H\"older Stability from Exact Uniqueness for Finite-Dimensional Analytic Inverse Problems

Abstract

We prove a stability theorem for finite-dimensional analytic inverse problems. Let \(U\subset\R^m\) be an open parameter set, let \(F(p)\) be a boundary measurement operator, and let \(R(p)\) be the finite-dimensional quantity to be recovered. If \(F\) is real analytic and \[ F(p)=F(q)\quad\Longrightarrow\quad R(p)=R(q), \] then \(R\) satisfies a H\"older stability estimate on every compact subset of \(U\). The proof uses a Hilbert--Schmidt scalarization of the operator equation \(F(p)=F(q)\) and the \L{}ojasiewicz distance inequality. We also prove that, after fixing countable dense families of boundary inputs and tests, finitely many scalar matrix elements of the data give the same H\"older recovery on compact parameter sets. This finite-measurement conclusion is qualitative: the proof does not give an effective measurement list, exponent, or constant. The finite-measurement statement follows from finite determinacy of real analytic zero sets. We apply the result to local Neumann-to-Dirichlet data for piecewise constant anisotropic conductivities and to localized Dirichlet-to-Neumann data for piecewise homogeneous anisotropic elasticity.