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

主要研究者（PI）将获得非线性两点边值问题（TPBVP）的“基本解”。TPBVP是这样的问题，其中给出关于系统的初始状态和关于终端状态的一些信息，目的是确定初始状态的剩余条件，使得满足给定的终端状态条件。例如，一个人可能有一些关于空间飞行器或小行星的初始状态（位置和速度）的数据，并且想知道初始数据的其他组成部分需要什么，以便出现特定的理想状态，或者可能是非常不理想的终端状态。基本解是非常有价值的;给定这样的解，不需要在每次初始或终端数据改变时重新求解问题。因此，可以非常快速地为不同的可能数据生成大量的解决方案。在天体动力学方面的应用包括行星际飞行任务的重力辅助轨道和从部分数据分析潜在的小行星/彗星撞击。这个一般的方法将得到扩展，以获得量子力学薛定谔方程的非常快速的解决方案的方法。薛定谔方程是一个偏微分方程（PDE）。求解偏微分方程的经典方法都受到著名的“维数灾难”的影响。具体地说，PDE必须求解的空间的维度随着每个粒子添加到问题中而增加三个，而这种维度的增加通常导致计算时间增加超过100，000倍。PI和合作者开发的“无量纲诅咒”（CODF）方法大大减少了某些类别高维问题的计算量。以前，这种方法只适用于一阶偏微分方程。利用这一突破，PI将构建一个非常快速的CODF方法适用于（二阶）薛定谔PDE。这将使研究人员能够研究量子系统中的非线性效应，而这些效应以前是我们的工具所无法达到的。该奖项支持的研究生将在PI的指导下积极参与与该项目各个方面相关的研究。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和合作者先前证明了最小作用方法可以用于生成保守系统中TPBVP的基本解，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。动态规划和哈密尔顿-雅可比理论将扩展到覆盖的情况下，寻求一个稳定点的回报。生成基本解需要静力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。此外，从扩散过程的控制推广工具，PI将获得一个扩展的应用程序的静态化复值，随机问题，产生一个基于静态化的表示有效的某些二阶Hamilton-Jacobi偏微分方程。特别是，表示薛定谔初值问题的解决方案，将通过控制复值扩散过程的复值作用泛函的静态化。这也将产生一个基本的解决方案和无量纲诅咒的方法，这样的问题。扩展的存在性和唯一性结果的解决方案的新类退化随机微分方程将获得作为一个必要的subtask.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.