A Stationarity-Based Operator, Associated Fundamental Solutions and a Curse-of-Dimensionality-Free Algorithm

主要研究者（PI）将获得一类非线性两点边值问题（tpbvp）的“基本解”。tpbvp是这样的问题：给定系统的初始状态和终端状态的一些信息，目标是确定初始状态的剩余条件，使给定的终端状态条件得到满足。例如，某人可能有一些关于太空飞行器或小行星的初始状态（位置和速度）的数据，并且想知道该初始数据的其他组成部分需要是什么，以便出现特定的理想或可能非常不希望的最终状态。基本解决方案非常有价值；有了这样的解决方案，就不需要每次初始数据或终端数据发生变化时都重新解决问题。因此，我们可以非常迅速地为各种可能的数据生成大量的解决方案集。天体动力学的应用包括行星际任务的重力辅助轨迹和从部分数据分析潜在的小行星/彗星撞击。这种一般方法将被推广到获得量子力学薛定谔方程的极快速解方法。薛定谔方程是一个偏微分方程（PDE）。经典的求偏微分方程解的方法受制于著名的“维数诅咒”。具体来说，随着每个粒子的加入，PDE必须解决的空间的维度增加了3，而这种三维的增加通常会导致计算时间的增长，其数量级超过100,000。由PI和合作者开发的“无维诅咒”（CODF）方法大大减少了某些高维问题的计算负荷。以前，这种方法只适用于一阶偏微分方程。利用这一突破，PI将构建一种适用于（二阶）薛定谔PDE的极快速CODF方法。这将使研究人员能够研究量子系统中的非线性效应，这在以前是我们的工具无法达到的。获资助的研究生将在项目负责人的指导下，积极参与与本项目有关的各个方面的研究。在动力学，控制理论，分析和随机过程领域的新理论和工具将被开发。虽然PI和合作者先前证明了最小作用方法可以用于产生保守系统中TPBVPs的基本解决方案，但这只适用于短时间。扩展到任意持续时间的TPBVPs需要扩展控制理论以涵盖平稳性问题（即，静态化）。这是该领域的一个全新方向。动态规划和汉密尔顿-雅可比理论将被扩展，以涵盖寻求收益的平稳点的情况。生成基本解要求静态算子具有一定的交换性，这是高度非平凡的。基本解可以存储为有限维的系数集；通过对编码边界数据的函数进行幂等卷积，从基本解得到特定问题数据的特解。此外，从扩散过程控制的推广工具，PI将获得静态应用于复值随机问题的扩展，产生对某些二阶Hamilton-Jacobi偏微分方程有效的基于静态的表示。特别地，薛定谔初值问题的解的表示将通过复值作用泛函在受控复值扩散过程上的静态化得到。这也将为这类问题提供一个基本的解决方案和一种不受维数限制的方法。新一类退化随机微分方程解的存在唯一性结果的推广将作为一个必要的子任务。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The principal investigator (PI) will obtain "fundamental solutions" for classes of nonlinear two-point boundary value problems (TPBVPs). TPBVPs are problems where one is given some information about the initial state of the system and about the terminal state with the goal of determining the remaining conditions on the initial state such that the given terminal state conditions are met. For example, one may have some data on the initial state (position and velocity) of a space vehicle or asteroid, and would like to know what the other components of that initial data need to be such that a specific desirable, or possibly highly undesirable, terminal state will occur. Fundamental solutions are extremely valuable; given such a solution, one does not need to re-solve the problem each time the initial or terminal data changes. Hence, one can generate large sets of solutions for varying possible data very rapidly. Applications in astrodynamics include gravity-assist trajectories for interplanetary missions and analysis of potential asteroid/comet impacts from partial data. This general approach will be extended to obtain extremely rapid solution methods for the Schrodinger equation of quantum mechanics. The Schrodinger equation is a partial differential equation (PDE). Classical methods for obtaining solutions of PDEs are subject to the famous "curse of dimensionality". Specifically, the dimension of the space over which the PDE must be solved grows by three with the addition of each particle to the problem, while such an increase of three in dimension typically results in a growth in computational time by a factor on the order of over 100,000. The "curse-of-dimensionality-free" (CODF) methods developed by the PI and collaborators have massively reduced the computational load for certain classes of high-dimensional problems. Previously, this approach was only useful for first-order PDEs. Using this breakthrough, the PI will construct an extremely rapid CODF method applicable to the (second-order) Schrodinger PDE. This will allow researchers to study nonlinear effects in quantum systems that were previously beyond the reach of our tools. The graduate students supported by this award will be actively involved in research related to various aspects of this project under the guidance of the PI. New theory and tools in the areas of dynamics, control theory, analysis and stochastic processes will be developed. Although the PI and collaborators previously demonstrated that the least-action approach can be used to generate fundamental solutions to TPBVPs in conservative systems, that was appropriate only for short duration. The extension to arbitrary-duration TPBVPs requires an extension of control theory to cover stationarity problems (i.e., staticization). This is an entirely new direction for the field. Dynamic programming and Hamilton-Jacobi theory will be extended to cover cases where one seeks a stationary point of the payoff. Generating fundamental solutions requires a certain commutativity of staticization operators, which is highly nontrivial. The fundamental solutions may be stored as finite-dimensional sets of coefficients; the particular solutions for specific problem data are obtained from the fundamental solutions through idempotent convolution against functions encoding the boundary data. Also, generalizing tools from control of diffusion processes, the PI will obtain an extension of the application of staticization to complex-valued, stochastic problems, yielding a staticization-based representation valid for certain second-order Hamilton-Jacobi PDEs. In particular, representations for solutions of Schrodinger initial value problems will be obtained via staticization of complex-valued action functionals over controlled complex-valued diffusion processes. This will also yield a fundamental solution and a curse-of-dimensionality-free method for such problems. Extension of existence and uniqueness results for solutions of new classes of degenerate stochastic differential equations will be obtained as a necessary subtask.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.