In this article, the mathematical study of the problem of identifying the space-dependent source term, in transport processes given by an -dimensional linear parabolic equation, from space-dependent noisy measurements taken at an arbitrary fixed time is conducted. The problem is analytically solved using Fourier techniques, and it is shown that this solution is not stable. Three families of uniparametric regularization operators are proposed to address the instability of the solution. Each of them is designed to compensate for the factor that causes the instability of the inverse operator. Additionally, a selection rule for the regularization parameter is included, and an error bound for each estimation of Hölder type is obtained. Numerical examples of different characteristics are presented to demonstrate the benefits of the proposed strategies.

中文翻译：

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使用空间相关噪声数据估计 n 维线性抛物线方程中空间相关源的正则化技术


在本文中，对从任意固定时间进行的空间相关噪声测量中识别空间相关源项的问题进行了数学研究，该传输过程由一维线性抛物线方程给出。使用傅里叶技术对该问题进行解析求解，结果表明该解不稳定。提出了三个单参数正则化算子系列来解决解决方案的不稳定性。它们中的每一个都是为了补偿导致逆算子不稳定的因素而设计的。此外，还包括正则化参数的选择规则，并获得 Hölder 类型的每个估计的误差界限。提出了不同特征的数值示例来证明所提出策略的好处。