Globally convergent techniques in nonlinear Newton-Krylov

Some convergence theory is presented for nonlinear Krylov subspace methods. The basic idea of these methods is to use variants of Newton's iteration in conjunction with a Krylov subspace method for solving the Jacobian linear systems. These methods are variants of inexact Newton methods where the approximate Newton direction is taken from a subspace of small dimensions. The main focus is to analyze these methods when they are combined with global strategies such as linesearch techniques and model trust region algorithms. Most of the convergence results are formulated for projection onto general subspaces rather than just Krylov subspaces

Improving Efficiency of Rational Krylov Subspace Methods

This thesis studies two classes of numerical linear algebra problems, approximating the product of a function of a matrix with a vector, and solving the linear eigenvalue problem $Av=\lambda Bv$ for a small number of eigenvalues. These problems are solved by rational Krylov subspace methods (RKSM). We present several improvements in two directions: pole selection and applying inexact methods. In Chapter 3, a flexible extended Krylov subspace method ($\mathcal{F}$-EKSM) is considered for numerical approximation of the action of a matrix function $f(A)$ to a vector $b$, where the function $f$ is of Markov type. $\mathcal{F}$-EKSM has the same framework as the extended Krylov subspace method (EKSM), replacing the zero pole in EKSM with a properly chosen fixed nonzero poles. For symmetric positive definite matrices, the optimal fixed pole is derived for $\mathcal{F}$-EKSM to achieve the lowest possible upper bound on the asymptotic convergence factor, which is lower than that of EKSM. The analysis is based on properties of Faber polynomials of $A$ and $(I-A/s)^{-1}$. For large and sparse matrices that can be handled efficiently by LU factorizations, numerical experiments show that $\mathcal{F}$-EKSM and a variant of RKSM based on a small number of fixed poles outperform EKSM in both storage and runtime, and they usually have advantage over adaptive RKSM in runtime. Chapter 4 concerns the theory and development of inexact RKSM for approximating the action of a function of matrix $f(A)$ to a column vector $b$. At each step of RKSM, a shifted linear system of equations needs to be solved to enlarge the subspace. For large-scale problems, arising from discretizations of PDEs in 3D domains, such a linear system is usually solved by an iterative method approximately. The main question is how to relax the accuracy of these linear solves without negatively affecting the convergence for approximating $f(A)b$. Our insight into this issue is obtained by exploring the residual bounds on the rational Krylov subspace approximations to $f(A)b$, based on the decaying behavior of the entries in the first column of the matrix function of the block Rayleigh quotient of $A$ with respect to the rational Krylov subspaces. The decay bounds on these entries for both analytic functions and Markov functions can be efficiently and accurately evaluated by appropriate quadrature rules. A heuristic based on these bounds is proposed to relax the tolerances of the linear solves arising from each step of RKSM. As the algorithm progresses toward convergence, the linear solves can be performed with increasingly lower accuracy and computational cost. Numerical experiments for large nonsymmetric matrices show the effectiveness of the tolerance relaxation strategy for the inexact linear solves of RKSM. In Chapter 5, inexact RKSM are studied to solve large-scale nonsymmetric eigenvalue problems. Similar to the problem setting in Chapter 4, each iteration (outer step) of RKSM requires solution to a shifted linear system to enlarge the subspace, but these linear solves by direct methods are prohibitive due to the problem scale. Errors are introduced at each outer step if these linear systems are solved approximately by iterative methods (inner step), and these errors accumulate in the rational Krylov subspace. In this thesis, we derive an upper bound on the errors that can be introduced at each outer step to maintain the same convergence as exact RKSM for approximating an invariant subspace. Since this bound is inversely proportional to the current eigenresidual norm of the desired invariant subspace, the tolerance of iterative linear solves at each outer step can be relaxed with the outer iteration progress. A restarted variant of the inexact RKSM is also proposed. Numerical experiments show the effectiveness of relaxing the inner tolerance to save computational cost

Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems

Based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods, a generalized preconditioned MHSS (GPMHSS) method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we have analyzed and implemented inexact GPMHSS (IGPMHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments are reported to confirm the efficiency of the proposed methods

A flexible and adaptive Simpler GMRES with deflated restarting for shifted linear systems

In this paper, two efficient iterative algorithms based on the simpler GMRES method are proposed for solving shifted linear systems. To make full use of the shifted structure, the proposed algorithms utilizing the deflated restarting strategy and flexible preconditioning can significantly reduce the number of matrix-vector products and the elapsed CPU time. Numerical experiments are reported to illustrate the performance and effectiveness of the proposed algorithms.Comment: 17 pages. 9 Tables, 1 figure; Newly update: add some new numerical results and correct some typos and syntax error

Adaptive Aggregation Based Domain Decomposition Multigrid for the Lattice Wilson Dirac Operator

In lattice QCD computations a substantial amount of work is spent in solving discretized versions of the Dirac equation. Conventional Krylov solvers show critical slowing down for large system sizes and physically interesting parameter regions. We present a domain decomposition adaptive algebraic multigrid method used as a precondtioner to solve the "clover improved" Wilson discretization of the Dirac equation. This approach combines and improves two approaches, namely domain decomposition and adaptive algebraic multigrid, that have been used seperately in lattice QCD before. We show in extensive numerical test conducted with a parallel production code implementation that considerable speed-up over conventional Krylov subspace methods, domain decomposition methods and other hierarchical approaches for realistic system sizes can be achieved.Comment: Additional comparison to method of arXiv:1011.2775 and to mixed-precision odd-even preconditioned BiCGStab. Results of numerical experiments changed slightly due to more systematic use of odd-even preconditionin