Many phenomena in physical and life sciences can be modeled by partial
differential equations that are parabolic in nature. Applications
range from the conception of semi-conductors in Materials science
(Stefan problem) to the treatment planning for cancer patients in
radiation oncology (image segmentation). The modeling and numerical
simulation of these equations share similar drawbacks, such as the
computational burden imposed by a stringent time step restriction.
In this talk we will discuss some new numerical algorithms that
address some of these issues. First a fourth order accurate finite
difference numerical discretization for the Laplace and heat equations
with Dirichlet boundary conditions on irregular domains will be
described. Then, we turn our focus to the Stefan problem and construct
a third order accurate implicit discretization. Multidimensional
computational results are presented to demonstrate the order of
accuracy of these numerical methods. An adaptive grid refinement for
the Poisson equation in the context of the incompressible Euler
equations of fluid dynamics will be briefly presented, noting that the
aim of this work is to construct an adaptive method for parabolic
equations. Finally, a fast hybrid level Set/k-Means algorithm for
image segmentation and its application to the segmentation of organs
in the context of radiation oncology will be presented.
|