Preconditioning and convergence in the right norm

The convergence of numerical approximations to the solutions of differential equations is a key aspect of Numerical Analysis and Scientific Computing. Iterative solution methods for the systems of linear(ised) equations which often result are also underpinned by analyses of convergence. In the function space setting, it is widely appreciated that there are appropriate ways in which to assess convergence and it is well-known that different norms are not equivalent. In the finite dimensional linear algebra setting, however, all norms are equivalent and little attention is often payed to the norms used. In this paper, we highlight this consideration in the context of preconditioning for minimum residual methods (MINRES and GMRES/GCR/ORTHOMIN) and argue that even in the linear algebra setting there is a ‘right’ norm in which to consider convergence: stopping an iteration which is rapidly converging in an irrelevant or highly scaled norm at some tolerance level may still give a poor answer

Composing Scalable Nonlinear Algebraic Solvers

Most efficient linear solvers use composable algorithmic components, with the most common model being the combination of a Krylov accelerator and one or more preconditioners. A similar set of concepts may be used for nonlinear algebraic systems, where nonlinear composition of different nonlinear solvers may significantly improve the time to solution. We describe the basic concepts of nonlinear composition and preconditioning and present a number of solvers applicable to nonlinear partial differential equations. We have developed a software framework in order to easily explore the possible combinations of solvers. We show that the performance gains from using composed solvers can be substantial compared with gains from standard Newton-Krylov methods.Comment: 29 pages, 14 figures, 13 table

A class of nonsymmetric preconditioners for saddle point problems

For iterative solution of saddle point problems, a nonsymmetric preconditioning is studied which, with respect to the upper-left block of the system matrix, can be seen as a variant of SSOR. An idealized situation where the SSOR is taken with respect to the skew-symmetric part plus the diagonal part of the upper-left block is analyzed in detail. Since action of the preconditioner involves solution of a Schur complement system, an inexact form of the preconditioner can be of interest. This results in an inner-outer iterative process. Numerical experiments with solution of linearized Navier-Stokes equations demonstrate efficiency of the new preconditioner, especially when the left-upper block is far from symmetric

On choice of preconditioner for minimum residual methods for nonsymmetric matrices

Existing convergence bounds for Krylov subspace methods such as GMRES for nonsymmetric linear systems give little mathematical guidance for the choice of preconditioner. Here, we establish a desirable mathematical property of a preconditioner which guarantees that convergence of a minimum residual method will essentially depend only on the eigenvalues of the preconditioned system, as is true in the symmetric case. Our theory covers only a subset of nonsymmetric coefficient matrices but computations indicate that it might be more generally applicable

Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices

We propose a Preconditioned Locally Harmonic Residual (PLHR) method for computing several interior eigenpairs of a generalized Hermitian eigenvalue problem, without traditional spectral transformations, matrix factorizations, or inversions. PLHR is based on a short-term recurrence, easily extended to a block form, computing eigenpairs simultaneously. PLHR can take advantage of Hermitian positive definite preconditioning, e.g., based on an approximate inverse of an absolute value of a shifted matrix, introduced in [SISC, 35 (2013), pp. A696-A718]. Our numerical experiments demonstrate that PLHR is efficient and robust for certain classes of large-scale interior eigenvalue problems, involving Laplacian and Hamiltonian operators, especially if memory requirements are tight

Domain Decomposition preconditioning for high-frequency Helmholtz problems with absorption

In this paper we give new results on domain decomposition preconditioners for GMRES when computing piecewise-linear finite-element approximations of the Helmholtz equation $-\Delta u - (k^2+ {\rm i} \varepsilon)u = f$, with absorption parameter $\varepsilon \in \mathbb{R}$. Multigrid approximations of this equation with $\varepsilon \not= 0$ are commonly used as preconditioners for the pure Helmholtz case ($\varepsilon = 0$). However a rigorous theory for such (so-called "shifted Laplace") preconditioners, either for the pure Helmholtz equation, or even the absorptive equation ($\varepsilon \not=0$), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left- or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a $k$- and $\varepsilon$-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if $|\varepsilon|\sim k^2$, then classical overlapping additive Schwarz will perform optimally for the absorptive problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case $\varepsilon = 0$. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about $\mathcal{O}(n^{4/3})$ for solving finite element systems of size $n=\mathcal{O}(k^3)$, where we have chosen the mesh diameter $h \sim k^{-3/2}$ to avoid the pollution effect. Experiments on problems with $h\sim k^{-1}$, i.e. a fixed number of grid points per wavelength, are also given