Non-iterative Numerical Methods for Boundary Value Problems

Cornell University, Ithaca NY

Investigators

Abstract

Static non-linear Partial Differential Equations (PDEs) describe a variety of problems in physics and engineering. Numerical schemes are commonly used to approximate the solution satisfying particular boundary conditions. Such schemes usually require solving a large system of coupled non-linear discretized equations. Iterative methods for such systems can be very expensive computationally, often leading practitioners to use alternative problem descriptions to avoid solving the full boundary value problem. The investigator proposes a family of fast (non-iterative) methods for a wide class of static PDEs, for which the direction of "information flow" defines a natural ordering on the discretized equations. In a joint work with J.A. Sethian, non-iterative Ordered Upwind Methods (OUMs) were introduced for Hamilton-Jacobi PDEs arising in anisotropic (& hybrid) control and in front propagation. The investigator proposes to extend OUMs to boundary value problems describing differential games and non-autonomous optimal control problems. In a joint work with J. Guckenheimer, the OUMs were previously applied to a special system of quasilinear PDEs to approximate invariant manifolds of vector fields. The investigator proposes to extend the invariant manifold approach to compute multi-valued solutions of boundary value problems. The efficiency of the proposed methods stems from the notion of "causality" -- unobvious yet natural ordering of the elements of computation. This approach is relevant for the applications as diverse as robotic navigation and photolithography, seismic imaging and computational geometry, optics and transient elastography, differential games and segmentation of images. Which trajectory is optimal for a rover traveling on the surface of Mars? With what delay and how strongly will an underground explosion be felt by a sensor at a given point on the surface? What are the optimal parameter values for etching and deposition in the integrated circuit manufacturing? What is the minimum safe distance for the aircraft collision avoidance? Will the electrical power system automatically recover after a "fault"? Answering these important practical questions in real time requires efficient and robust numerical methods for solving the corresponding partial differential equations.