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

这项建议的一部分是制定统计和统计力学工具，以解决极为复杂的问题，此外，在这些问题的制定方面可能存在不确定性或缺乏数据。惊人的事实是，在适当的规范中估计这些问题的最佳解决方案与不可逆统计力学中困难问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主义来表达，有时称为Mori-Zwanzig形式主义。使用这种形式主义带来了与重整化，朗之万方程和缩放程序，这必须更加严格，现在已经导致新的近似程序和新的配方的最佳程序。最初的应用是在建模问题，这可以验证新的算法，现在的重点是发展到流体力学，包括粘弹性流动，在等离子体物理学和生物学中的应用。另一部分的工作是连接的离散网络算法的几何观点，以获得算法，有效地解决问题的连续偏微分方程。这导致了，到目前为止，有序逆风方法计算问题的最优控制和各向异性前传播，静态相空间解决方案的欧拉公式多到达波传播，天线设计和地震学。通过利用由信息流沿着特性确定的解的构造中的潜在排序，可以开发“一次通过”方法，其构造这些问题的解而无需迭代，并且具有基本上线性地取决于计算域中的网格点的数量的计算复杂度。这些技术将被扩展到最小/最大的问题，出现在非凸游戏中，与应用程序复杂的控制，开发自适应版本，使我们能够计算六维机器人导航问题的计算机辅助加工，最重要的是，在应用程序中的多个到达技术的逆问题在断层扫描。该项目的目标是设计新的方法来使用计算机解决非常复杂的问题，这些问题可能包含各种不确定性来源，因为缺乏数据，影响解决方案的因素的信息不完整，计算机时间的过度要求，或者因为它们涉及固有的混乱行为。从实用的角度来看，迄今为止的工作已经导致了更准确的成像技术，用于预测地下石油储量，以及心脏成像和电子显微镜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的情况下也能避免飞机碰撞，并更可靠地预测气候。

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.