Abstract
In this paper we study the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type methods. We carry out a convergence analysis in the sense of regularization methods and discuss applicability to the problem of identifying the spatially varying diffusivity in an elliptic PDE from different sets of observations. Among these is a novel hybrid imaging technology known as impedance acoustic tomography, for which we provide numerical experiments.
Highlights
Inverse problems usually consist of a model A(x, u) = 0 (1)where the operator A acts on the state u of a system and contains unknown parameters x, and an observation equation C(x, u) = y (2)quantifying the available information that is supposed to allow for identifying the parameters x; by a slight notation overload, we will often summarize (x, u) into a single element, which we again call x.The classical formulation of an inverse problem is as an operator equation1 3 Vol.:(0123456789)B
The message of this paper is supposed to be two fold: First of all, we show that for inverse problems formulated by constrained minimization, besides the approach of regularizing and applying state-of-the-art iterative optimization tools there is the option of applying iterative methods to the un- or only partly regularized problem
In this paper we have provided convergence results on the iterative solution methods for minimization based formulations of inverse problems
Summary
Where the operator A acts on the state u of a system and contains unknown parameters x, and an observation equation. All-at-once approaches have been studied for PDE constrained optimization already for many years in, e.g., [25–28, 30, 31] due to their computational advantages: The iterates need not be feasible with respect to the PDE constraint which safes computational effort and potentially allows for larger steps This looseness can lead to convergence problems and we will see this in the most challenging of our numerical test cases, namely the severely ill-posed problem of electric impedance tomography EIT. Assuming its uniform positivity amounts to demanding bounded invertibitily of F (x†) , which usually does not hold for ill-posed problems Along with these two paradigms concerning the search direction, we will consider two approaches for guaranteeing feasibility of the sequence, namely projection onto the admissible set in the context of gradient methods in Sect. By means of the mentioned diffusion identification problems we wish to demonstrate the large variety of possible minimization formulations arising even in the context of a single elliptic PDE, and to highlight some of the chances and limitations related to these various formulations
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