Regularization of inverse problems via box constrained minimization

Abstract

In the present paper we consider minimization based formulations of inverse problems (x, Φ) ∈ argmin {J (x, Φ; y) : (x, Φ) ∈ M ad (y)} for the specific but highly relevant case that the admissible set M δ ad (y δ ) is defined by pointwise bounds, which is the case, e.g., if L ∞ constraints on the parameter are imposed in the sense of Ivanov regularization, and the L ∞ noise level in the observations is prescribed in the sense of Morozov regularization. As application examples for this setting we consider three coefficient identification problems in elliptic boundary value problems.Discretization of (x, Φ) with piecewise constant and piecewise linear finite elements, respectively, leads to finite dimensional nonlinear box constrained minimization problems that can numerically be solved via Gauss-Newton type SQP methods. In our computational experiments we revisit the suggested application examples. In order to speed up the computations and obtain exact numerical solutions we use recently developed active set methods for solving strictly convex quadratic programs with box constraints as subroutines within our Gauss-Newton type SQP approach.

Some application examples of minimization based formulations of inverse problems and their regularization

“…• Assumption 1(i),(j) adjusted by Remark 1 follow from Assumption 3(i), noting that Q E (x † ,û † ; y) = 0, (x † ,û † ) ∈ M and "the suitable subset of Y " in (6) is Z. • The last one, Assumption 1(k), is verified by Assumption 3(ii) and (13), (14) as below…”

“…Also differentiability of J δ is an asset in this sense and holds for the examples below. On the constraints side, it is pointwise bounds that on one hand can be very efficiently implemented see, e.g., [13], on the other hand are practically relevant in view of known a prior bounds on the searched for quantities. Morover, in the spirit of the Kohn-Vogelius functional, we strive for first order least squares formulations of the PDE models.…”

“…A numerical implementation for the EIT problem together with numerical results is subject of ongoing work. For further numerical evidence on the efficiency of the minimization based and all-at-once approach, we point to, e.g., [13,20,17]. Future research in this context will be on the question of optimally selecting this right inverse operator, as well as on further applications.…”

Iterative regularization for constrained minimization formulations of nonlinear inverse problems

“…Indeed, while nonnegativity of the Hessian would be enforced by assuming J ≥ 0 on all of X, we only assume this to hold on M𝛿 cf. (11).…”

On the incorporation of box-constraints for ensemble Kalman inversion

“…Our work will build on the theory by Bertsekas [6,7] and others [33,34] for projected preconditioned gradient methods. For inverse problems, box-constrained optimization has been applied but in a different context and for different applications, we refer the reader to [29].…”