of the functional with respect to the model parameters. We compute the gradient with the adjoint-state technique. Preconditioners can improve the convergence of the inversion. Diagonal preconditioner based on a Born approximation are commonly used. In the context of CSEM inversion, the Born approximation is not really accurate, this limits the possibility of estimating a correct approximation of the Hessian in a smooth medium or, in fact, in any reference background that does not roughly account for the resistors. We hence rely on the limited memory BFGS approximation of the inverse of the Hessian and we improve the inversion convergence with the help of a heuristic data and depth weighting. Based on a numerical example, we show that a simple exponential depth weighting combined with an offset or frequency data weighting significantly improves the convergence rate of a deep-water controlled-source electromagnetic data inversion.

A New Petrov-Galerkin Smoothed Aggregation Preconditioner for Nonsymmetric Linear Systems

Abstract: We propose a new variant of smoothed aggregation (SA) suitable for nonsymmetric linear systems. The new algorithm is based on two key generalizations of SA: restriction smoothing and local damping. Restriction smoothing refers to the smoothing of a tentative restriction operator via a damped Jacobi-like iteration. Restriction smoothing is analogous to prolongator smoothing in standard SA and in fact has the same form as the transpose of prolongator smoothing when the matrix is symmetric. Local damping refers to damping parameters used in the Jacobi-like iteration. In standard SA, a single damping parameter is computed via an eigenvalue computation. Here, local damping parameters are computed by considering the minimization of an energy-like quantity for each individual grid transfer basis function. Numerical results are given showing how this method performs on highly nonsymmetric systems.

On the relationship between the multiscale finite-volume method and domain decomposition preconditioners

Abstract: In this paper, we review the classical nonoverlapping domain decomposition (NODD) preconditioners, together with the newly developed multiscale control volume (MSCV) method. By comparing the formulations, we observe that the MSCV method is a special case of a NODD preconditioner. We go on to suggest how the more general framework of NODD can be applied in the multiscale context to obtain improved multiscale estimates.

Limited‐memory preconditioners, with application to incremental four‐dimensional variational data assimilation

Abstract: Incremental four-dimensional variational assimilation (4D-Var) is an algorithm that approximately solves a nonlinear minimization problem by solving a sequence of linearized (quadratic) minimization problems of the form where x is the control vector, A is a symmetric positive-definite matrix, b is a vector containing data and prior information, and c is a constant. This paper proposes a family of limited-memory preconditioners (LMPs) for accelerating the convergence of conjugate-gradient (CG) methods used to solve quadratic minimization problems such as those encountered in incremental 4D-Var. The family is constructed from a set of vectors {si : i = 1, …, l}, where each si is assumed to be conjugate with respect to the (Hessian) matrix A. In incremental 4D-Var, approximate LMPs from this family can be built using conjugate vectors generated during the CG minimization on previous outer iterations. The spectral and quasi-Newton LMPs employed in many operational 4D-Var systems are shown to be special cases of the family of LMPs proposed here. In addition, a new LMP based on Ritz vectors (approximate eigenvectors) is derived. The Ritz LMP can be interpreted as a stabilized version of the spectral LMP. Numerical experiments performed with a realistic global ocean 4D-Var system are presented, to test the impact of the three different preconditioners. The Ritz LMP is shown to be more effective than, or at least as effective as, the spectral and quasi-Newton LMPs in our 4D-Var experiments. Our experiments also demonstrate the importance of limiting the number of CG (inner) iterations on certain outer iterations to avoid possible divergence of the cost function on the outer loop. The optimal number of CG iterations will depend on the specific preconditioner used, and can be computed a priori, albeit at the expense of several evaluations of the cost function on the outer loop. In a cycled 4D-Var system, it may be necessary to perform this computation periodically to account for changes in the Hessian matrix arising from changes in the observing system and background-flow field. Copyright © 2008 Royal Meteorological Society

