Regularization Schemes for Linear Inverse Problems



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Recent advances in machine learning have led to breakthrough developments in many areas of the applied sciences. In many cases, traditional learning approaches are driven by data only, and typically ignore any prior physical knowledge. This can lead to overfitting or lack of generalizability, both of which result in poor predictions. Moreover, human observers cannot readily interpret the obtained results. To mitigate these obstacles, we propose to explore the integration of data-driven methods of learning with model-driven approaches. Our work is concerned with linear inverse problems, i.e., the inverse of the direct problem. In direct problems, we are concerned with the solution of a model that yields outputs for given inputs. In the inverse problem, we seek the inputs given measurable model outputs. This problem poses significant mathematical challenges, some of which can be alleviated by incorporating regularization schemes that stipulate prior knowledge about an expected solution. In the present work, we investigate how to construct appropriate regularization operators from data. We derive optimality conditions and implement different direct and iterative solvers. Specifically, we consider first and second order (curvature) information and compare the numerical solution of our schemes to a direct solution based on the pseudo-inverse of the forward operator obtained via a truncated singular value decomposition (TSVD). We then develop methodology to estimate an optimal regularization parameter. In conclusion, we demonstrate that a regularization scheme based on the right-singular vectors of the forward operator yields optimal results that are in quality similar to those obtained by TSVD.