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Solving partial differential equations via sparse SDP relaxations of polynomial optimization problems

A vast number of problems arising from various areas, as for instance physics, economics and engineering, can be expressed as partial differential equations (PDE). As problems involving PDE are difficult to solve analytically, numerical solvers for fairly general classes of PDE are in high demand. By discretizing a class of PDE problems via finite differences, we formulate PDE problems as polynomial optimization problems. We encounter and examine correlative sparsity in those polynomial optimization problems. Their sparsity is exploited by applying the sparse semidefinite program (SDP) relaxations of Waki, Kojima, Kim and Muramatsu to them, in order to obtain a solution of the PDE problem. ODE, PDE, differential algebraic and optimal control problems are solved by employing a multigrid method, that is based on a combination of sparse SDP relaxations and Newton's method. In order to select particular solutions of PDE problems, appropriate objective functions are chosen and additional inequality constraints are imposed on the corresponding polynomial optimization problems.

Type of Seminar:
Public Seminar
Martin Mevissen
Jun 01, 2007   10am

Contact Person:

Dr. C. Jones
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