Efficient learning methods for large-scale optimal inversion design
Efficient learning methods for large-scale optimal inversion design
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DOI：
10.3934/naco.2022036
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发表时间：
2021-10
期刊：
影响因子：
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通讯作者：
Julianne Chung;Matthias Chung;S. Gazzola;M. Pasha
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文献类型：
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作者：
Julianne Chung;Matthias Chung;S. Gazzola;M. Pasha
在这项工作中，我们研究了使用从训练数据中学习来解决逆问题的各种方法，遵循两级学习方法。我们考虑了最优反演设计的一般框架，其中训练数据可用于学习最优正则化参数、数据保真项和正则化，从而产生更好的变分正则化方法。特别地，我们描述了L正则化的最优p范数和q正则化的最优范数的学习方法以及协方差核定义的正则化矩阵的最优参数的学习方法。我们开发了基于Krylov投影方法的高效算法来解决正则化问题，无论是在训练阶段还是验证阶段，使这些方法非常适合于大规模问题。我们的实验表明，即使在正演算子中存在一些不准确，导致模型和测量误差混合的情况下，学习的正则化方法也能很好地执行。
In this work, we investigate various approaches that use learning from training data to solve inverse problems, following a bi-level learning approach. We consider a general framework for optimal inversion design, where training data can be used to learn optimal regularization parameters, data fidelity terms, and regularizers, thereby resulting in superior variational regularization methods. In particular, we describe methods to learn optimal $p$ and $q$ norms for ${\rm L}^p-{\rm L}^q$ regularization and methods to learn optimal parameters for regularization matrices defined by covariance kernels. We exploit efficient algorithms based on Krylov projection methods for solving the regularized problems, both at training and validation stages, making these methods well-suited for large-scale problems. Our experiments show that the learned regularization methods perform well even when there is some inexactness in the forward operator, resulting in a mixture of model and measurement error.