Sparse SDP relaxation and moment methods for approximating solutions of
nonlinear differential equations
Martin Mevissen
Tokyo Institute of Technology
Abstract:
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.