Pseudospectral methods and iterative solvers for optimization problems from multiscale particle dynamics

Abstract

We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we investigate problems where the control acts as an advection ‘flow’ vector or a source term of the partial differential equation, and the constraint is equipped with boundary conditions of Dirichlet or no-flux type. After deriving continuous first-order optimality conditions for such problems, we solve the resulting systems by developing a link with computational methods for statistical mechanics, deriving pseudospectral methods in space and time variables, and utilizing variants of existing fixed-point methods as well as a recently developed Newton–Krylov scheme. Numerical experiments indicate the effectiveness of our approach for a range of problem set-ups, boundary conditions, as well as regularization and model parameters, in both two and three dimensions. A key contribution is the provision of software which allows the discretization and solution of a range of optimization problems constrained by differential equations describing particle dynamics.