Numerical and Computational Challenges
in Science and Engineering Program
LECTURES
Benren Zhu
College of Mathematics and System Sciences, Shandong University, P.R.
China (Visiting the Fields Institute)
November 15, 2001, 10am, Room 210
Inverse Problems for Elliptic PDEs
In this talk, we consider inverse problems associated with the
elliptic PDE
\[
\nabla ( k \nabla \varphi ) - \gamma \varphi = f
\]
where $k = k(x,y) > 0$, $f = f(x,y)$, $\gamma = \gamma(x,y)$,
$(x,y) \in \Omega \subset (-1,1) \times (-1,1)$. These inverse
problems include the Inverse Coefficient ($k$) Problem (ICP), the
Inverse Source ($f$) Problem (ISP) and the Inverse Potential ($\gamma$)
Problem (IPP). The major difficulty in solving these problems is
ill-posedness. Therefore, various regularization techniques are used
in solving them. However, we note that in some cases it is possible
to
reconstruct the unknown function from data at the boundary of the
region $\Omega$ or outside it without using regularization. In other
cases it is not possible to do so even for the same unknown function.
October 12, 2001
An Adaptive Algorithm for Computing the Propagation
of Singularities of Wave Equations
In many applied sciences, one needs to know how singularities propagate
in the solution of a wave equation, including how the wave front moves
and when a singularity first hits a specified location. A computation
of this type usually leads to a nonlinear hyperbolic system.
Unfortunately the numerical solution of such systems is often polluted
by dissipation. We will present an algorithm that appears interesting
for the following reasons:
1. It clearly shows how the wave front moves as time evolves.
2. It gives good resolution of the solution, which other methods
could not.
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