The research presented in this thesis is about inverse problems, or more precisely, distributed parameter identification problems for two electrocardiology models: the FitzHugh-Nagumo model and the bidomain model. These models describe the evolution of electrical potentials in heart tissues. The objective of these inverse problems is to reconstruct coefficients in these electrocardiology models from electrical potential measurements.
The first type of algorithm we consider is a full nonlinear reconstruction algorithm of Newton-Krylov-Schur-Schwarz type. This is an iterative algorithm that combines Newton's method for numerical optimization with Krylov subspace solvers for the resulting reduced Karnsh-Kuhn-Tucker (KKT) system. Schwarz-type methods are used to solve the partial differential equations that are involved in the inversion procedure. The algorithm is implemented on parallel processors so that we can solve the reconstruction problem in large-scale parallel environments.
The second type of reconstruction algorithm is a linear method that, is based on Born-type linearization of the original nonlinear inverse problem. We linearize the inverse problem around a background coefficient that can be either a function of space and assumed known or an unknown constant that can be reconstructed accurately with a convex PDE-constrained optimization algorithm. We then try to reconstruct small perturbations from the background coefficient. The true coefficient is thus approximated by the superposition of the background and the small amplitude perturbation.
The two types of algorithms we have developed have been implemented for both problems with time-dependent measurements and problems with time-independent measurements. We show by numerical simulations that parameter reconstruction can be performed from measurements at various locations of the domain, including interior, boundary, and the combination. We discuss the effects of various model parameters on the quality of reconstructions, and show that reconstructions with stationary data are in general unstable with respect to high-frequency components in the unknown field, which are easily lost in the reconstructions. We thus should not expect to reconstruct the parameters very accurately. We propose to parameterize the inverse problems with a priori information to obtain more stable reconstructions.
|Adviser||David E. Keyes|
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