## Collective dynamics, control and imaging

### Parameter estimation for elliptic problems

Ronald DeVore

Texas A&M University
[SLIDES]

Abstract:

We consider a family of parametric elliptic PDEs \$-\div( a\grad u_a ))= f\$ on a domain \$D\$ with zero boundary conditions and diffusion coefficient \$a\$ satisfying the ellipticity condition \$r ? a ? R\$ on \$D\$ with \$r>0\$. If \$a\$ varies continuously over a parameter set \$\sc A\$, then the solution \$u_a\$ describes a solution manifold. We study the question of whether we can determine \$a\$ when \$u_a\$ is known and the related question of how well we can approximate \$a\$ when we have partial information of \$u_a\$ given by data observations. In the case that \$a\$ is uniquely determined by \$u_a\$, we study the related question of how smooth is the inverse map \$u_a\$?\$a\$.