Fast algorithms for inverse problems with elliptic and parabolic PDE constraints: applications to cardiac electrophysiology.

Santi Swaroop Adavani
Ph.D. Candidate
Advisor: Professor George Biros
Mechanical Engineering and Applied Mechanics
University of Pennsylvania

Abstract
The main goal of this project is to design and implement fast algorithms to solve inverse problems that arise in cardiac electrophysiology. We investigate the computational challenges involved in solving inverse electrophysiology problems and propose numerical techniques to address them. We consider the following two formulations : 1) a source identification problem with an elliptic PDE constraint, and 2) an inverse medium problem with a parabolic PDE constraint. We use L2 Tikhonov regularization in both the problems for stability. We use a reduced space approach in which we eliminate the state and the adjoint variables and we iterate in the inversion parameter space using Conjugate Gradients (CG). We propose SVD based preconditioners to accelerate the convergence of CG to solve the source identification problem. The overall complexity of reconstructing the source is O(N logN), where N is the number of grid points. We propose multigrid based preconditioners to accelerate the convergence of CG to solve the inverse medium problem. The overall complexity of recovering the medium properties is O(NtN +N log^2 N), where N is the number of grid points and Nt is the number of time steps. We present numerical experiments to show mesh-independent convergence of our algorithms-even in the case of no regularization. This feature makes these methods algorithmically robust to the value of the regularization parameter, and thus, useful for the cases in which we seek high-fidelity reconstructions.