Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks
This addresses computational efficiency in inverse problems for fields like medical imaging, though it appears incremental as it combines existing methods (finite elements and ELM).
The paper tackles parameter-dependent PDEs in inverse problems by developing a reduced-order modeling framework combining finite element discretization with parameter approximation, achieving substantial computational savings without accuracy loss in quantitative photoacoustic tomography applications.
We develop an interpolation-based reduced-order modeling framework for parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The solution is discretized in the physical domain using finite element methods, while the dependence on a finite-dimensional parameter is approximated separately. We establish existence, uniqueness, and regularity of the parametric solution and derive rigorous error estimates that explicitly quantify the interplay between spatial discretization and parameter approximation. In low-dimensional parameter spaces, classical interpolation schemes yield algebraic convergence rates based on Sobolev regularity in the parameter variable. In higher-dimensional parameter spaces, we replace classical interpolation by extreme learning machine (ELM) surrogates and obtain error bounds under explicit approximation and stability assumptions. The proposed framework is applied to inverse problems in quantitative photoacoustic tomography, where we derive potential and parameter reconstruction error estimates and demonstrate substantial computational savings compared to standard approaches, without sacrificing accuracy.