A Hybrid Segmentation and D-bar Method for Electrical Impedance Tomography

The Regularized D-bar method for Electrical Impedance Tomography provides a rigorous mathematical approach for solving the full nonlinear inverse problem directly, i.e. without iterations. It is based on a low-pass filtering in the (nonlinear) frequency domain. However, the resulting D-bar reconstructions are inherently smoothed leading to a loss of edge distinction. In this paper, a novel approach that combines the rigor of the D-bar approach with the edge-preserving nature of Total Variation regularization is presented. The method also includes a data-driven contrast adjustment technique guided by the key functions (CGO solutions) of the D-bar method. The new TV-Enhanced D-bar Method produces reconstructions with sharper edges and improved contrast while still solving the full nonlinear problem. This is achieved by using the TV-induced edges to increase the truncation radius of the scattering data in the nonlinear frequency domain thereby increasing the radius of the low pass filter. The algorithm is tested on numerically simulated noisy EIT data and demonstrates significant improvements in edge preservation and contrast which can be highly valuable for absolute EIT imaging

A SIMPLE NUMERICAL METHOD FOR COMPLEX GEOMETRICAL OPTICS SOLUTIONS TO THE CONDUCTIVITY EQUATION

This paper concerns numerical methods for computing complex geometrical optics (CGO) solutions to the conductivity equation del . sigma del u(., k) = 0 in R(2) for piecewise smooth conductivities sigma, where k is a complex parameter. The key is to solve an R-linear singular integral equation defined in the unit disk. Recently, Astala et al. [Appl. Comput. Harmon. Anal., 29 (2010), pp. 2-17] proposed a complicated method for numerical computation of CGO solutions by solving a periodic version of the R-linear integral equation in a rectangle containing the unit disk. In this paper, based on the fast algorithms in [P. Daripa and D. Mashat, Numer. Algorithms, 18 (1998), pp. 133-157] for singular integral transforms, we propose a simpler numerical method which solves the R-linear integral equation in the unit disk directly. For the resulting R-linear operator equation, a minimal residual iterative method is proposed. Numerical examples illustrate the accuracy and efficiency of the new method

Computational Inverse Problems for Partial Differential Equations

The problem of determining unknown quantities in a PDE from measurements of (part of) the solution to this PDE arises in a wide range of applications in science, technology, medicine, and finance. The unknown quantity may e.g. be a coefficient, an initial or a boundary condition, a source term, or the shape of a boundary. The identification of such quantities is often computationally challenging and requires profound knowledge of the analytical properties of the underlying PDE as well as numerical techniques. The focus of this workshop was on applications in phase retrieval, imaging with waves in random media, and seismology of the Earth and the Sun, a further emphasis was put on stochastic aspects in the context of uncertainty quantification and parameter identification in stochastic differential equations. Many open problems and mathematical challenges in application fields were addressed, and intensive discussions provided an insight into the high potential of joining deep knowledge in numerical analysis, partial differential equations, and regularization, but also in mathematical statistics, homogenization, optimization, differential geometry, numerical linear algebra, and variational analysis to tackle these challenges