## Rebecca C. McKay

# Research

I am interested in applied mathematics: applied analysis, asymptotic
analysis, ODEs/PDEs,
dynamical systems, numerical
analysis and mathematical biology.
Curriculum Vitae

Research Statement

## Preprints

R. C. McKay, T. Kolokolnikov and P. Muir, Interface oscillations in reaction-diffusion
systems beyond the Hopf bifurcation, to appear, DCDS-B
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.

## Presentations

Instability
thresholds and
dynamics of mesa patterns in reaction-diffusion systems, CAIMS
meeting
July 2010, St. John's, Newfoundland
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.

## MSc research

My Masters research was with Dr. John Clements on the inverse problem of
electrocardiography.
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.

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