Ubiquitous Doubling Algorithms for Nonlinear Matrix Equations and Applications

University Of Texas At Arlington, Arlington TX

Investigators

Abstract

The principal investigator's (PI's) research is aimed at the solution of practical problems of optimal control theory from areas as diverse as vibration analysis for high speed trains and quantum transport in nano research. Underlying these applications are formulations that require the efficient and accurate solution of important nonlinear matrix equations which critically influence the overall performance and fidelity of the entire simulations. Past experience has shown that practically relevant simulations requiring solutions of nonlinear matrix equations are notoriously challenging to the point that computed solutions may be completely erroneous. This project aims at changing the status quo by developing a complete theory, devising innovative algorithms to better reflect problem structures, and designing more robust implementations. In addition to advancing research in these nonlinear matrix equations, the PI will recruit and train graduate students in computational mathematics and interdisciplinary studies. This project will advance the understanding and solution techniques for high impact nonlinear matrix equations in the context of mathematical theory, computational methods, and software. For decades, computational scientists and engineers have been struggling to compute trustworthy numerical solutions to some of these nonlinear matrix equations with limited success. For example, the entries of the solution matrices to M-matrix algebraic Riccati equations from Markov-modulated fluid flow theory represent probabilities of events, and tiny entries indicate events that are rare but still important, and thus need to be computed accurately. The eigenvalues from vibration analysis of high speed trains vary widely in magnitude beyond the ranges that the IEEE double precision can handle if not dealt with correctly. The doubling algorithm may not converge at all on the unperturbed nonlinear matrix equations for quantum transport in nano research. These difficulties are often not caused by the large sizes of the problems but rather are more due to their inherent mathematical properties. With the successful completion of this project, a complete and coherent unifying framework of doubling algorithms, along with the relevant theory, will be developed. A much deeper understanding of doubling algorithms will be gained, and it is possible that important applications will be identified where doubling algorithms can be utilized and are faster than the current state-of-the-art methods. More significantly, the theory developed will assure that the new algorithms will produce more trustworthy accurate numerical results than is currently possible.