762a

Optimization problems involving the solution of partial differential equations (PDE) are often encountered in the context of optimal design, optimal control and parameter estimation. Based on the reduced gradient method, a general strategy was proposed to solve these problems by using mostly existing software. As an illustration, this strategy was employed to solve a PDE-based optimization problem that arises from the optimal design of the network of pore channels in hierarchically structured porous catalysts. A Fortran implementation was developed by combining a gradient-based optimization package, NLPQL, a multigrid solver, MGD9V, and a limited amount of in-house coding. The value and gradient of the objective function were computed by solving the discretized PDE and another system of linear equations using the MGD9V, and were fed into the NLPQL. The PDE was discretized in terms of a finite volume method on a matrix of computational cells. The number of cells ranged from 129x129 to 513x513, and the number of optimization variables from 41 to 201. Numerical tests were carried out on a Dell laptop with a 2.16 GHz Intel Core2 Duo processor. The results showed that the optimization typically converged within a limited number (i.e., 9-48) of iterations. The CPU time was from 2.52 to 211.52 seconds. The PDE was solved 36-201 times in each of the numerical tests. This study calls for the use of our strategy to solve PDE-based optimization problems.

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