Efficiently preconditioned inexact Newton methods for large symmetric eigenvalue problems

2017

Summary In this paper, we present preconditioning techniques to accelerate the convergence of Krylov solvers at each step of an Inexact Newton's method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices arising in large‐scale scientific computations. We propose a two‐stage spectral preconditioning strategy: The first stage produces a very rough approximation of a number of the leftmost eigenvectors. The second stage uses these approximations as starting vectors and also to construct the tuned preconditioner from an initial inverse approximation of the coefficient matrix, as proposed by Martínez. In the framework of the Implicitly Restarted Lanczos method. The action of this spectral preconditioner results in clustering a number of the eigenvalues of the preconditioned matrices close to one. We also study the combination of this approach with a BFGS‐style updating of the proposed spectral preconditioner as described by Bergamaschi and Martínez. Extensive numerical testing on a set of representative large SPD matrices gives evidence of the acceleration provided by these spectral preconditioners.

“…The sequence of correction equations

J_{k}^{false(j false)} s_{k} = - r_{k}

is preconditioned by means of a sequence of low‐rank updates of a given approximate inverse of A which takes the following form, in the computation of the smallest eigenvalue:

\begin{matrix} right {\hat{P}}_{0} & left = P_{italicchol} & right \\ right P_{0} & left = (I - {bold-italicu}_{0} {bold-italicu}_{0}^{⊤}) {\hat{P}}_{0} (I - {bold-italicu}_{0} {bold-italicu}_{0}^{⊤}); \end{matrix}

\begin{matrix} right {\hat{P}}_{k + 1} & left = - \frac{{bold-italics}_{k} {bold-italics}_{k}^{⊤}}{{bold-italics}_{k}^{⊤} {bold-italicr}_{k}} + (I - \frac{{bold-italics}_{k} {bold-italicr}_{k}^{⊤}}{{bold-italics}_{k}^{⊤} {bold-italicr}_{k}}) {\hat{P}}_{k} (I - \frac{{bold-italicr}_{k} {bold-italics}_{k}^{⊤}}{{bold-italics}_{k}^{⊤} r}), & right \\ right & left k = 0, ⋯, \\ right P_{k + 1} & left = (I - {bold-italicu}_{k + 1} {bold-italicu}_{k + 1}^{⊤}) {\hat{P}}_{k + 1} (I - {bold-italicu}_{k + 1} {bold-italicu}_{k}^{⊤}) . \end{matrix}

The following theorem is proved, which bounds the norm of the preconditioned matrix in terms of t...…”

Section: Bfgs Low‐rank Update Of Given Preconditionersmentioning

confidence: 99%

\begin{matrix} right J_{k} {bold-italicu}_{k} & left = - {bold-italicr}_{k}, where & right \\ right J_{k} & left = (I - {bold-italicu}_{k} {bold-italicu}_{k}^{⊤}) (A - {normalθ}_{k} I) (I - {bold-italicu}_{k} {bold-italicu}_{k}^{⊤}) and \\ right {bold-italicr}_{k} & left = - (A {bold-italicu}_{k} - {normalθ}_{k} {bold-italicu}_{k}), {normalθ}_{k} = \frac{{bold-italicu}_{k}^{⊤} A {bold-italicu}_{k}}{{bold-italicu}_{k}^{⊤} {bold-italicu}_{k}} \end{matrix}

to be solved at each step of this projected Newton method.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Two‐stage spectral preconditioners for iterative eigensolvers

Bergamaschi

2017

Tuned preconditioners for the eigensolution of large SPD matrices arising in engineering problems

2016

In this paper, we study a class of tuned preconditioners that will be designed to accelerate both the DACG-Newton method and the implicitly restarted Lanczos method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices arising in large-scale scientific computations. These tuning strategies are based on low-rank modifications of a given initial preconditioner. We present some theoretical properties of the preconditioned matrix. We experimentally show how the aforementioned methods benefit from the acceleration provided by these tuned/deflated preconditioners. Comparisons are carried out with the Jacobi-Davidson method onto matrices arising from various large realistic problems arising from finite element discretization of PDEs modeling either groundwater flow in porous media or geomechanical processes in reservoirs. The numerical results show that the Newton-based methods (which includes also the Jacobi-Davidson method) are to be preferred to the -yet efficiently implemented -implicitly restarted Lanczos method whenever a small to moderate number of eigenpairs is required.

Compact quasi‐Newton preconditioners for symmetric positive definite linear systems

Bergamaschi

Marı́n

2020

Summary In this paper, preconditioners for the conjugate gradient method are studied to solve the Newton system with symmetric positive definite Jacobian. In particular, we define a sequence of preconditioners built by means of Symmetric Rank one (SR1) and Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) low‐rank updates. We develop conditions under which the SR1 update maintains the preconditioner symmetric positive definite. Spectral analysis of the SR1 preconditioned Jacobians shows an improved eigenvalue distribution as the Newton iteration proceeds. A compact matrix formulation of the preconditioner update is developed which reduces the cost of its application and is more suitable to parallel implementation. Some notes on the implementation of the corresponding Inexact Newton method are given and some numerical results on a number of model problems illustrate the efficiency of the proposed preconditioners.