An optimal linear solver for the Jacobian system of the extreme type-II Ginzburg--Landau problem

This paper considers the extreme type-II Ginzburg--Landau equations, a nonlinear PDE model for describing the states of a wide range of superconductors. Based on properties of the Jacobian operator and an AMG strategy, a preconditioned Newton--Krylov method is constructed. After a finite-volume-type discretization, numerical experiments are done for representative two- and three-dimensional domains. Strong numerical evidence is provided that the number of Krylov iterations is independent of the dimension $n$ of the solution space, yielding an overall solver complexity of O(n)

Preconditioning for Allen-Cahn variational inequalities with non-local constraints

The solution of Allen-Cahn variational inequalities with mass constraints is of interest in many applications. This problem can be solved both in its scalar and vector-valued form as a PDE-constrained optimization problem by means of a primal-dual active set method. At the heart of this method lies the solution of linear systems in saddle point form. In this paper we propose the use of Krylov-subspace solvers and suitable preconditioners for the saddle point systems. Numerical results illustrate the competitiveness of this approach

Analyzing the effect of local rounding error propagation on the maximal attainable accuracy of the pipelined Conjugate Gradient method

Pipelined Krylov subspace methods typically offer improved strong scaling on parallel HPC hardware compared to standard Krylov subspace methods for large and sparse linear systems. In pipelined methods the traditional synchronization bottleneck is mitigated by overlapping time-consuming global communications with useful computations. However, to achieve this communication hiding strategy, pipelined methods introduce additional recurrence relations for a number of auxiliary variables that are required to update the approximate solution. This paper aims at studying the influence of local rounding errors that are introduced by the additional recurrences in the pipelined Conjugate Gradient method. Specifically, we analyze the impact of local round-off effects on the attainable accuracy of the pipelined CG algorithm and compare to the traditional CG method. Furthermore, we estimate the gap between the true residual and the recursively computed residual used in the algorithm. Based on this estimate we suggest an automated residual replacement strategy to reduce the loss of attainable accuracy on the final iterative solution. The resulting pipelined CG method with residual replacement improves the maximal attainable accuracy of pipelined CG, while maintaining the efficient parallel performance of the pipelined method. This conclusion is substantiated by numerical results for a variety of benchmark problems.Comment: 26 pages, 6 figures, 2 tables, 4 algorithm

MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems

CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length (pseudoinverse) solution. This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems. MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned systems (singular or not), MINRES-QLP can give more accurate solutions than MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm, and the condition number.Comment: 26 pages, 6 figure

Preconditioned Recycling Krylov subspace methods for self-adjoint problems

The authors propose a recycling Krylov subspace method for the solution of a sequence of self-adjoint linear systems. Such problems appear, for example, in the Newton process for solving nonlinear equations. Ritz vectors are automatically extracted from one MINRES run and then used for self-adjoint deflation in the next. The method is designed to work with arbitrary inner products and arbitrary self-adjoint positive-definite preconditioners whose inverse can be computed with high accuracy. Numerical experiments with nonlinear Schr\"odinger equations indicate a substantial decrease in computation time when recycling is used

Analysis of rounding error accumulation in Conjugate Gradients to improve the maximal attainable accuracy of pipelined CG

Pipelined Krylov solvers typically offer better scalability in the strong scaling limit compared to standard Krylov methods. The synchronization bottleneck is mitigated by overlapping time-consuming global communications with useful computations in the algorithm. However, to achieve this communication hiding strategy, pipelined methods feature multiple recurrence relations on additional auxiliary variables to update the guess for the solution. This paper aims at studying the influence of rounding errors on the convergence of the pipelined Conjugate Gradient method. It is analyzed why rounding effects have a significantly larger impact on the maximal attainable accuracy of the pipelined CG algorithm compared to the traditional CG method. Furthermore, an algebraic model for the accumulation of rounding errors throughout the (pipelined) CG algorithm is derived. Based on this rounding error model, we then propose an automated residual replacement strategy to reduce the effect of rounding errors on the final iterative solution. The resulting pipelined CG method with automated residual replacement improves the maximal attainable accuracy of pipelined CG to a precision comparable to that of standard CG, while maintaining the efficient parallel performance of the pipelined method

Differences in the Effects of Rounding Errors in Krylov Solvers for Symmetric Indefinite Linear Systems

The 3-term Lanczos process for a symmetric matrix leads to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving a reduced system in one way or another. This leads to well-known methods: MINRES, GMRES, and SYMMLQ. We will discuss in what way and to what extent these approaches differ in their sensitivity to rounding errors. In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the different methods, and we will not consider the errors in the Lanczos process itself. We will show that the method of solution may lead, under certain circumstances, to large additional errors, that are not corrected by continuing the iteration process. Our findings are supported and illustrated by numerical examples

Three-dimensional finite element analysis of earth pressure balance tunnelling

Ph.DDOCTOR OF PHILOSOPH