Moment and SDP relaxation techniques for smooth approximations of nonlinear differential equations

Martin Mevissen (1) , Jean-Bernard Lasserre (2) , Didier Henrion (2, 3)

1 Department of Mathematical and Computing Sciences
2 LAAS - Laboratoire d'analyse et d'architecture des systèmes
3 FEL CTU - Faculty of Electrical Engineering [Prague]

Combining recent moment and sparse semidefinite programming (SDP) relaxation techniques, we propose an approach to find smooth approximations for solutions of nonlinear differential equations. Given a system of nonlinear differential equations, we apply a technique based on finite differences and sparse SDP relaxations for polynomial optimization problems (POP) to obtain a discrete approximation of its solution. In a second step we apply maximum entropy estimation (using moments of a Borel measure associated with the discrete solution) to obtain a smooth closed-form approximation. The approach is illustrated on a variety of linear and nonlinear ordinary differential equations (ODE) and partial differential equations (PDE) and preliminary numerical results are reported.

Mots clés: ordinary differential equations; partial differential equations; moment problems; semidefinite programming

Domaines: Optimisation et contrôle [math.OC]

Martin Mevissen, Jean-Bernard Lasserre, Didier Henrion. Moment and SDP relaxation techniques for smooth approximations of nonlinear differential equations. 2010. ⟨hal-00462301v1⟩