Computational Techniques from Geometry and Statistical Physics for Optimal Prediction, Control and Wave Propagation

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One part of this proposal is the formulation of statistical and statistical mechanics tools for solving problems of great complexity, where, in addition, there may be uncertainties in the formulation of the problems or a lack of data. The striking fact is that estimating the best solution in an appropriate norm to such problems is very closely related to the solution of difficult problems in irreversible statistical mechanics, a connection that can be expressed through a formalism of statistical projection, sometimes knows as the Mori-Zwanzig formalism. The use of this formalism brings in connections with renormalization, Langevin equations, and scaling procedures, which had to be made more rigorous, and which have now led to new approximation procedures and new formulations of optimal procedures. The first applications were in modeling problems, which could validate the new algorithms, and now the focus is evolving to applications in fluid mechanics, including viscoelastic flows, in plasma physics and in biology. The other part of the work is the linking of discrete network algorithms to geometric perspectives to obtain algorithms for efficiently solving problems in continuous partial differential equations. This has led, so far, to Ordered Upwind Methods for computing problems in optimal control and anisotropic front propagation, and static phase space solutions to Eulerian formulations for multiple arrivals in wave propagation, antenna design, and seismology. By exploiting an underlying ordering in the construction of the solution, determined by the flow of information along characteristics, "one pass" methods can be developed which construct the solution to these problems without iteration, and with a computational complexity that depends essentially linearly on the number of mesh points in the computational domain. These techniques will be extended to min/max problems that arise in non-convex games, with applications to complex control, to developing adaptive versions which allows us to compute six-dimensional robotic navigation problems in computer-aided machining, and, most importantly, in the application of multiple arrival techniques to inverse problems in tomography. The goal of this project is to devise new ways to use computers in the solution of problems which are very complex and that may contain various sources of uncertainty, because of lack of data, incomplete information about the factors that affect the solution, excessive requirements of computer time, or because they involve inherent chaotic behavior. From a practical point of view, the work so far has led to more accurate imaging techniques for predicting underground oil reserves and new techniques for cardiac imaging and automatic analysis of cell irregularities in electron microscopy. The coming methods will make it possible to interpret medical images more reliably, design computer chips even more efficiently, gain a better understanding of human physiology, avoid aircraft collisions even when the skies become very crowded, and predict climate more reliably.