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Sparse SDP relaxation and moment methods for approximating solutions of nonlinear differential equations

Martin Mevissen
Tokyo Institute of Technology


To solve a system of nonlinear differential equations numerically, we formulate it as a polynomial optimization problem (POP) by discretizing it via a finite difference approximation. The resulting POP satisfies a structured sparsity, which we can exploit to apply the sparse semidefinite programming (SDP) relaxation of Waki et al. to the POP to obtain a discrete approximation for a solution of the differential equation. The main features of this approach are: (a) we can choose an appropriate objective function, and (b) we can add inequality constraints on the unknown variables and their derivatives, in order to compute specific solutions of the system of differential equations. High resolution grid discretizations of differential equations result in high dimensional POPs and large-scale SDP relaxations. We discuss techniques to reduce the size of sparse SDP relaxations by exploiting sparsity and structure of the POPs. Finally, we propose a further method, which is based on sparse SDP relaxations, moments and maximum entropy estimation to detect smooth approximations for solutions of nonlinear differential equations. This method utilizes the SDP relaxation method for finding discrete approximations and can be understood as an attempt to overcome the curse of dimensionality in grid based approaches for solving differential equations.

Tuesday, October 5, 2010
11:00AM AP&M 2402