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Overcoming the Curse of Dimensionality for Hamilton-Jacobi equations with Applications to Control and Differential Games

Speaker: Stanley Osher

Seminar: CITRIS People and Robots Initiative, “Design of Robotics and Embedded systems, Analysis, and Modeling” Seminars (DREAMS) | Monday, March 13 | 4 – 5 PM | 250 Sutardja Dai Hall | Webcast

It is well known that certain Hamilton-Jacobi partial differential equations (HJ PDE’s) play an important role in analyzing control theory and differential games. The cost of standard numerical algorithms for HJ PDE’s is exponential in the space dimension and time, with huge memory requirements. Here we propose and test methods for solving a large class of these problems without the use of grids or significant numerical approximation. We begin with the classical Hopf and Hopf-Lax formulas which enable us to solve state independent problems via variational methods originating in compressive sensing with remarkable results. We can evaluate the solution in 10^(-4) to 10^(-8) seconds per evaluation on a laptop. The method is embarrassingly parallel and has low memory requirements. Recently, with a slightly more complicated, but still embarrassingly parallel method, we have extended this in great generality to state dependent HJ equations, apparently, with the help of parallel computers, overcoming the curse of dimensionality for these problems. The term, “curse of dimensionality” was coined by Richard Bellman in 1957 when he did his classic work on dynamic optimization.

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Stanley Osher is a Professor of Mathematics & Computer Science in the Department of Electrical Engineering & Chemical and Biomolecular Engineering at the University of California, Los Angeles. He is also the Director of Special Projects at the Institute for Pure and Applied Mathematics (IPAM). Osher’s research consists of developing innovative numerical methods to solve partial differential equations, especially those whose solutions have steep gradients, analysis of these algorithms and the underlying P.D.E.’s and applications to various areas of Engineering, Physics and recently, image processing.