Preconditioners for Generalized Saddle-Point Problems

Generalized saddle point problems arise in a number of applications, ranging from optimization and metal deformation to fluid flow and PDE-governed optimal control. We focus our discussion on the most general case, making no assumption of symmetry or definiteness in the matrix or its blocks. As these problems are often large and sparse, preconditioners play a critical role in speeding the convergence of Krylov methods for these problems. We first examine two types of preconditioners for these problems, one block-diagonal and one indefinite, and present analyses of the eigenvalue distributions of the preconditioned matrices. We also investigate the use of approximations for the Schur complement matrix in these preconditioners and develop eigenvalue analysis accordingly. Second, we examine new developments in probing methods, inspired by graph coloring methods for sparse Jacobians, for building approximations to Schur complement matrices. We then present an analysis of these techniques and their accuracy. In addition, we provide a mathematical justification for their use in approximating Schur complements and suggest the use of approximate factorization techniques to decrease the computational cost of applying the inverse of the probed matrix. Finally, we consider the effect of our preconditioners on four applications. Two of these applications come from the realm of fluid flow, one using a finite element discretization and the other using a spectral discretization. The third application involves the stress relaxation of aluminum strips at low stress levels. The final application involves mesh parameterization and flattening. For these applications, we present results illustrating the eigenvalue bounds on our preconditioners and demonstrating the theoretical justification of these methods. We also present convergence and timing results, showing the effectiveness of our methods in practice. Specifically the use of probing methods for approximating the Schur compliment matrices in our preconditioners is empirically justified. We also investigate the $h$-dependence of our preconditioners one model fluid problem, and demonstrate empirically that our methods do not suffer from a deterioration in convergence as the problem size increases

Spatially Adaptive Stochastic Methods for Fluid-Structure Interactions Subject to Thermal Fluctuations in Domains with Complex Geometries

We develop stochastic mixed finite element methods for spatially adaptive simulations of fluid-structure interactions when subject to thermal fluctuations. To account for thermal fluctuations, we introduce a discrete fluctuation-dissipation balance condition to develop compatible stochastic driving fields for our discretization. We perform analysis that shows our condition is sufficient to ensure results consistent with statistical mechanics. We show the Gibbs-Boltzmann distribution is invariant under the stochastic dynamics of the semi-discretization. To generate efficiently the required stochastic driving fields, we develop a Gibbs sampler based on iterative methods and multigrid to generate fields with $O(N)$ computational complexity. Our stochastic methods provide an alternative to uniform discretizations on periodic domains that rely on Fast Fourier Transforms. To demonstrate in practice our stochastic computational methods, we investigate within channel geometries having internal obstacles and no-slip walls how the mobility/diffusivity of particles depends on location. Our methods extend the applicability of fluctuating hydrodynamic approaches by allowing for spatially adaptive resolution of the mechanics and for domains that have complex geometries relevant in many applications

Preconditioners for Generalized Saddle-Point Problems

We examine block-diagonal preconditioners and efficient variants of indefinite preconditioners for block two-by-two generalized saddle-point problems. We consider the general, nonsymmetric, nonsingular case. In particular, the (1,2) block need not equal the transposed (2,1) block. Our preconditioners arise from computationally efficient splittings of the (1,1) block. We provide analyses for the eigenvalue distributions and other properties of the preconditioned matrices. We extend the results of [de Sturler and Liesen 2003] to matrices with non-zero (2,2) block and to allow for the use of inexact Schur complements. To illustrate our eigenvalue bounds, we apply our analysis to a model Navier-Stokes problem, computing the bounds, comparing them to actual eigenvalue perturbations and examining the convergence behavior

Probing Methods for Generalized Saddle-Point Problems

Several Schur complement-based preconditioners have been proposed for solving (generalized) saddle-point problems. We consider probing-based methods for approximating those Schur complements in the preconditioners of the type proposed by [Murphy, Golub and Wathen '00], [de Sturler and Liesen '03] and [Siefert and de Sturler '04]. This approach can be applied in similar preconditioners as well. We discuss the implementation of probing-based approximations to Schur complements. We consider the application of those approximations in preconditioners for Navier-Stokes problems and metal deformation problems. Finally, we present eigenvalue clustering for the preconditioned matrices, and convergence and timing results. These demonstrate the effectiveness of the proposed preconditioners with probing-based approximate Schur complements

Preconditioners for Generalized Saddle-Point Problems

We examine block-diagonal preconditioners and e#cient variants of indefinite preconditioners for block two-by-two generalized saddle-point problems. We consider the general, nonsymmetric, nonsingular case

Probing methods for generalized saddle-point problems

Several Schur complement-based preconditioners have been proposed for solving (generalized) saddlepoint problems. We consider probing-based methods for approximating those Schur complements in the preconditioners of the type proposed by [Murphy, Golub and Wathen ’00], [de Sturler and Liesen ’03] and [Siefert and de Sturler ’04]. This approach can be applied in similar preconditioners as well. We discuss the implementation of probing-based approximations to Schur complements. We consider the application of those approximations in preconditioners for Navier-Stokes problems and metal deformation problems. Finally, we present eigenvalue clustering for the preconditioned matrices, and convergence and timing results. These demonstrate the effectiveness of the proposed preconditioners with probing-based approximate Schur complements.

Preconditioners for generalized saddle-point problems

Abstract. We propose and examine block-diagonal preconditioners and variants of indefinite preconditioners for block two-by-two generalized saddle-point problems. That is, we consider the nonsymmetric, nonsingular case where the (2,2) block is small in norm, and we are particularly concerned with the case where the (1,2) block is different from the transposed (2,1) block. We provide theoretical and experimental analyses of the convergence and eigenvalue distributions of the preconditioned matrices. We also extend the results of [de Sturler and Liesen, SIAM J. Sci. Comput., 26 (2005), pp. 1598–1619] to matrices with nonzero (2,2) block and to the use of approximate Schur complements. To demonstrate the effectiveness of these preconditioners we show convergence results, spectra, and eigenvalue bounds for two model Navier–Stokes problems