Recent Developments in Inverse Problems - Abstract

Weidling, Frederic

Verification of a variational source condition for inverse medium scattering problems

We study scattering of time-harmonic acoustic waves by a medium characterized by its contrast $fin mathcal D:=fin L^infty(mathbb R^3) colon Im (f)leq 0, f(x)=0 text for lvert xrvert>1 $. The forward problem is then given by finding the emphtotal fields $u=u^mathrmi u^mathrms$ solving beginequation* Delta u kappa^2 u=kappa^2 f u quad textin mathbb R^3 labeleq:scattering endequation* where $kappa>0$ and the emphincident fields $u^mathrmi$ solve the Helmholtz equation $Delta u^mathrmi kappa^2 u^mathrmi=0$ and the emphscattered fields $u^mathrms$ fulfill the Sommerfeld radiation condition. The inverse problem we consider is to recover $f$ from near field measurements $w_f(x,y)=u_y(x)$ for all $(x,y)in partial B(R)timespartial B(R)$ where $u_y$ is the total field corresponding to the incident point source wave $u^mathrmi_y(x)=operatornameexp(mathrmilvert x-y rvert)/(4pi lvert x-yrvert)$ for a $R>1$. The problem above can be formulated as an operator equation $F(f)=g$ where $F$ is the near field operator that maps $fin mathcal D$ to $w_fin L^2((partial B(R))^2)$. Since the problem is ill-posed we use nonlinear Tikhonov regularization beginequation* f^delta_alpha in operatorname*argmin_fin mathcal Dcap X left[lVert F(f)-g^deltarVert_L^2((partial B(R))^2)^2 alpha Omega(f)right] labeleq:tikhonov endequation* to obtain a stable reconstruction from noisy measurement $g^delta$ assuming that the solution is in some Sobolev space $X$ with $Omega$ as a $X$-norm power. Using weak closedness of $F$ and a suitable parameter choice rule this is a regularization method.?hough the convergence is in general arbitrarily slow. One way to obtain convergence rate is by requiring that the true solution $f^dagger$ fulfills for a $betain [0,1)$ and a concave index function $psi$ the emphvariational source condition beginequation langle f^*,f^dagger-frangleleq beta Delta_Omega(f,f^dagger) psi left( lVert F(f^dagger)-F(f)rVert_L^2((partial B(R))^2)^2right)quad forall fin mathcal D cap X labeleq:varsorctagVSC endequation where $f^*inpartial Omega(f^dagger)$ and $Delta_Omega(cdot,cdot)$ is the Bregman distance with respect to $Omega$. With a suitable parameter choice rule this leads to the convergence rate beginequation* (1-beta) Delta_Omega(f_alpha^delta,f^dagger)leq 4 psi(delta^2). labeleq:convergence endequation* Variational source conditions have become popular in regularization theory due to a number of advantages over spectral source conditions. However, only a few results on the verification of such conditions exist, except via spectral source conditions. Using emphgeometrical optics solution, which have previously been used to establish stability estimates for inverse scattering problems, we show that the described problem fulfills eqrefeq:varsorc with $psi(t)=Cln(t^-1)^-mu$ under the assumption that $f^dagger$ is in a higher order Sobolev space, where $mu>0$ depends on the smoothness difference. This gives the first proof of logarithmic convergence rates under Sobolev smoothness for the given problem. The result can be extended to far field data.