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 $u_a$?$a$. |