Preconditioners and Tensor Product Solvers for Optimal Control Problems from Chemotaxis
For researchers solving large-scale optimal control problems in chemotaxis, this provides efficient numerical methods that scale sublinearly, though the approach is domain-specific.
This work develops preconditioners and low-rank tensor-train solvers for optimal control problems in chemotaxis, achieving sublinear computing time and memory costs relative to problem size, with GMRES iterations depending mildly on model parameters.
In this paper, we consider the fast numerical solution of an optimal control formulation of the Keller--Segel model for bacterial chemotaxis. Upon discretization, this problem requires the solution of huge-scale saddle point systems to guarantee accurate solutions. We consider the derivation of effective preconditioners for these matrix systems, which may be embedded within suitable iterative methods to accelerate their convergence. We also construct low-rank tensor-train techniques which enable us to present efficient and feasible algorithms for problems that are finely discretized in the space and time variables. Numerical results demonstrate that the number of preconditioned GMRES iterations depends mildly on the model parameters. Moreover, the low-rank solver makes the computing time and memory costs sublinear in the original problem size.