Efficient Multiscale Methods for Optimal In Situ Bioremediation Design

This paper presents a multiscale derivative method for solving a successive approximation linear quadratic regulator model for optimal in situ bioremediation design. An efficient one-sided forward divided difference numerical derivatives calculation was implemented as the first stage of the method, which only required assembling the right-hand-side vector of the linear systems of equations of the simulation model and performing backward substitution. The derivative calculation was reduced from $O\left(N^3\right)$ to nearly $O\left(N^2\right)\text{,}$ where N is the number of non-Dirichlet state variables. A V-cycle multiscale derivatives approximation was implemented as the second stage, which used coarser mesh derivatives to interpolate finer mesh derivatives. Implementing the numerical derivatives method in a case study with over 1,600 state variables caused a reduction of more than two-thirds in computing time over the previous analytical derivatives method without loss of accuracy. Using the V-cycle multiscale derivatives approximation further reduced computing time by 29%, resulting in an overall 77% reduction compared to the previous analytical derivatives method. The reduction will be even greater for applications with more state variables, enabling the solution of much larger-scale problems than was previously possible.