Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise
The authors are concerned with the parameter identification problems where \(a\) is an unknown parameter in an elliptic differential equation (assumed to belong to a separable Hilbert space \(\mathcal{X}\) which is not directly observable) and \(u\) is the solution to a boundary value problem associated with the given equation. It is only possible to access \(u,\) which belongs to another Hilbert space \(\mathcal{Y},\) and the relation between \(a\) and \(u\) is described in terms of so-called parameter-to-solution (nonlinear) operator \(F:D( F) \to\mathcal{Y},\) \(F(a)=u,\) where \(D( F) \subset\mathcal{X}.\) The exact solution to the problem is \(a_{\dag }\in D( F) \) and the exact data is \(u_{\dag}=F( a_{\dag }) .\) It is assumed that \(u_{\dag}\) is not given explicitly, but through a random variable \(\hat{u}\in\mathcal{Y}\) satisfying \[ \sqrt{\mathbf{E}\| \hat{u}-u_{\dag}\| _{\mathcal{Y}}^{2}} \leq\delta. \] The estimators are defined by means of the Tikhonov functional \[ \hat{a}=\arg\min_{a\in D(F)}( \| F(a)-\hat{u}\| _{\mathcal{Y}}^{2}+\alpha\| a-a_{0}\| _{\mathcal{X}} ^{2}), \] where \(\alpha>0\) is a regularization parameter and \(a_{0}\in\mathcal{X}\) stands for some initial guess of \(a_{\dag}.\) The authors derive estimates on the expected squared error under certain assumptions on the forward operator. They study reconstruction of a reaction coefficient and a diffusion coefficient in two inverse elliptic boundary value problems and demonstrate that assumptions of their convergence result are met in both cases. Numerical experiments illustrating the effect of the smoothness of the parameter \(a_{\dag}\) on the convergence rate of \(\hat{u}\) are provided.
inverse elliptic boundary value problems
scientific article; zbMATH DE number 5500621