A data-driven model reduction method for parabolic inverse source problems and its convergence analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems with uncertain data efficiently. Our consists of offline and online stages. the stage, explore low-dimensional structures in solution space partial differential equations (PDEs) forward given class functions construct small number proper orthogonal decomposition (POD) basis achieve significant dimension reduction. Equipped POD functions, can extremely fast stage. Thus, develop algorithm optimization problem problems, which is referred as method. Moreover, design an posteriori find optimal regularization parameter using proposed without knowing noise level. Under weak regularity assumption on PDEs, prove convergence solving PDEs. addition, obtain error estimate for problems. Finally, present numerical examples demonstrate accuracy efficiency Numerical results show that provides considerable computational savings over finite element while maintaining same accuracy.