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

这一建议的一部分是制定统计和统计力学工具来解决非常复杂的问题，此外，在这些问题的制定中可能存在不确定性或缺乏数据。惊人的事实是，在适当的范数中估计这些问题的最佳解决方案与不可逆统计力学中难题的解决方案密切相关，这种联系可以通过统计投影的形式来表达，有时被称为Mori-Zwanzig形式主义。这种形式的使用带来了与重整化、朗之万方程和缩放过程的联系，这些过程必须更加严格，现在已经导致了新的近似过程和最优过程的新公式。第一个应用是建模问题，可以验证新的算法，现在的重点是发展到流体力学，包括粘弹性流动，等离子体物理学和生物学的应用。另一部分工作是将离散网络算法与几何观点联系起来，以获得有效解决连续偏微分方程问题的算法。到目前为止，这导致了用于最优控制和各向异性锋面传播计算问题的有序逆风方法，以及波传播、天线设计和地震学中多次到达的欧拉公式的静态相空间解。通过利用解决方案构建中的潜在顺序，由沿着特征的信息流决定，可以开发“一次通过”方法，该方法无需迭代即可构建这些问题的解决方案，并且计算复杂性基本上线性依赖于计算域中的网格点数量。这些技术将扩展到非凸游戏中出现的最小/最大问题，应用于复杂控制，开发自适应版本，使我们能够在计算机辅助加工中计算六维机器人导航问题，最重要的是，在多层到达技术应用于断层扫描中的逆问题。这个项目的目标是设计新的方法来使用计算机来解决非常复杂的问题，这些问题可能包含各种不确定性的来源，因为缺乏数据，关于影响解决方案的因素的不完整信息，对计算机时间的过度要求，或者因为它们涉及固有的混沌行为。从实际的角度来看，迄今为止的工作已经导致了更准确的成像技术来预测地下石油储量，以及心脏成像和电子显微镜下细胞不规则性自动分析的新技术。未来的方法将使更可靠地解释医学图像、更有效地设计计算机芯片、更好地了解人体生理学、即使在天空变得非常拥挤时也能避免飞机碰撞、更可靠地预测气候成为可能。

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.