Minimization Based Formulations of Inverse Problems and Their Regularization
本文提出一种更一般的反问题最小化表述，涵盖传统简化形式、全同时形式和Kohn-Vogelius变分方法，并探讨其正则化方面，特别针对电阻抗成像问题引入箱约束和偏差原理。
The conventional way of formulating inverse problems such as identification of a (possibly infinite dimensional) parameter is via some forward operator, which is the concatenation of the observation operator with the parameter-to-state-map for the underlying model. Recently, all-at-once formulations have been considered as an alternative to this reduced formulation, avoiding the use of a parameter-to-state map, which would sometimes lead to overly restrictive conditions. Here the model and the observation are considered simultaneously as one large system, with the state and the parameter as unknowns. A still more general formulation of inverse problems, containing not only the reduced and all-at-once formulations but also the well-known and highly versatile variational approach (also called the Kohn--Vogelius functional approach) as special cases, is to formulate the inverse problem as a minimization problem---instead of an equation---for the state and parameter. Regularization can be incorporated via imposing constraints and/or adding regularization terms to the objective. In this paper, after giving a motivation by formulating the electrical impedance tomography (EIT) problem by means of the classical Kohn--Vogelius functional, we explore the regularization aspects for such variational formulations in an abstract setting. Indeed, combination of regularization by constraints and by penalization leads to new methods that are applicable without solving forward problems. In particular, for the EIT problem we will consider a method employing box constraints in a very natural manner to incorporate the discrepancy principle for regularization parameter choice as well as a priori information on the searched for conductivity.