A SPECTRAL-IN-TIME NEWTON-KRYLOV METHOD FOR NONLINEAR PDE-CONSTRAINED OPTIMIZATION

We devise a method for nonlinear time-dependent PDE-constrained optimization problems that uses a spectral-in-time representation of the residual, combined with a Newton-Krylov method to drive the residual to zero. We also propose a preconditioner to accelerate this scheme. Numerical results indicate that this method can achieve fast and accurate solution of nonlinear problems for a range of mesh sizes and problem parameters, the numbers of outer Newton and inner Krylov iterations required to reach the attainable accuracy of a spatial discretization are robust with respect to the number of collocation points in time, and also do not change substantially when other problem parameters are varied.