An IE-ODDM-MLFMA Scheme With DILU Preconditioner for Analysis of Electromagnetic Scattering From Large Complex Objects

Abstract: For electrically large complex electromagnetic (EM) scattering problems, huge memory is often required for most EM solvers, which is too difficult to be handled by a personal computer (PC) even a workstation. Although the multilevel fast multipole algorithm (MLFMA) effectively deals with electrically large problems to some extent, it is still time and memory consuming for very large objects. In order to further reduce the CPU time and the memory requirement, a hybrid algorithm, based on the overlapped domain decomposition method for integral equations (IE-ODDM), MLFMA and block-diagonal, incomplete lower and upper triangular matrices (DILU) preconditioner, is proposed for the analysis of electrically large problems. The dominant memory requirement for plane wave expansions in the three processes of aggregation, translation and disaggregation in the MLFMA is drastically reduced by the first two techniques. The iterative procedure for each overlapped subdomain solved by the MLFMA is effectively sped up by the DILU preconditioner. After integrating these techniques, the proposed hybrid algorithm is more efficient in computing time and memory requirement compared to the conventional MLFMA and is more suitable for analyzing very large EM scattering problems. Enough accurate solution can be obtained within quite a few outer iterations, where an outer iteration means a complete sweep for all the subdomains. Some numerical examples are presented to demonstrate its validity and efficiency.

Deflation and Balancing Preconditioners for Krylov Subspace Methods Applied to Nonsymmetric Matrices

Abstract: For quite some time, the deflation preconditioner has been proposed and used to accelerate the convergence of Krylov subspace methods. For symmetric positive definite linear systems, the convergence of conjugate gradient methods combined with deflation has been analyzed and compared with other preconditioners, e.g., with the abstract balancing preconditioner [R. Nabben and C. Vuik, SIAM J. Sci. Comput., 27 (2006), pp. 1742-1759]. In this paper, we extend the convergence analysis to nonsymmetric linear systems in the context of GMRES iteration and compare it with the abstract nonsymmetric balancing preconditioner. We are able to show that many results for symmetric positive definite matrices carry over to arbitrary nonsymmetric matrices. First we establish that the spectra of the preconditioned systems are similar. Moreover, we show that under certain conditions, the 2-norm of residuals produced by GMRES combined with deflation is never larger than the 2-norm of residuals produced by GMRES combined with the abstract balancing preconditioner. Numerical experiments are done to nonsymmetric linear systems arising from a finite volume discretization of the convection-diffusion equation, and the numerical results confirm our theoretical results.

A Parallel Direct/Iterative Solver Based on a Schur Complement Approach

Abstract: In this paper, we present HIPS (hierarchical iterative parallel solver) a parallel sparse linear solver that combines effectively direct and iterative methods through a Schur complement approach. The corner stone of our method is to use a special decomposition and ordering of the matrix that allows to construct a reduced system and a robust preconditioner at low memory cost. The parallelization scheme we describe is original for this type of solver and provide a natural way to find a good trade-off between memory and convergence. Eventually, we give some results obtained by our solver on large referenced test cases.

Augmentation block preconditioners for saddle point‐type matrices with singular (1, 1) blocks

Abstract: We consider the use of block preconditioners for the application of the preconditioned Krylov subspace iterative methods to the solution of large saddle point-type systems with singular (1, 1) blocks. Two block triangular preconditioners are introduced and the block diagonal preconditioner in Greif and Schotzau (Electron. Trans. Numer. Anal. 2006; 22:114–121) is extended to nonsymmetric saddle point systems. All these preconditioners are based on augmentation, using nonsingular weight matrices. If the nullity of the (1, 1) block takes its highest possible value, the preconditioned matrix with either block triangular preconditioner has precisely three distinct eigenvalues, and the preconditioned matrix with the block diagonal preconditioner has precisely two distinct eigenvalues, giving rise to immediate convergence of preconditioned GMRES. Finally, numerical experiments that validate the analysis are reported. Copyright © 2008 John Wiley & Sons, Ltd.

