非局部吉洪诺夫正则化：希尔伯特尺度、显式收敛率与经典极限
Nonlocal Tikhonov Regularization: Hilbert Scales, Explicit Rates, and the Classical Limit
浏览论文内容
中文总结 AI 辅助
该研究针对有界Lipschitz区域的线性逆问题，提出非局部吉洪诺夫正则化方法，推导显式收敛率，分析其经典极限，并通过数值实验验证了该方法的有效性。
中文摘要 AI 辅助
我们在有界Lipschitz区域上研究线性逆问题的分数阶Sobolev吉洪诺夫正则化。正则化罚项由受限狄利克雷分数阶拉普拉斯算子生成，相关变分问题存在唯一极小元，且该极小元对数据的依赖是利普希茨连续的。定义正自伴算子 $A_s=I+(-\triangle)^s,\, D(A_s^{1/2})=H_0^s(\triangle)$，我们将问题等距变换为带有观测算子 $B=KA_s^{-1/2}$ 的经典希尔伯特空间吉洪诺夫问题。在赫尔德型源条件下，这给出了显式均方误差界与最优阶的先验和后验参数选取规则。该框架通过部分观测与反向分数阶热方程进行示例说明：在后一情形中，$B^*B=A_s^{-1}e^{-2tA_s}$，这可对源条件、奇异值衰减及有效重建带宽进行逐模描述。我们还研究了局部极限 $s\ o1^-$：经Bourgain-Brezis-Mironescu归一化后，分数阶泛函在 $L^2(\triangle)$ 中Γ-收敛到经典 $H_0^1$-吉洪诺夫泛函，对应极小元在 $L^2(\triangle)$ 中强收敛。反向分数阶热问题的数值实验展示了重建过程与罚项阶数的影响，通过蒙特卡洛模拟确认预测的均方收敛率在几个百分点内，且Morozov偏差原理在后验情形下达到相同的最优阶率。
英文摘要
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 $$A_s=I+(-Δ)^s,\, D(A_s^{1/2})=H_0^s(Ω),$$ we transform the problem isometrically into a classical Hilbert-space Tikhonov problem with observation operator $B=KA_s^{-1/2}$. 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, $$ B^*B=A_s^{-1}e^{-2tA_s}, $$ 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 $s\to1^-$: after Bourgain--Brezis--Mironescu normalization, the fractional functionals $Γ$-converge in $L^2(Ω)$ to the classical $H_0^1$-Tikhonov functional, and the corresponding minimizers converge strongly in $L^2(Ω)$. 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.
