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The terms “inverse problems” and “ill-posed problems” have been steadily and surely gaining popularity in modern science since the middle of the 20th century. A little more than fifty years of studying problems of this kind have shown that a great number of problems from various branches of classical mathematics (computational algebra, differential and integral equations, partial differential equations, functional analysis) can be classified as inverse or ill-posed, and they are among the most complicated ones (since they are unstable and usually nonlinear). At the same time, inverse and ill-posed problems began to be studied and applied systematically in physics, geophysics, medicine, astronomy, and all other areas of knowledge where mathematical methods are used. The reason is that solutions to inverse problems describe important properties of media under study, such as density and velocity of wave propagation, elasticity parameters, conductivity, dielectric permittivity and magnetic permeability, and properties and location of inhomogeneities in inaccessible areas, etc . In this paper we consider definitions and classification of inverse and ill-posed problems and describe some approaches which have been proposed by outstanding Russian mathematicians A. N. Tikhonov, V. K. Ivanov and M. M. Lavrentiev.

In this paper we solve (locally in time and under suitable assumptions on the data) an identification problem related to a linear parabolic equation when the additional information is time-dependent and nonlocal in space . More exactly, our problem consists in recovering a (positive) time-dependent coefficient β in front of the time derivative. We prove a local in time existence, uniqueness and stability result when the data belong to suitable function spaces. Our basic tool is the Semigroup Theory of linear bounded operators.

The Magnetic resonance electrical impedance tomography (MREIT) is a new medical imaging method combining electrical impedance tomography (EIT) and current injection MRI technique. In this paper, we show the uniqueness in MREIT problem with an orthotropic conductivity under the hypothesis that the ratios of conductivities are known. Based on an effective numerical differentiation method and an approach to detect discontinuity, we also propose an iterative reconstruction algorithm for the orthotropic conductivity reconstruction. The resulting numerical algorithm is accurate and stable against the noise, and numerical examples are made to illustrate the performance of our algorithm.

We recover a non-negative time-dependent function, vanishing only at t = 0, in a linear multi-dimensional parabolic equation with a power degeneration. First we prove an existence and uniqueness result for an identification problem in a general Banach space via Semigroup Theory. Then we apply such a result to our specific linear parabolic equation related to a smooth bounded domain in ℝ n .