Rebecca McKay and Theodore Kolokolnikov, Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion system in one space dimension, DCDS-B, Vol. 17, no. 1, January 2012.
Mesa-type patterns in reaction-diffusion systems, Bluenose Numerical Analysis Day, July 2009.
Stability of a Reaction-Diffusion Model with Mesa-type Patterrns, Canada-France Congress, June 2008, Montreal, Quebec.
The inverse problem of electrocardiography---requiring the calculation of the heart surface potentials from the body surface potentials---presents a challenge because it is mathematically ill-posed. The currently accepted way of overcoming this is to use Tikhonov regularization. A key component of an effective regularization method is choosing a suitable value for the regularization parameter $\lambda$. Four commonly used methods (L-curve, CRESO, zero-crossing and discrepancy principle) as well as the new norm summation method are examined in this work. By calculating the optimal regularization parameter, using an a priori solution, the effectiveness of these methods is compared. An alternative way of solving the inverse problem is using a Krylov subspace method called the Generalized Minimal Residual (GMRES) method. This method forms an orthogonal basis of the sequence of successive matrix powers times the initial residual. The approximations to the solution are then obtained by minimizing the residual over the subspace formed. The GMRES method is able to solve the inverse problem without oversmoothing the solution, so as not to lose valuable localized electrical behaviour of the heart. The GMRES method and the Tikhonov regularization are compared and it is found that they perform similarly. Using both methods (one as a verification of the other) may help obtain more accurate assessments of the epicardial potentials.