Full Multiscale Approach for Optimal Control of In Situ Bioremediation

Solving field-scale optimal groundwater remediation design problems is a challenge, especially when computationally intensive reactive transport models are needed. In this paper, a full multiscale approach to partial differential equation (PDE) constrained optimization is developed and is used to solve a successive approximation linear quadratic regulator model for optimal control of in situ bioremediation. The method starts the search for optimal designs from the coarsest mesh and solves for the optimal solution at that level, then uses the optimal solution obtained as the initial guess for the finer mesh. While at the finer mesh, the method switches back to the coarser mesh to solve for the derivatives and uses those derivatives to interpolate back to the finer mesh. This procedure continues until convergence is achieved at the finest level. This approach exploits important interactions between PDE discretization and optimization and achieves significant computational saving by using approximations early in the search when a broad search of the decision space is being performed. As the solution becomes more refined, more accurate estimates are needed to fine-tune the solution, and finer spatial discretizations are used. Application of the method to a bioremediation case study with about 6,500 state variables converges in about 8.8 days, compared to nearly 1 year using the previous model. This substantial improvement will enable much more realistic bioremediation design problems to be solved than was previously possible, particularly once the model is implemented in parallel.