Numerical integration of large stiff nonlinear systems of ODEs over
long time intervals is a challenging task since the stiffness of the
system makes solving them using standard explicit and implicit
methods computationally expensive. Exponential propagation-type
techniques offer significant advantages for these problems compared
to standard integrators. We will discuss new exponential propagation
iterative (EPI) methods, which are constructed by approximating the
integral form of the solution to a nonlinear autonomous system of
ODEs by an expansion in terms of products between special functions
of matrices and vectors. The matrix functions-vector products are
then approximated using Krylov subspace projections. For problems
where no good preconditioner is available, the EPI integrators can
outperform standard methods since they possess superior stability
properties compared to explicit schemes and offer computational
savings compared to implicit Newton-Krylov integrators by requiring
fewer Arnold iterations per time step. As an application, for which
exponential propagation methods offer computational savings, I will
discuss modeling large-scale plasma behavior using resistive MHD
equations. I will present results of a numerical model which
describes plasma as a time-dependent driven system. The results of
the simulations suggest new structure of plasma configurations that
form in the course of evolution of solar coronal arcades or
laboratory spheromak-type plasmas.

We present a variational framework for shape optimization
problems that hinges on devising energy decreasing flows based on
shape differential calculus followed by suitable space and time
discretizations (discrete gradient flows). A key ingredient is
the flexibility in choosing appropriate descent directions by
varying the scalar products, used for computation of normal
velocity, on the deformable domain boundary. We discuss
applications to image segmentation, optimal shape design for PDE,
and surface diffusion, along with several simulations exhibiting
large deformations as well as pinching and topological changes in
finite time. This work is joint with E. Baensch, G. Dogan, P.
Morin, and M. Verani.