Nonlocal Tikhonov Regularization: Hilbert Scales, Explicit Rates, and the Classical Limit

Abstract

We study fractional-Sobolev Tikhonov regularization for linear inverse problems on a bounded Lipschitz domain. The regularization penalty is generated by the restricted Dirichlet fractional Laplacian, and the associated variational problem is shown to admit a unique minimizer that depends Lipschitz continuously on the data. Identifying the positive self-adjoint operator $$A_s=I+(-\Delta {)}^s,D(A_s^{1/2})=H_0^s(\Omega ),$$ we transform the problem isometrically into a classical Hilbert-space Tikhonov problem with observation operator $B=K{A}_s^{-1/2}$. This yields explicit mean-square error bounds and an order-optimal \emph{a priori} and \emph{a posteriori} parameter rules under Hölder-type source conditions. The framework is illustrated by partial observations and by the backward fractional heat equation. In the latter case, $$B^{\ast }B=A_s^{-1}{e}^{-2t{A}_s},$$ which permits a mode-wise description of the source condition, the singular-value decay, and the effective reconstruction bandwidth. We also study the local limit $s\to 1^{-}$: after Bourgain--Brezis--Mironescu normalization, the fractional functionals $\Gamma$-converge in $L^2(\Omega )$ to the classical $H_0^1$-Tikhonov functional, and the corresponding minimizers converge strongly in $L^2(\Omega )$. Numerical experiments for the backward fractional heat problem illustrate the reconstruction procedure and the influence of the penalty order, and confirm the predicted mean-square convergence rate to within a few percent via Monte Carlo simulation, with Morozov's discrepancy principle attaining the same order-optimal rate a posteriori.

30 pages