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.iii 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.iv AcknowledgmentsThere are many people deserving of thanks for their assistance in making this thesis a reality, but none more so than my advisor, Eric de Sturler.His sage advice, thoughtful insights and all-around good company played no small role in making this work happen. I am deeply grateful to him for everything he has done! I would also like to thank Michael Heath for his role as the Numerical Analysis group "patriarch." As the unofficial advisor to all of the graduate students, Mike's wisdom (and critique, when warranted) has helped me to hone my presentation skills and helped me to succinctly explain my work to my professional colleagues.The other members of my committee, Robert Skeel and Armand Beaudoin, are also deserving of special thanks, Bob for coming all the way b...

MueLu is a library within the Trilinos software project [An overview of Trilinos, Technical Report SAND2003-2927, Sandia National Laboratories, 2003] and provides a framework for parallel multigrid preconditioning methods for large sparse linear systems. While providing efficient implementations of modern multigrid methods based on smoothed aggregation and energy minimization concepts, MueLu is designed to be customized and extended. This article gives an overview of design considerations for the MueLu package: user interfaces, internal design, data management, usage of modern software constructs, leveraging Trilinos capabilities, linear algebra operations and advanced application. J. Gaidamour et al. / Design considerations for a flexible multigrid preconditioning library // Linear problem RCP< Tpetra::CrsMatrix > A = ...; // Scalar matrix // MueLu operator. Default view is 'point' (1x1 block) RCP< MueLu::Operator > Op = rcp (new CrsOperator(A)); // Create a 2x2 block representation of the matrix Op.CreateConstBlkView("2x2", 2, 2); // The 2x2 can be used everywhere by default by using: // Op.SetDefaultView('2x2'); // Alternatively, it can be done on a case by case basis: // Multigrid hierarchy Hierarchy H(Op); // The 2x2 block view is used during the aggregation andSaPFactory aggregationFact = rcp(new UCAggregationFactory()); tentativePFact = rcp(new TentativePFactory(aggregationFact)); prolongationFact = rcp(new SaPFactory(tentativePFact)); prolongationFact->SetDiagonalView("2x2"); prolongationFact->SetView("2x2");

Four adaptations of the smoothed aggregation algebraic multigrid (SA-AMG) method are proposed with an eye towards improving the convergence and robustness of the solver in situations when the discretization matrix contains many weak connections. These weak connections can cause higher than expected levels of fill-in within the coarse discretization matrices and can also give rise to sub-optimal smoothing within the prolongator smoothing phase. These smoothing drawbacks are due to the relatively small size of some diagonal entries within the filtered matrix that one obtains after dropping the weak connections. The new algorithms consider modifications to the Jacobi-like step that defines the prolongator smoother, modifications to the filtered matrix, and also direct modifications to the resulting grid transfer operators. Numerical results are given illustrating the potential benefits of the proposed adaptations.