Mathematical Sciences: Iterative Methods for Large-Scale NonLinear and Linear Systems

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The investigator develops more effective and efficient Newton iterative (truncated Newton) methods, i.e., implementations of Newton's method that use iterative linear algebra methods. A concomitant objective is to develop improved Krylov subspace methods and associated techniques for linear problems; this work supports the primary objective and is important in its own right. The project centers around recently developed globally convergent inexact Newton algorithms, improved implementations of GMRES, and new applications of Lanczos-based methods and "residual smoothing" techniques. A specific goal is to construct high-quality codes for Newton iterative methods and for GMRES that are suitable for public distribution. Newton iterative methods and Krylov subspace methods have many important scientific and industrial applications; examples include aircraft dynamics modeling, semiconductor modeling, climate modeling, oil reservoir simulation, and contaminant transport in aquifers. The large-scale problems to which they are usually applied often require high-performance vector and parallel computers and, indeed, have provided significant incentive for their development. This research contributes to improved computational modeling techniques for scientific and industrial applications and more effective utilization of high-performance computers.