Optimal Control for Chemotaxis Systems and Adjoint-Based Optimization with Multiple-Relaxation-Time Lattice Boltzmann Models

Abstract

This paper is devoted to continuous and discrete adjoint-based optimization approaches for optimal control problems governed by an important class of Nonlinear Coupled Anisotropic Convection-Diffusion Chemotaxis-type System (NCACDCS). This study is motivated by the fact that the considered complex systems (with complex geometries) appear in diverse biochemical, biological and biosocial criminology problems. To solve numerically the corresponding nonlinear optimization problems, the primal problem NCACDCS is discretised by a coupled Lattice Boltzmann Method with a general Multiple-Relaxation-Time collision operators (MRT) while for the adjoint problem, an Adjoint Multiple-Relaxation-Time lattice Boltzmann model (AMRT) is proposed and investigated. First, the optimal control problems are formulated and first-order necessary optimality conditions are established by using sensitivity and adjoint calculus. The resulting problems are discretised by the coupled MRT and AMRT models and solved via gradient descent methods. First of all, an efficient and stable modified MRT model for NCACDCS is developed, and through the Chapman-Enskog analysis we show that NCACDCS can be correctly recovered from the proposed MRT model. For the adjoint problem, the discretisation strategy is based on AMRT model, which is found to be as simple as MRT model with also highly-efficient parallel nature. The derivation of AMRT model and the discrete cost functional gradient are derived mathematically in detail using the developed MRT model. The obtained method is reliable, efficient, practical to implement and can be easily incorporated into any existing MRT code.