Convergence Estimates of Regularized Solutions to Inverse Space-Dependent Source Problems with Time-Dependent Boundary Measurement

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2025-07-04 DOI:10.1137/24m1692885

Chunlong Sun, Wenlong Zhang

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Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1369-1397, August 2025.
Abstract. In this work, we investigate the Tikhonov-type regularized solutions and their finite element solutions to the inverse space-dependent source problem from boundary measurement data. First, with the classical source condition, we establish the convergence of regularized solutions and their finite element solutions under the standard [math] norm. The error estimates present explicit dependence on the critical parameters like noise level, regularization parameter, mesh size, and time step size. Next, based on a proposed weak norm, we get the stability of Lipschitz type for the inverse problem, and then the first order convergence of regularized solutions can be derived in the sense of weak norm. We get this convergence without any source condition. Moreover, this work is carried out for the discrete data. We suppose that the observation points are discrete and the pointwise measurement data come with independent sub-Gaussian random noises. Then we give the stochastic convergence of regularized solutions and propose an efficient iterative algorithm to determine the optimal regularization parameter. Numerical experiments are presented to demonstrate the effectiveness of the proposed algorithms.

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具有时变边界测量的逆空间依赖源问题正则解的收敛性估计

SIAM数值分析杂志，第63卷，第4期，1369-1397页，2025年8月。摘要。在这项工作中，我们研究了基于边界测量数据的逆空间依赖源问题的tikhonov型正则解及其有限元解。首先，利用经典的源条件，建立了正则解及其在标准范数下的有限元解的收敛性。误差估计明确地依赖于关键参数，如噪声水平、正则化参数、网格大小和时间步长。其次，在弱范数的基础上，我们得到了反问题的Lipschitz型的稳定性，并在弱范数的意义上推导了正则化解的一阶收敛性。我们在没有任何源条件的情况下得到这个收敛性。此外，这项工作是针对离散数据进行的。我们假设观测点是离散的，逐点测量数据具有独立的亚高斯随机噪声。然后给出了正则化解的随机收敛性，并提出了一种确定最优正则化参数的有效迭代算法。数值实验验证了所提算法的有效性。

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.