**Numerical and Computational Challenges
in Science and Engineering Program**

*LECTURE*S

### 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.

Back to Thematic
Year Index