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

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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.