Serge Gratton

Algorithm 842: A set of GMRES routines for real and complex arithmetics on high performance computers

In this article we describe our implementations of the GMRES algorithm for both real and complex, single and double precision arithmetics suitable for serial, shared memory and distributed memory computers. For the sake of portability, simplicity, flexibility and efficiency the GMRES solvers have been implemented in Fortran 77 using the reverse communication mechanism for the matrix-vector product, the preconditioning and the dot product computations. For distributed memory computation, several orthogonalization procedures have been implemented to reduce the cost of the dot product calculation, which is a well-known bottleneck of efficiency for the Krylov methods. Either implicit or explicit calculation of the residual at restart are possible depending on the actual cost of the matrix-vector product. Finally the implemented stopping criterion is based on a normwise backward error.

Recursive Trust-Region Methods for Multiscale Nonlinear Optimization

A class of trust-region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations. The algorithms in this class make use of the discretization level as a means of speeding up the computation of the step. This use is recursive, leading to true multilevel/multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partial differential equations. A simple algorithm of the class is then described and its numerical performance is shown to be numerically promising. This observation then motivates a proof of global convergence to first-order stationary points on the fine grid that is valid for all algorithms in the class.

On A Class of Limited Memory Preconditioners For Large Scale Linear Systems With Multiple Right-Hand Sides

This work studies a class of limited memory preconditioners (LMPs) for solving linear (positive-definite) systems of equations with multiple right-hand sides. We propose a class of (LMPs), whose construction requires a small number of linearly independent vectors. After exploring the theoretical properties of the preconditioners, we focus on three particular members: spectral-LMP, quasi-Newton-LMP, and Ritz-LMP. We show that the first two are well known, while the third is new. Numerical tests indicate that the Ritz-LMP is efficient on a real-life nonlinear optimization problem arising in a data assimilation system for oceanography.

A Partial Condition Number for Linear Least Squares Problems

We consider here the linear least squares problem

Flexible GMRES with Deflated Restarting

\min_{y \in \mathbb{R}^n}\|Ay-b\|_2

An active-set trust-region method for derivative-free nonlinear bound-constrained optimization

, where

Flexible Variants of Block Restarted GMRES Methods with Application to Geophysics

b \in \mathbb{R}^m

Additive and Multiplicative Two-Level Spectral Preconditioning for General Linear Systems

and

Constrained Regional Recovery of Continental Water Mass Time-variations from GRACE-based Geopotential Anomalies over South America

A \in \mathbb{R}^{m\times n}

An improved two-grid preconditioner for the solution of three-dimensional Helmholtz problems in heterogeneous media‡

is a matrix of full column rank