In this paper, we propose novel proper orthogonal decomposition (POD)--based model reduction methods that effectively address the issue of inverse crime in solving parabolic inverse problems. Both the inverse initial value problems and inverse source problems are studied. By leveraging the inherent low-dimensional structures present in the data, our approach enables a reduction in the forward model complexity without compromising the accuracy of the inverse problem solution. Besides, we prove the convergence analysis of the proposed methods for solving parabolic inverse problems. Through extensive experimentation and comparative analysis, we demonstrate the effectiveness of our method in overcoming inverse crime and achieving improved inverse problem solutions. The proposed POD model reduction method offers a promising direction for improving the reliability and applicability of inverse problem-solving techniques in various domains.

翻译：本文提出了一种基于本征正交分解(POD)的新型模型降阶方法，可有效解决抛物型反问题求解中的"反演犯罪"问题。研究同时涵盖了反初值问题与反源项问题。通过利用数据中固有的低维结构，本方法能够在保持反问题求解精度的前提下降低正演模型的复杂度。此外，我们证明了所提方法在求解抛物型反问题中的收敛性。通过大量实验与对比分析，验证了本方法在克服反演犯罪现象及提升反问题求解精度方面的有效性。所提出的POD模型降阶方法为提升各领域反问题求解技术的可靠性与适用性提供了新的研究方向。