Stand-alone Solver Research Articles

A Parallel Wavelet‐Based Algebraic Multigrid Black‐Box Solver and Preconditioner

This work introduces a new parallel wavelet‐based algorithm for algebraic multigrid method (PWAMG) using a variation of the standard parallel implementation of discrete wavelet transforms. This new approach eliminates the grid coarsening process in traditional algebraic multigrid setup phase simplifying its implementation on distributed memory machines. The PWAMG method is used as a parallel black‐box solver and as a preconditioner in some linear equations systems resulting from circuit simulations and 3D finite elements electromagnetic problems. The numerical results evaluate the efficiency of the new approach as a standalone solver and as preconditioner for the biconjugate gradient stabilized iterative method.

Multi-stage solvers optimized for damping and propagation

Explicit multi-stage solvers are routinely used to solve the semi-discretized equations that arise in Computational Fluid Dynamics (CFD) problems. Often they are used in combination with multi-grid methods. In that case, the role of the multi-stage solver is to efficiently reduce the high frequency modes on the current grid and is called a smoother. In the past, when optimizing the coefficients of the scheme, only the damping characteristics of the smoother were taken into account and the interaction with the remainder of the multi-grid cycle was neglected. Recently it had been found that coefficients that result in less damping, but allow for a higher Courant–Friedrichs–Lewy (CFL) number are often superior to schemes that try to optimize damping alone. While this is certainly true for multi-stage schemes used as a stand-alone solver, we investigate in this paper if using higher CFL numbers also yields better results in a multi-grid setting. We compare the results with a previous study we conducted and where a more accurate model of the multi-grid cycle was used to optimize the various parameters of the solver. We show that the use of the more accurate model results in better coefficients and that in a multi-grid setting propagation is of little importance. We also look into the gains to be made when we allow the parameters to be different for the pre- and post-smoother and show that even better coefficients can be found in this way.

FLEXMG: A new library of multigrid preconditioners for a spectral/finite element incompressible flow solver

AbstractA new library called FLEXMG has been developed for a spectral/finite element incompressible flow solver called SFELES. FLEXMG allows the use of various types of iterative solvers preconditioned by algebraic multigrid methods. Two families of algebraic multigrid preconditioners have been implemented, namely smooth aggregation‐type and non‐nested finite element‐type. Unlike pure gridless multigrid, both of these families use the information contained in the initial fine mesh. A hierarchy of coarse meshes is also needed for the non‐nested finite element‐type multigrid so that our approaches can be considered as hybrid. Our aggregation‐type multigrid is smoothed with either a constant or a linear least‐square fitting function, whereas the non‐nested finite element‐type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand‐alone solvers or coupled with a GMRES method. After analyzing the accuracy of the solutions obtained with our solvers on a typical test case in fluid mechanics, their performance in terms of convergence rate, computational speed and memory consumption is compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in the study. Copyright © 2010 John Wiley & Sons, Ltd.

A High Throughput FPGA-Based Floating Point Conjugate Gradient Implementation for Dense Matrices

Recent developments in the capacity of modern Field Programmable Gate Arrays (FPGAs) have significantly expanded their applications. One such field is the acceleration of scientific computation and one type of calculation that is commonplace in scientific computation is the solution of systems of linear equations. A method that has proven in software to be very efficient and robust for finding such solutions is the Conjugate Gradient (CG) algorithm. In this article we present a widely parallel and deeply pipelined hardware CG implementation, targeted at modern FPGA architectures. This implementation is particularly suited for accelerating multiple small-to-medium-sized dense systems of linear equations and can be used as a stand-alone solver or as building block to solve higher-order systems. In this article it is shown that through parallelization it is possible to convert the computation time per iteration for an order n matrix from Θ ( n 2 ) clock cycles on a microprocessor to Θ ( n ) on a FPGA. Through deep pipelining it is also possible to solve several problems in parallel and maximize both performance and efficiency. I/O requirements are shown to be scalable and convergent to a constant value with the increase of matrix order. Post place-and-route results on a readily available VirtexII-6000 demonstrate sustained performance of 5 GFlops, and results on a Virtex5-330 indicate sustained performance of 35 GFlops. A comparison with an optimized software implementation running on a high-end CPU demonstrate that this FPGA implementation represents a significant speedup of at least an order of magnitude.

Improving benders decomposition using a genetic algorithm

We develop and investigate the performance of a hybrid solution framework for solving mixed-integer linear programming problems. Benders decomposition and a genetic algorithm are combined to develop a framework to compute feasible solutions. We decompose the problem into a master problem and a subproblem. A genetic algorithm along with a heuristic are used to obtain feasible solutions to the master problem, whereas the subproblem is solved to optimality using a linear programming solver. Over successive iterations the master problem is refined by adding cutting planes that are implied by the subproblem. We compare the performance of the approach against a standard Benders decomposition approach as well as against a stand-alone solver (Cplex) on MIPLIB test problems.

Efficient d-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations

Fast and efficient solution techniques are developed for high-dimensional parabolic partial differential equations (PDEs). In this paper we present a robust solver based on the Krylov subspace method Bi-CGSTAB combined with a powerful, and efficient, multigrid preconditioner. Instead of developing the perfect multigrid method, as a stand-alone solver for a single problem discretized on a certain grid, we aim for a method that converges well for a wide class of discrete problems arising from discretization on various anisotropic grids. This is exactly what we encounter during a sparse grid computation of a high-dimensional problem. Different multigrid components are discussed and presented with operator construction formulae. An option-pricing application is focused and presented with results computed with this method.

