On the Connection Between Forward and Optimization Problem in One-shot One-step Methods

In certain applications of PDE constrained optimization one would like to base an optimization method on an already existing contractive method (solver) for the forward problem. The forward problem consists of finding a feasible point with some parts of the variables (e.g., design variables) held fixed. This approach often leads to so-called simultaneous, all-at-once, or oneshot optimization methods. If only one iteration of the forward method per optimization iteration is necessary, a simultaneous method is called one-step. We present three illustrative linear examples in four dimensions with two constraints which highlight that in general there is only little connection between contraction of forward problem method and simultaneous one-step optimization method. We analyze the asymptotics of three prototypical regularization strategies to possibly recover convergence and compare them with Griewank’s One-Step One-Shot projected Hessian preconditioners. We present de facto loss of convergence for all of these methods, which leads to the conclusion that, at least for fast contracting forward methods, the forward problem solver must be used with adaptive accuracy controlled by the optimization method.