Volume 15, Issue 5

We review the state of the art and the current open problems regarding the stability of the inverse boundary value problem of Calderón, also known as the inverse conductivity problem or EIT.

The problem to determine the location and shape of a perfect conductor within a conducting homogeneous host medium from measured current and voltages on the accessible exterior boundary of the host medium can be modelled by an inverse Dirichlet boundary value problem for the Laplace equation. For this, recently Kress and Rundell suggested a novel inverse algorithm based on nonlinear integral equations arising from the reciprocity gap principle. The present paper extends this approach to the problem to recover the location and shape of a rigid body immersed in a fluid from the measured velocity and traction at the exterior boundary of the fluid, that is, to an inverse boundary value problem for the Stokes equation. The mathematical foundation of this extension is provided and numerical examples illustrate the feasibility of the method.

We consider the scattering of time harmonic electromagnetic plane waves by a bounded, inhomogeneous dielectric medium that is partially coated by a thin metallic layer in 2 . We use far field pattern of the scattered waves at a fixed frequency as data to determine the support D of the inhomogeneous obstacle, the surface conductivity that characterizes the coating and the relative permittivity. No a prior information on the material properties of the scatterer is needed. The support D is determined by the linear sampling method which is based on the approximate solution of the far field equation. This solution is also used to obtain lower bounds for the surface conductivity and relative permittivity. The techniques for solving this inverse scattering problem rely on the analysis of a non standard boundary value problem known as the interior transmission problem.

In this paper we study an inverse problem for a parabolic partial differential equation. The parabolic partial differential equation considered is the Fokker Planck equation associated to a system of stochastic differential equations and the inverse problem studied consists in finding from suitable data the values of the parameters that appear in the coefficients of this Fokker Planck equation. The data used in the reconstruction of the parameters are observations made at discrete times of the stochastic process solution of the system of stochastic differential equations. That is, the data of the inverse problem are a “sample” taken at discrete times of some of the components of the random vector solution of the stochastic differential equations and not, as usual, observations made on the solution of the parabolic equation. The choice of the system of stochastic differential equations and of the data used in the inverse problem are motivated by applications in mathematical finance. The stochastic differential equations presented can be used to model the dynamics of the log-returns of the index of some classes of hedge funds, such as, for example, the so called “long short equity” hedge funds and of some auxiliary variables. The solution of the inverse problem proposed is obtained through the solution of a filtering and of an estimation problem. The solution of these last two problems is based on the knowledge of the joint probability density function of the state variables of the model conditioned to the observations made and to the initial condition. This joint probability density function is solution of an initial value problem for the Kushner equation that in the circumstances considered here can be written as a sequence of initial value problems for the Fokker Planck equation associated to the system of stochastic differential equations with appropriate initial conditions. An integral representation formula for this probability density function is derived and used to develop a numerical procedure to solve the estimation problem using the maximum likelihood method. The Kushner equation provides the relation between the data and the Fokker Planck equation used to solve the inverse problem considered. The computational method proposed has been tested on synthetic data and the results obtained are presented. Some auxiliary material useful to understand this paper including some animations and some numerical experiments can be found in the website http://www.econ.univpm.it/recchioni/finance/w5. A more general reference to the work in mathematical finance of the authors and of their coauthors is the website http://www.econ.univpm.it/recchioni/finance.

The authors study regularization of non-linear inverse problems by projection methods. Non-linearity is controlled by some range invariance assumption. Emphasis is on approximation theoretic properties of the discretization which determine the convergence rates. Instead of using source conditions for the true solution to represent the error, the authors show how distance functions with respect to some benchmark smoothness are able to replace this. Some examples indicate how the results can be applied.