Victoria E. Howle

A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations

In recent years, considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible Navier-Stokes equations based on preconditioned Krylov methods. These include physics-based methods, such as SIMPLE, and purely algebraic preconditioners based on the approximation of the Schur complement. All these techniques can be represented as approximate block factorization (ABF) type preconditioners. The goal is to decompose the application of the preconditioner into simplified sub-systems in which scalable multi-level type solvers can be applied. In this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the Navier-Stokes system first presented in A. Quarteroni, F. Saleri, A. Veneziani, Factorization methods for the numerical approximation of Navier-Stokes equations, Computational Methods in Applied Mechanical Engineering 188 (2000) 505-526]. This taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain Schur complement operators. We then present a parallel computational study that examines the performance of these methods and compares them to an additive Schwarz domain decomposition (DD) algorithm. Results are presented for two and three-dimensional steady state problems for enclosed domains and inflow/outflow systems on both structured and unstructured meshes. The numerical experiments are performed using MPSalsa, a stabilized finite element code.

Least Squares Preconditioners for Stabilized Discretizations of the Navier-Stokes Equations

This paper introduces two stabilization schemes for the least squares commutator (LSC) preconditioner developed by Elman, Howle, Shadid, Shuttleworth, and Tuminaro [SIAM J. Sci. Comput., 27 (2006), pp. 1651-1668] for the incompressible Navier-Stokes equations. This preconditioning methodology is one of several choices that are effective for Navier-Stokes equations, and it has the advantage of being defined from strictly algebraic considerations. It has previously been limited in its applicability to div-stable discretizations of the Navier-Stokes equations. This paper shows how to extend the same methodology to stabilized low-order mixed finite element approximation methods.

A parallel block multi-level preconditioner for the 3D incompressible Navier--Stokes equations

The development of robust and efficient algorithms for both steady-state simulations and fully implicit time integration of the Navier-Stokes equations is an active research topic. To be effective, the linear subproblems generated by these methods require solution techniques that exhibit robust and rapid convergence. In particular, they should be insensitive to parameters in the problem such as mesh size, time step, and Reynolds number. In this context, we explore a parallel preconditioner based on a block factorization of the coefficient matrix generated in an Oseen nonlinear iteration for the primitive variable formulation of the system. The key to this preconditioner is the approximation of a certain Schur complement operator by a technique first proposed by Kay, Loghin, and Wathen [SIAM J. Sci. Comput., 2002] and Silvester, Elman, Kay, and Wathen [J. Comput. Appl. Math. 128 (2001) 261]. The resulting operator entails subsidiary computations (solutions of pressure Poisson and convection-diffusion subproblems) that are similar to those required for decoupled solution methods; however, in this case these solutions are applied as preconditioners to the coupled Oseen system. One important aspect of this approach is that the convection-diffusion and Poisson subproblems are significantly easier to solve than the entire coupled system, and a solver can be built using tools developed for the subproblems. In this paper, we apply smoothed aggregation algebraic multigrid to both subproblems. Previous work has focused on demonstrating the optimality of these preconditioners with respect to mesh size on serial, two-dimensional, steady-state computations employing geometric multi-grid methods; we focus on extending these methods to large-scale, parallel, three-dimensional, transient and steady-state simulations employing algebraic multigrid (AMG) methods. Our results display nearly optimal convergence rates for steady-state solutions as well as for transient solutions over a wide range of CFL numbers on the two-dimensional and three-dimensional lid-driven cavity problem.

An Iterative Method for Solving Complex-Symmetric Systems Arising in Electrical Power Modeling

We propose an iterative method for solving a complex-symmetric linear system arising in electric power networks. Our method extends Gremban, Miller, and Zaghas [in Proceedings of the International Parallel Processing Symposium, IEEE Computer Society, Los Alamitos, CA, 1995] support-tree preconditioner to handle complex weights and vastly different admittances. Our underlying iteration is a modification to transpose-free QMR to enhance accuracy. Computational results are described.

Block Preconditioners for Coupled Physics Problems

Finite element discretizations of multiphysics problems frequently give rise to block-structured linear algebra problems that require effective preconditioners. We build two classes of preconditioners in the spirit of well-known block factorizations [M. F. Murphy, G. H. Golub, and A. J. Wathen, SIAM J. Sci. Comput., 21 (2000), pp. 1969--1972; I. C. F. Ipsen, SIAM J. Sci. Comput., 23 (2001), pp. 1050--1051] and apply these to the diffusive portion of the bidomain equations and the Benard convection problem. An abstract generalized eigenvalue problem allows us to give application-specific bounds for the real parts of eigenvalues for these two problems. This analysis is accompanied by numerical calculations with several interesting features. One of our preconditioners for the bidomain equations converges in five iterations for a range of problem sizes. For Benard convection, we observe mesh-independent convergence with reasonable robustness with respect to physical parameters, and offer some preliminary para...

Playa: High-performance programmable linear algebra

This paper introduces Playa, a high-level user interface layer for composing algorithms for complex multiphysics problems out of objects from other Trilinos packages. Among other features, Playa provides very high-performance overloaded operators implemented through an expression template mechanism. In this paper, we give an overview of the central Playa objects from a users perspective, show application to a sequence of increasingly complex solver algorithms, provide timing results for Playas overloaded operators and other functions, and briefly survey some of the implementation issues involved.

Some Parallel Extensions to Optimization Methods in OPT

OPT++ provides an array of optimization tools for solving scientific and engineering design problems. While these tools are useful, all of the code is serial. With increasingly easy access to multiprocessor machines and clusters of workstations, this results in unnecessarily long times to solution. In order to correct this problem, we have implemented a number of parallel techniques in OPT++. In particular, we have incorporated a speculative gradient algorithm that drastically reduces the time to solution for standard trust-region and line search algorithms. In addition, we have implemented a new version of the Trust-Region Parallel Direct Search (TRPDS) algorithm of Hough and Meza that yields a significant reduction in solution time for problems with expensive function evaluations.

Augmented Lagrangian-based preconditioners for steady buoyancy driven flow

Abstract In this paper, we apply the augmented Lagrangian (AL) approach to steady buoyancy driven flow problems. Two AL preconditioners are developed based on the variable’s order, specifically whether the leading variable is the velocity or the temperature variable. Correspondingly, two non-augmented Lagrangian (NAL) preconditioners are also considered for comparison. An eigenvalue analysis for these two pairs of preconditioners is conducted to predict the rate of convergence for the GMRES solver. The numerical results show that the AL preconditioner pair is insensitive with respect to the mesh size, Rayleigh number, and Prandtl number in terms of GMRES iterations, resulting in a significantly more robust preconditioner pair compared to the NAL pair. Accordingly, the AL pair performs much better than the NAL pair in terms of computational time. For the AL pair, the preconditioner with velocity as the leading variable gives slightly better efficiency than the one with temperature as the leading variable.

Experience with Approximations in the Trust-Region Parallel Direct Search Algorithm

Recent years have seen growth in the number of algorithms designed to solve challenging simulation-based nonlinear optimization problems. One such algorithm is the Trust-Region Parallel Direct Search (TRPDS) method developed by Hough and Meza. In this paper, we take advantage of the theoretical properties of TRPDS to make use of approximation models in order to reduce the computational cost of simulation-based optimization. We describe the extension, which we call m TRPDS, and present the results of a case study for two earth penetrator design problems. In the case study, we conduct computational experiments with an array of approximations within the m TRPDS algorithm and compare the numerical results to the original TRPDS algorithm and a trust-region method implemented using the speculative gradient approach described by Byrd, Schnabel, and Shultz. The results suggest new ways to improve the algorithm.