Logarithmic stability for recovering initial data of fractional heat equations from thin-set observations

RESUMO

We study the inverse problem of recovering initial data for fractional heat equations on bounded domains from observations taken on two types of possibly Lebesgue-null sets: the first consists of general thin sets whose Hausdorff dimension exceeds \(n-1\); the second consists of specially constructed sets of zero Hausdorff dimension, built from algebraic irrational points and rapidly accumulating this http URL extending thin-set observability inequalities from the classical heat equation to the fractional setting and employing a unified spectral inequality with exponent \(\beta\in(0,s)\), we establish a quantitative observability estimate for every fractional exponent \(s>1/2\). For general thin sets one has \(\beta=1/2\); for the special zero-dimensional sets any \(\beta\in(1/2,s)\) is admissible, yet the short-time observability cost always remains of exponential type, with the rate governed by \(\beta\) relative to \(s\). Under an a priori smoothness assumption, we also prove a logarithmic stability this http URL further design a regularised least-squares reconstruction algorithm and provide a conditional convergence analysis, showing that the reconstruction error is bounded by the sum of a spectral truncation term and a logarithmic term dictated by the continuous stability, and that the error decays logarithmically as the noise level tends to zero. Numerical simulations using very few observation points confirm the feasibility of the proposed approach.