A Tensor-Krylov Method for Solving Large-scale Systems of Nonlinear Equations

Brett W. Bader
Robert B. Schnabel

Department of Computer Science
University of Colorado at Boulder
Boulder, CO 80303-0430


Abstract

Traditional tensor methods for nonlinear equations have performed especially well on problems where the Jacobian matrix is rank deficient or ill-conditioned at the solution. This success originates from a special, restricted form of the quadratic term included in the local tensor model that provides information lacking in a (near) singular Jacobian. This talk will present an iterative tensor method for solving nonlinear equations based on Krylov subspace projection methods. The new iterative tensor method intends to capitalize on the success of direct tensor methods by developing a Krylov-based iterative method for solving the local tensor model for use within an inexact tensor framework, similar to an inexact Newton method. The proposed tensor-Krylov method requires no more function nor extra derivative evaluations, and hardly more storage or arithmetic per iteration, than a Newton-Krylov method. The tensor-Krylov method still retains faster convergence on certain ill-conditioned systems; thus, the tensor-Krylov method intends to target large-scale discretizations of PDE problems that exhibit near singularities. Preliminary results appear promising that this method will outperform Newton-GMRES in addition to competing with other tensor-Krylov methods. Its ability to capitalize on the superlinear convergence behavior of tensor methods and its ability to solve the tensor model to a specified accuracy with very little extra arithmetic beyond standard GMRES are the advantages over existing methods. We will present numerical results showing comparisons with Newton-GMRES.