Solving inverse problems without forwar...

An inverse problem is a problem of finding unknown quantities from a set of observations related via a forward operator F, for example, the problems of imaging an internal body organ from measurements on the skin, identifying sound sources from sound vibrations recorded by microphones, etc. Because of their wide application in science and technology, inverse problems receive great interest from scientists. Many regularization methods have been invented for various types of inverse problems.

The classical methods directly use the forward operator F and approximately invert F on the data. This slows down the speed of computations and is sometimes difficult to calculate, especially for inverse problems for which the forward operator is only defined under very restrictive conditions or computationally expensive to evaluate. In recent years, methods of solving inverse problems without using forward operators have attracted the attention of researchers. This thesis follows these methods, gives a new idea of using data inversion and some application examples as well as iterative solvers and numerical results.

The first contribution of this thesis is to discuss a new approach for minimization based regularization methods using data inversion. The key idea for the new approach is to use a right inverse operator of the observation operator. There are many ways to choose such a right inverse operator, but we do not focus on discussing them in general. We only present some important examples, namely, electrical impedance tomography, determination of magnetic permeability, and determination of sound sources to demonstrate that our new approach is applicable. The second contribution is also about the minimization based formulations, but follows an iterative approach. We present iterative regularization methods for constrained minimization problems which can be considered as generalizations of gradient and Newton-type methods. Then, we provide several options for the choice of the cost function in problems of diffusion/impedance identification. Finally, we show some numerical examples.