Recent Developments in Inverse Problems - Abstract
Nair, M. Thamban
This work is in collaboration with my student Samprita Das Roy. We consider the ill-posed inverse problem of recovering the parameter function $q$ which satisfies the Neumann boundary value problem begineqnarray* - nabla . (q nabla u) &=& f,quad xin Omega
q fracpartial upartial eta &=& gquadmboxonquad partial Omega, endeqnarray* where $Omega subset mathbbR^2$ is a bounded open set with smooth boundary, $fin L^2(Omega),, gin H^-1/2(Omega)$ and $uin W^1, infty(Omega)$. For obtaining regularized approximations for this problem, we investigate the weak formulation of the problem as a linear equation $$ T_u(q) = Phi $$ for $qin H^1(Omega), uin W^1, infty(Omega)$, where $T_u(q)$ and $Phi$ are in $H^1(Omega)^*$, the dual of $H^1(Omega)$, defined by $$ T_u(q)(v):= int_Omega q(x) nabla u(x). nabla v(x) dxquad mboxandquad Phi(v) := int_Omega f(x) v(x) dx int_partial Omega g(x) v(x) dx, $$ respectively. We show that $T_u$ is a compact opeator from $H^1(Omega)$ into its dual $H^1(Omega)^*$ so that the linear equation is also ill-posed. The Tikhonov regularizatiion of this ill-posed equation and its finite dimensional realization using sequences of projections $(P_n)$ and $(Q_n)$ on $H^1(Omega)$ and $H^1(Omega)^*$, respectively, are considered when the available data is inexact, namely, $zin W^1,infty$ with $ u-z _W^1,inftyleq varepsilon$. This approach facilitates to obtain error estimates corresponding to the approximations $q_alpha, delta^(n)$, as has been done in Nair citelinop, which leads to order optimal rate under appropriate choise of the parameters $alpha$ and $n$ depending on the error level $delta>0$ of the data $u$. It is also observed that the Tikhonov regularization considered above is same as the equation error method consdered by Al-Jamal and Gockenbach citegock, wherein the error estimate is given in terms of a quotient norm which does not reflect the ill-posed nature of the problem. beginthebibliography99 bibitemlinop M.T. Nair, sl Linear Operator Equations: Approximation and Regularization, World Scientific, 2009. bibitemgock Mohammad F Al-Jamal and Mark S Gockenbach, sl Stability and error estimates for an equation error method for elliptic equations, Inverse Problems, 2012. endthebibliography