带积分约束的线性抛物型问题源项的适定性与数值重建
Well-posedness and numerical reconstruction of a source term for linear parabolic problems with an integral constraint
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中文总结 AI 辅助
该研究针对带积分约束的线性抛物型方程源项识别问题，建立解对的适定性，开发结合有限元与隐式时间步长的数值算法，采用导数罚项Tikhonov正则化等实现准确鲁棒的源项重建。
中文摘要 AI 辅助
本研究针对定义在$\boldsymbol{R}^d$（$d\boldsymbol{\u2265}1$）区域内、受积分约束和Neumann边界条件约束的线性抛物型方程，探究与时变源项相关的识别问题。我们在抛物型Hölder空间中建立了解对的适定性与更高阶正则性。随后开发了一种基于空间有限元离散化和隐式时间步长格式的数值算法，用于重建未知源项。所得的离散逆问题涉及一个良态算子。我们表明，在该场景下，恒等Tikhonov正则化仅提供均匀、非选择性的收缩；而通过广义奇异值分解分析的、基于导数罚项的Tikhonov正则化则提供了有效的去噪策略。正则化参数采用Morozov偏差原理选取。数值误差通过完整的抛物型Hölder范数进行评估，该范数通过纳入解、其导数及相关Hölder半范数的误差，为重建提供了更全面的评估。针对光滑源项和分段常数源项的数值实验表明，在噪声水平不断提高的情况下，仍能实现准确且鲁棒的重建。
英文摘要
This work investigates a time-dependent source identification problem for linear parabolic equations subject to an integral constraint and Neumann boundary conditions in a domain of $\mathbb{R}^d$, $d\ge 1$. We establish well-posedness and higher regularity of the solution pair in parabolic Hölder spaces. A numerical algorithm based on a finite element discretization in space and an implicit time-stepping scheme is then developed for the reconstruction of the unknown source. The resulting discrete inverse problem involves a well-conditioned operator. We show that identity Tikhonov regularization provides only uniform, nonselective shrinkage in this setting, whereas Tikhonov regularization with derivative-based penalties, analyzed through the generalized singular value decomposition, provides an effective denoising strategy. The regularization parameter is selected using the Morozov discrepancy principle. Numerical errors are evaluated using full parabolic Hölder norms, which provide a more comprehensive assessment of the reconstruction by incorporating errors in the solution, its derivatives, and the associated Hölder seminorms. Numerical experiments for smooth and piecewise constant sources demonstrate accurate and robust reconstructions under increasing levels of noise.