A comparison of preconditioners for incompressible Navier–Stokes solvers

Abstract: We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier-Stokes equations. These systems are of the so-called saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For medium-sized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software.

Inertia-Revealing Preconditioning For Large-Scale Nonconvex Constrained Optimization

Abstract: Fast nonlinear programming methods following the all-at-once approach usually employ Newton's method for solving linearized Karush-Kuhn-Tucker (KKT) systems. In nonconvex problems, the Newton direction is guaranteed to be a descent direction only if the Hessian of the Lagrange function is positive definite on the nullspace of the active constraints; otherwise some modifications to Newton's method are necessary. This condition can be verified using the signs of the KKT eigenvalues (inertia), which are usually available from direct solvers for the arising linear saddle point problems. Iterative solvers are mandatory for very large-scale problems, but in general they do not provide the inertia. Here we present a preconditioner based on a multilevel incomplete $LBL^T$ factorization, from which an approximation of the inertia can be obtained. The suitability of the heuristics for application in optimization methods is verified on an interior point method applied to the CUTE and COPS test problems, on large-scale three-dimensional (3D) PDE-constrained optimal control problems, and on 3D PDE-constrained optimization in biomedical cancer hyperthermia treatment planning. The efficiency of the preconditioner is demonstrated on convex and nonconvex problems with $150^3$ state variables and $150^2$ control variables, both subject to bound constraints.

Combination Preconditioning and the Bramble-Pasciak$^{+}$ Preconditioner

Abstract: It is widely appreciated that the iterative solution of linear systems of equations with large sparse matrices is much easier when the matrix is symmetric. It is equally advantageous to employ symmetric iterative methods when a nonsymmetric matrix is self-adjoint in a nonstandard inner product. Here, general conditions for such self-adjointness are considered. A number of known examples for saddle point systems are surveyed and combined to make new combination preconditioners which are self-adjoint in different inner products. In particular, a new method related to the Bramble-Pasciak CG method is introduced and it is shown that a combination of the two outperforms the widely used classical method on a number of examples. Furthermore, we combine Bramble and Pasciak's method with a recently introduced method by Schoberl and Zulehner. The result gives a new preconditioner and inner product that can outperform the original methods of Bramble-Pasciak and Schoberl-Zulehner.

Multilevel Projection-Based Nested Krylov Iteration for Boundary Value Problems

Abstract: We propose a multilevel projection-based method for acceleration of Krylov subspace methods. The projection is constructed in a similar way as in deflation but shifts small eigenvalues to the largest one instead of to zero. In contrast with deflation, however, the convergence rate of a Krylov method combined with this new projection method is insensitive to the inaccurate solve of the Galerkin matrix, which with some particular choice of deflation subspaces is closely related to the coarse-grid solve in multigrid or domain decomposition methods. Such an insensitivity allows the use of inner iterations to solve the Galerkin problem. An application of a Krylov subspace method to the associated Galerkin system with the projection preconditioner leads to a multilevel, nested Krylov iteration. In this multilevel projection Krylov subspace method, information about small eigenvalues to be projected is contained implicitly in the Galerkin system associated with the matrix of the linear system to be solved. These small eigenvalues, from a Krylov method point of view, are responsible for slow convergence. In terms of projection methods, this is conceptually similar to multigrid but different in the sense that in multigrid the projection is done by the smoother. Furthermore, with the only condition being that the deflation matrices are full rank, we have in principle more freedom in choosing the deflation subspace. Intergrid transfer operators used in multigrid are some of the possible candidates. We present numerical results from solving the Poisson equation and the convection-diffusion equation, both in two dimensions. The latter represents the case where the related matrix of coefficients is nonsymmetric. By using a simple piecewise constant interpolation as the basis for constructing the deflation subspace, we obtain the following results: (i) $h$-independent convergence for the Poisson equation and (ii) almost independent of $h$ and the Peclet number for the convection-diffusion equation.