Seminars Academic Year 2014-2015

Source reconstruction from final data in the heat equation

The inverse ill-posed problem of reconstructing a heat source in the parabolic heat equation from given knowledge of the solution at a final time is considered. An overview of a uniqueness result for spacewise dependent heat sources will be given together with a discussion about some counter examples showing that the conditions obtained for uniqueness cannot be relaxed. A numerical procedure for the stable determination of the source together with some numerical results will also be presented. The results obtained is joint work with prof. M. Slodicka.

Introduction to fractional calculus

The introduction to the theory of fractional integration and differentiation will be given. We consider the Riemann-Liouville and the Caputo fractional derivative. Basic properties of the considered fractional derivatives will be introduced, including the use of the Laplace transform, and the numerical approximation of a fractional derivative. The Mittag-Leffler function and its important role will be highlighted. Examples of applications in various branches will be given.

Global convergence for inverse problems and phaseless inverse problems

In this talk three topics will be discussed. Corresponding papers were published in 2008-2014, also see These topic are:
1. A globally convergent numerical method of the first type for coefficient inverse problems with single measurement data. Both the theory and numerical results will be presented. Numerical results will be focused on the most challenging case of blind backscattering experimental data for buried targets.
2. A globally convergent numerical method of the second type type for coefficient inverse problems will be presented. This method is based on the construction of a globally strictly convex cost functional. The key element of this functional is the Carleman Weight Function. 
3. The first solution of a long standing problem (since 1977). This is uniqueness of the 3-d coefficient inverse scattering problem in the case when only the modulus of the complex valued scattering wave field is measured, whereas the phase is unknown. In quantum inverse scattering only the differential cross-section is measured, which means the modulus. On the other hand, the entire theory of quantum inverse scattering is constructed for the case when both the modulus and phase of the scattering wave field are measured.

Numerical Method for solving an boundary value inverse heat conduction problem

In this talk two different approaches based on the Laplace and Fourier transforms will be discussed. Corresponding papers were published in 2013-2014.
Application of the Laplace transform makes it possible to obtain an integral equation describing the explicit dependence of the unknown boundary value function on the initial data at the other boundary. Regularization methods are then used to solve this equation. This eliminates the unstable procedure of numerical inversion of the Laplace transform in the computational process. The proposed method was used in a computational experiment to obtain a numerical solution of the inverse problem. Experimental error estimates of the obtained solutions show sufficient stability of these solutions.
The approach based on the projection regularization method for the direct and inverse Fourier transforms with respect to the time variable provides regularized solutions with guaranteed accuracy. The estimation of errors of these solutions are the exact with respect to the order. This property provided the basis for comparative analysis of the solutions obtained by the Laplace and Fourier transform methods.
The proposed methods were employed to carry out a computational experiment. The objectives of this experiment were to test effectiveness of the proposed approaches and to evaluate the errors of the regularized solutions provided by each approach. The computational results confirm the stability of the solutions obtained by these methods.