Adjoint-based optimization of PDE systems with alternative gradients

In this work we investigate a technique for accelerating convergence of adjoint–based optimization of PDE systems based on a nonlinear change of variables in the control space. This change of variables is accomplished in the differentiate–then–discretize approach by constructing the descent directions in a control space not equipped with the Hilbert structure. We show how such descent directions can be computed in general Lebesgue and Besov spaces, and argue that in the Besov space case determination of descent directions can be interpreted as nonlinear wavelet filtering of the adjoint field. The freedom involved in choosing parameters characterizing the spaces in which the steepest descent directions are constructed can be leveraged to accelerate convergence of iterations. Our computational examples involving state estimation problems for the 1D Kuramoto–Sivashinsky and 3D Navier–Stokes equations indeed show significantly improved performance of the proposed method as compared to the standard approaches.