In science and engineering, many applications require the solution of a sequence of linear systems. There are many ways to solve linear systems and we always look for methods that are faster and/or require less storage. In this dissertation, we focus on solving these systems with Krylov subspace methods and how to obtain effective preconditioners inexpensively. We first present an application for electronic structure calculation. A sequence of slowly changing linear systems is produced in the simulation. The linear systems change by rank-one updates. Properties of the system matrix are analyzed. We use Krylov subspace methods to solve these linear systems. Krylov subspace methods need a preconditioner to be efficient and robust. This causes the problem of computing a sequence of preconditioners corresponding to the sequence of linear systems. We use recycling preconditioners, which is to update and reuse existing preconditioner. We investigate and analyze several preconditioners, such as ILU(0), ILUTP, domain decomposition preconditioners, and inexact matrix-vector products with inner-outer iterations. Recycling preconditioners produces cumulative updates to the preconditioner. To reduce the cost of applying the preconditioners, we propose approaches to truncate the cumulative preconditioner updates, which is a low-rank matrix. Two approaches are developed. The first one is to truncate the low-rank matrix using the best approximation given by the singular value decomposition (SVD). This is effective if many singular values are close to zero. If not, based on the ideas underlying GCROT and recycling, we use information from an Arnoldi recurrence to determine which directions to keep. We investigate and analyze their properties. We also prove that both truncation approaches work well under suitable conditions. We apply our truncation approaches on two applications. One is the Quantum Monte Carlo (QMC) method and the other is a nonlinear second order partial differential equation (PDE). For the QMC method, we test both truncation approaches and analyze their results. For the PDE problem, we discretize the equations with finite difference method, solve the nonlinear problem by Newton's method with a line-search, and utilize Krylov subspace methods to solve the linear system in every nonlinear iteration. The preconditioner is updated by Broyden-type rank-one updates, and we truncate the preconditioner updates by using the SVD finally. We demonstrate that the truncation is effective. In the last chapter, we develop a matrix reordering algorithm that improves the diagonal dominance of Slater matrices in the QMC method. If we reorder the entire Slater matrix, we call it global reordering and the cost is O(N^3), which is expensive. As the change is geometrically localized and impacts only one row and a modest number of columns, we propose a local reordering of a submatrix of the Slater matrix. The submatrix has small dimension, which is independent of the size of Slater matrix, and hence the local reordering has constant cost (with respect to the size of Slater matrix). / Ph. D.

Science and engineering problems frequently require solving a sequence of single linear systems or a sequence of dual linear systems. We develop algorithms that recycle Krylov subspaces and preconditioners from one system (or pair of systems) in the sequence to the next, leading to efficient solutions. Besides the benefit of only having to store few Lanczos vectors, using BiConjugate Gradients (BiCG) to solve dual linear systems may have application-specific advantages. For example, using BiCG to solve the dual linear systems arising in interpolatory model reduction provides a backward error formulation in the model reduction framework. Using BiCG to evaluate bilinear forms -- for example, in the variational Monte Carlo (VMC) algorithm for electronic structure calculations -- leads to a quadratic error bound. Since one of our focus areas is sequences of dual linear systems, we introduce recycling BiCG, a BiCG method that recycles two Krylov subspaces from one pair of dual linear systems to the next pair. The derivation of recycling BiCG also builds the foundation for developing recycling variants of other bi-Lanczos based methods like CGS, BiCGSTAB, BiCGSTAB2, BiCGSTAB(l), QMR, and TFQMR. We develop a generalized bi-Lanczos algorithm, where the two matrices of the bi-Lanczos procedure are not each other's conjugate transpose but satisfy this relation over the generated Krylov subspaces. This is sufficient for a short term recurrence. Next, we derive an augmented bi-Lanczos algorithm with recycling and show that this algorithm is a special case of generalized bi-Lanczos. The Petrov-Galerkin approximation that includes recycling in the iteration leads to modified two-term recurrences for the solution and residual updates. We generalize and extend the framework of our recycling BiCG to CGS, BiCGSTAB and BiCGSTAB2. We perform extensive numerical experiments and analyze the generated recycle space. We test all of our recycling algorithms on a discretized partial differential equation (PDE) of convection-diffusion type. This PDE problem provides well-known test cases that are easy to analyze further. We use recycling BiCG in the Iterative Rational Krylov Algorithm (IRKA) for interpolatory model reduction and in the VMC algorithm. For a model reduction problem, we show up to 70% savings in iterations, and we also demonstrate that solving the problem without recycling leads to (about) a 50% increase in runtime. Experiments with recycling BiCG for VMC gives promising results. We also present an algorithm that recycles preconditioners, leading to a dramatic reduction in the cost of VMC for large(r) systems. The main cost of the VMC method is in constructing a sequence of Slater matrices and computing the ratios of determinants for successive Slater matrices. Recent work has improved the scaling of constructing Slater matrices for insulators, so that the cost of constructing Slater matrices in these systems is now linear in the number of particles. However, the cost of computing determinant ratios remains cubic in the number of particles. With the long term aim of simulating much larger systems, we improve the scaling of computing determinant ratios in the VMC method for simulating insulators by using preconditioned iterative solvers. The main contribution here is the development of a method to efficiently compute for the Slater matrices a sequence of preconditioners that make the iterative solver converge rapidly. This involves cheap preconditioner updates, an effective reordering strategy, and a cheap method to monitor instability of ILUTP preconditioners. Using the resulting preconditioned iterative solvers to compute determinant ratios of consecutive Slater matrices reduces the scaling of the VMC algorithm from O(n^3) per sweep to roughly O(n^2), where n is the number of particles, and a sweep is a sequence of n steps, each attempting to move a distinct particle. We demonstrate experimentally that we can achieve the improved scaling without increasing statistical errors. / Ph. D.