Algebraic Multigrid Preconditioner for the Cardiac Bidomain Model

The bidomain equations are considered to be one of the most complete descriptions of the electrical activity in cardiac tissue, but large scale simulations, as resulting from discretization of an entire heart, remain a computational challenge due to the elliptic portion of the problem, the part associated with solving the extracellular potential. In such cases, the use of iterative solvers and parallel computing environments are mandatory to make parameter studies feasible. The preconditioned conjugate gradient (PCG) method is a standard choice for this problem. Although robust, its efficiency greatly depends on the choice of preconditioner. On structured grids, it has been demonstrated that a geometric multigrid preconditioner performs significantly better than an incomplete LU (ILU) preconditioner. However, unstructured grids are often preferred to better represent organ boundaries and allow for coarser discretization in the bath far from cardiac surfaces. Under these circumstances, algebraic multigrid (AMG) methods are advantageous since they compute coarser levels directly from the system matrix itself, thus avoiding the complexity of explicitly generating coarser, geometric grids. In this paper, the performance of an AMG preconditioner (BoomerAMG) is compared with that of the standard ILU preconditioner and a direct solver. BoomerAMG is used in two different ways, as a preconditioner and as a standalone solver. Two 3-D simulation examples modeling the induction of arrhythmias in rabbit ventricles were used to measure performance in both sequential and parallel simulations. It is shown that the AMG preconditioner is very well suited for the solution of the bidomain equation, being clearly superior to ILU preconditioning in all regards, with speedups by factors in the range 5.9-7.7.

Improved Filters and Randomized Drivers for Multi-start Global Optimization

We describe and test 2 types of improvements to the multistart heuristic framework for smooth global optimization presented in [8]. This starts a gradient based local NLP solver from a filtered subset of points generated by a starting point generator or driver, and has been available as both a standalone and GAMS solver since 2003. Changes to the distance and merit filters make them theoretically sounder and more adaptive to the problem instance. Stochastic drivers are proposed whose performance is comparable to the OptQuest scatter search implementation, if reasonably tight bounds on all variables are imposed. Tests on two large and diverse sets of test problems (56 problems with over 100 variables, and 18 with over 1000) show that best known solutions to almost all problems can be obtained with a median of about 30 solver calls. The best solution is located in the first ten calls in about 93% of the problems, making the procedure attractive within a branch and bound framework. Further, allowing more time and iterations usually yields improved solutions to problems not solved with default limits.

A Moment-Parity Multigrid Preconditioner for the First-Order System Least-Squares Formulation of the Boltzmann Transport Equation

This paper describes a preconditioned conjugate gradient scheme for the Pn-h finite element discretization of a first-order system least-squares (FOSLS) formulation of the Boltzmann transport equation. The preconditioner is based on the norm equivalence between the FOSLS functional and a V norm. Its realization is an inexact inversion of the system of partial differential equations corresponding to this V norm. This preconditioner is essentially a moment-parity multigrid solver. It involves a 2 × 2 block diagonal system with each block describing only the interior like-parity coupling (even-even or odd-odd) of the spherical harmonic coefficients or moments. The interior cross-parity coupling is handled in the outer conjugate gradient iteration. Since the like-parity coupling consists of only second- and zeroth-order differential terms, whereas the full Pn system consists also of cross-parity, first-order coupling terms, the construction of a robust multigrid algorithm for each diagonal block is easier than the construction for the full Pn system. Numerical results indicate that this preconditioned conjugate gradient algorithm is more robust than a stand-alone multigrid solver for the full Pn system.

On fast multigrid iteration techniques for the solution of normal equations in satellite gravity recovery

Dedicated satellite-to-satellite tracking (SST) or gradiometry missions like GRACE and GOCE will provide gravity field information with unprecedented resolution and precision. It has been recognized that better gravity field models and estimates of the geoid are useful for a wide range of research and application, including ocean circulation and climate change studies, physics of the Earth's interior and height datum connection and unification. The computation of these models will require the solution of large and non-sparse normal equation systems, especially if “brute force” approaches are applied. Evidently, there is a need for fast solvers. The multigrid approach is not only an extremely fast iterative solution technique, it yields, en passant, a well-defined sequence of coarser approximations as a byproduct to the final gravity field solution. We investigate the implementation of multigrid methods to satellite data analysis using space-domain representations of the anomalous gravity field. Multigrid algorithms are considered as stand-alone solvers as well as for the construction of preconditioners in the conjugate gradient technique. Our numerical results, concerning two regional gravity inversions from simulated GRACE and GOCE data, show that multigrid solvers run much faster than conjugate gradient solvers with conventional preconditioners.

Implementation of multigrid solvers for satellite gravity anomaly recovery

Dedicated SST – or gradiometry missions like GRACE and GOCE will provide gravity field information of unprecedented resolution and precision. It has been recognized that better gravity field models and estimates of the geoid are useful for a wide range of research and application, including ocean circulation and climate change studies, physics of the earth's interior and height datum connection and unification. The computation of these models will require the solution of large and non-sparse normal equation systems, especially if “brute force” approaches are applied. Evidently there is a need for fast solvers. The multigrid approach is not only an extremely fast iterative solution technique, it yields en passant a well-defined sequence of coarser approximations as a byproduct to the final gravity field solution. We investigate the implementation of multigrid methods to satellite data analysis using space-domain representations of the anomalous gravity field. Theoretical and numerical aspects are covered. Multigrid algorithms are applied as stand-alone solvers as well as for the construction of preconditioners in the conjugate gradient technique. Our numerical results, concerning a regional gravity inversion from simulated GRACE data, show that multigrid solvers run much faster than conjugate gradient solvers with conventional preconditioners.