Hybrid Symbolic-Numeric Algorithms for Complex Nonlinear Systems

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I specialize in the creation and testing of computational algorithms for the approximate solution of continuous nonlinear mathematical problems. Such algorithms provide essentially the only way to get modern theoretical descriptions of physical, economic, or engineering phenomena to make detailed predictions and to allow the design of effective devices and strategies. I have an extensive record of significant algorithmic work in each of three major overlapping research areas: computational linear and polynomial algebra, computational dynamical systems, and computational special functions. Much of my work is abstract, because abstraction gives leverage, while computation gives power. But I maintain contact with applications, because applications frequently give greater challenges than introspection does. The applications of my work include simulation of nonlinear models of the electrical behaviour of the human heart, blood flow, pricing of financial options, heat transfer in fluids, the chemical kinetics of dark adaptation in the human eye, and many others. Nearly every engineer, scientist, and social scientist relies on such algorithms, implemented in software for them to use, in order to do their work. Many useful, indeed critical, algorithms have been developed in the last century, and enormous resources (money, time, energy and carbon cost) are devoted to executing these algorithms in simulations of physical or biological or social phenomena. But our algorithmic knowledge is incomplete: e.g. we do not even know the fastest/most economical way to multiply matrices together. Our knowledge of algorithms for nonlinear problems is even less complete. The research program described in this proposal addresses critical issues in this area. Specifically, this application proposes work on new algorithms for matrix polynomial eigenvalue problems, new algorithms for solving continuous dynamical systems, and new algorithms for computing certain special functions (the inverse Gamma function, and the irregular case of the Mathieu functions). I also propose a new fast "divide-and-conquer" algorithm for computing eigenvalues of a special class of matrices. Approximate eigenvalues are needed in studies of vibration in nearly all fields, and this special class of matrices occurs often enough that fast algorithms for their computation is desirable. Recently, I have identified a new class of discrete problems, that has wide applicability, which I have called Bohemian problems, for BOunded Height Matrices of Integers (BOHEMI). This new field of study has connections to graph theory, to compressed sensing, to random matrices, and combinatorics. We have created a web page and a GitHub showcasing our results (search for "Bohemian Matrices"). We will study special Bohemian families with useful matrix structures, such as Metzler matrices and correlation matrices, which have applications in biology and in neuroscience, for example.