Gunay Dogan

Postdoc
Email: gunay
Phone: +1.215.898.3144

research

My research is on inverse problems and image processing. The emphasis is on the formulation of fast and effective numerical algorithms to address various problems in mathematical imaging. Currently I am working on efficient methods for image reconstruction in diffuse optical tomography.

publications

Biros G., Dogan G., "A multilevel algorithm for inverse problems with elliptic PDE constraints",accepted for publication in Inverse Problems

Dogan G., Biros., "A fast inversion algorithm for linearized diffuse optical tomography with many measurements", under preparation

projects

Multilevel algorithms for inverse problems with elliptic PDE constraints:

We formulate a novel multilevel algorithm for the solution of Tikhonov-regularized first-kind Fredholm equations. An example is the source identification problem, in which the forward problem is an elliptic partial differential equation. Our method assumes the availability of an approximate Hessian operator for which, first, the spectral decomposition is known, and second, there exists a fast algorithm that can perform the spectral transform. Based on this decomposition we propose a Conjugate Gradients (CG) solver which we precondition with a multilevel subspace projection scheme. Our novel preconditioner significantly improves the performance of CG. Not only do we obtain mesh-independent performance, but also very good scalability in the case of small regularization parameter.

Fast inversion schemes for diffuse optical tomography:

We consider a problem where a scattering medium is excited by numerous light sources at the boundary. We would like to identify the absorption coefficient of the medium from given large number of boundary measurements. An example of this problem is diffuse optical tomography. We work out the precise analytical structure of the problem in the case of constant background coefficient and devise a fast inversion scheme that allows us to infer the absorption coefficient. Our current work is on taking advantage of this scheme to implement an efficient algorithm that works for nonconstant background coefficient as well.