A Novel Model Reduction Method for Parabolic Inverse Problems Without Inverse Crime

Abstract

<p>In this paper, we propose a novel proper orthogonal decomposition (POD)-based method that effectively mitigates the issue of inverse crime in solving parabolic inverse problems using model reduction methods. We apply this approach to both inverse source and initial value problems. By leveraging the low-dimensional structures inherent in the solution space of parabolic equations and constructing POD basis functions, our method significantly reduces computational costs while maintaining accuracy. We also provide a convergence analysis of the proposed methods for these two types of parabolic inverse problems. Finally, we conduct numerical experiments to demonstrate the accuracy and efficiency of the proposed method. Numerical results show that our method efficiently solves parabolic inverse problems and overcomes the inverse crime issues associated with traditional model reduction methods for such problems.<br></p>