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In this paper, we consider the fast numerical solution of an optimal controlformulation of the Keller--Segel model for bacterial chemotaxis. Upondiscretization, this problem requires the solution of huge-scale saddle pointsystems to guarantee accurate solutions. We consider the derivation ofeffective preconditioners for these matrix systems, which may be embeddedwithin suitable iterative methods to accelerate their convergence. We alsoconstruct low-rank tensor-train techniques which enable us to present efficientand feasible algorithms for problems that are finely discretized in the spaceand time variables. Numerical results demonstrate that the number ofpreconditioned GMRES iterations depends mildly on the model parameters.Moreover, the low-rank solver makes the computing time and memory costssublinear in the original problem size.

Preconditioners and Tensor Product Solvers for Optimal Control Problems from Chemotaxis