A preconditioning technique for a class of PDE-constrained optimization problems

Abstract

We investigate the use of a preconditioning technique for solving linear systems of saddle point type arising from the application of an inexact Gauss-Newton scheme to PDE-constrained optimization problems with a hyperbolic constraint. The preconditioner is of block triangular form and involves diagonal perturbations of the (approximate) Hessian to insure nonsingularity and an approximate Schur complement. We establish some properties of the preconditioned saddle point systems and we present the results of numerical experiments illustrating the performance of the preconditioner on a model problem motivated by image registration.

Robust Iterative Solution of a Class of Time‐Dependent Optimal Control Problems

“…A recently developing area of research in the area of optimal control is the construction of preconditioners for time-dependent PDE-constrained optimization problems [3,[9][10][11]. It is desirable to design these preconditioners such that the computation time of the corresponding iterative method grows close to linearly with the problem size, although constructing methods as such often results in a lack of robustness with respect to the regularization term involved in the formulation of the problem.…”

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

“…We will follow the discretize-then-optimize approach for PDE-constrained optimization problems [11,35] (details on the numerical treatment of FDEs can be found in [50,51,69,71,72]). We consider Ω = [0, 1]d and assume that Ω u and Ωȳ are also cubes.…”

“…where p corresponds to the discretized adjoint variable p. The matrix V is used to associate p with a grid function (see [11]), and is written similarly to (12), V = τ h 1 · · · hd · I N . To obtain the required optimality conditions, we differentiate Λ with respect to y, u and p. This leads to the following first order, Karush-Kuhn-Tucker (KKT), system:…”

All-at-once solution of time-dependent Stokes control

“…We here want to employ the so-called all-at-once approach, which is a technique previously used in [23,24,5,44]. In detail, the discretization of the problem is constructed in the space-time domain and then solved for all time-steps at once.…”