Utilizing sparsity in time-varying optimal control of aquifer cleanup

A method for exploiting the sparsity structure of a finite-element simulation model in a linked dynamic optimal control model is developed. The linked optimization/simulation model has been used for the computation of time-varying optimal pumping rates for the pump-and-treat remediation of contaminated ground water. The methodology presented reduces the computational effort involved in the determination of time-varying optimal pumping rates by an order of n, from n³ to n², where n is the number of non-Dirichlet nodes used in the simulation of the aquifer. The method presented uses the characteristic banded structure of finite element model matrices in derivative computations used by the optimal control algorithm, and also within the algorithmic computations of the optimization method itself. Timing results demonstrating the improvement of this method for problems having n = 100 to n = 1,575 (state dimension 2n = 200 to 2n = 3,150) are presented. It is shown that the efficiency of the sparse algorithm is highest when the length of management periods equals the length of simulation time periods. The results indicate that for a problem with 2n = 1,000, the sparse algorithm is as much as 98% faster than an algorithm neglecting sparsity.