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

首席研究员 (PI) 将获得非线性两点边值问题 (TPBVP) 类的“基本解”。 TPBVP 是一种问题，其中给出了有关系统的初始状态和最终状态的一些信息，其目标是确定初始状态的剩余条件，以便满足给定的最终状态条件。例如，人们可能拥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什么，以便发生特定的期望的或可能非常不期望的最终状态。基本的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，我们可以非常快速地为变化的可能数据生成大量解决方案。天体动力学中的应用包括行星际任务的重力辅助轨迹以及根据部分数据分析潜在的小行星/彗星影响。这种通用方法将被扩展以获得量子力学薛定谔方程的极快速求解方法。薛定谔方程是一个偏微分方程（PDE）。获得偏微分方程解的经典方法受到著名的“维数灾难”的影响。具体来说，随着每个粒子添加到问题中，必须求解偏微分方程的空间维度会增加三倍，而维度增加三倍通常会导致计算时间增加超过 100,000 倍。 PI 和合作者开发的“无维数诅咒”（CODF）方法极大地减少了某些类别的高维问题的计算负载。以前，这种方法仅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）薛定谔偏微分方程的极其快速的 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，而这些效应以前超出了我们的工具的范围。获得该奖项资助的研究生将在PI的指导下积极参与本项目各方面的相关研究。将开发动力学、控制理论、分析和随机过程领域的新理论和工具。尽管 PI 和合作者之前证明，最小行动方法可用于在保守系统中生成 TPBVP 的基本解决方案，但这仅适用于短期。扩展到任意持续时间的 TPBVP 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-雅可比理论将扩展到涵盖寻求收益平稳点的情况。生成基本解需要静态化算子具有一定的交换性，这是非常重要的。基本解可以存储为有限维系数集；特定问题数据的特定解是通过对边界数据编码的函数进行幂等卷积从基本解中获得的。此外，通过推广扩散过程控制工具，PI 将获得静态化应用到复值随机问题的扩展，从而产生对某些二阶 Hamilton-Jacobi PDE 有效的基于静态化的表示。特别是，薛定谔初始值问题的解决方案的表示将通过受控复值扩散过程上的复值作用泛函的静态化来获得。这也将为此类问题提供一个基本的解决方案和无维数诅咒的方法。作为必要的子任务，将获得新类简并随机微分方程解的存在性和唯一性结果的扩展。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力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.