分数阶热方程初值从薄集观测恢复的对数稳定性

中文摘要 AI 辅助

我们研究了在有界域上从两类可能为Lebesgue零集的观测中恢复分数阶热方程初值的反问题：第一类由Hausdorff维数超过\\(n-1\\)的一般薄集组成；第二类由特殊构造的零Hausdorff维数集合组成，这些集合基于代数无理点并快速累积。通过将薄集可观测性不等式从经典热方程推广到分数阶情形，并采用指数为\\(\beta\in(0,s)\\)的统一谱不等式，我们为每个分数阶指数\\(s>1/2\\)建立了定量可观测性估计。对于一般薄集，有\\(\beta=1/2\\)；对于特殊的零维集合，任何\\(\beta\in(1/2,s)\\)都是允许的，但短时可观测性代价始终保持指数型，其速率由\\(\beta\\)相对于\\(s\\)决定。在先验光滑性假设下，我们还证明了对数稳定性结果。我们进一步设计了一个正则化最小二乘重建算法，并提供了条件收敛性分析，表明重建误差由谱截断项和由连续稳定性决定的对数项之和界定，且当噪声水平趋于零时误差对数衰减。使用极少观测点的数值模拟证实了所提出方法的可行性。

英文摘要

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 sequences.By extending thin-set observability inequalities from the classical heat equation to the fractional setting and employing a unified spectral inequality with exponent \(β\in(0,s)\), we establish a quantitative observability estimate for every fractional exponent \(s>1/2\). For general thin sets one has \(β=1/2\); for the special zero-dimensional sets any \(β\in(1/2,s)\) is admissible, yet the short-time observability cost always remains of exponential type, with the rate governed by \(β\) relative to \(s\). Under an a priori smoothness assumption, we also prove a logarithmic stability estimate.We 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.

发表机构

• School of Mathematics and Statistics, Ningbo University（宁波大学数学与统计学院）

机构由 AI 辅助整理，请以论文原文为准。