Iterative Methods for Nonlinear Equations and Optimization

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Computation simulation is pervasive in society. Automobiles, aircraft, and mobile phones, for example, need high-quality simulators not only for their design, but also for their operation. In this project the PI will investigate nonlinear solvers, which are core components of many simulators. He will consider classes of problems and solvers for which theory is incomplete or missing. A theoretical understanding of these problems and solvers will lead to improved performance and reliability of the solvers and the simulators which use those solvers. The PI will apply his findings to problems in physics and chemistry basic to the design of materials. This project addresses some unresolved algorithmic questions in numerical analysis theory. These problems have important applications. The work on optimization and nonlinear solvers with embedded Monte Carlo methods in the objective function or residual is a very new line of research. The work is particularly timely in view of Monte Carlo methods robustness and resiliency in a massively parallel environment. Progress in this topic will lead to better understanding of how to couple existing solvers to Monte Carlo models. The research on Anderson acceleration will explore a class of Jacobian-free solvers which are very important when Jacobians or Jacobian vector products are unavailable. Anderson is already widely used in electronic structure computations which, in turn, are core components in computational chemistry software and the design of materials. While there is wide literature on the use and operation of Anderson acceleration, including cases where it fails to converge, there is very little theory. The object of this part of the project is to develop that theory.