Ensemble Kalman inversion is a parallelizable methodology for solving inverse or parameter estimation problems. Although it is based on ideas from Kalman filtering, it may be viewed as a derivative-free optimization method. In its most basic form it regularizes ill-posed inverse problems through the subspace property: the solution found is in the linear span of the initial ensemble employed. In this work we demonstrate how further regularization can be imposed, incorporating prior information about the underlying unknown. In particular we study how to impose Tikhonov-like Sobolev penalties. As well as introducing this modified ensemble Kalman inversion methodology, we also study its continuous-time limit, proving ensemble collapse; in the language of multi-agent optimization this may be viewed as reaching consensus. We also conduct a suite of numerical experiments to highlight the benefits of Tikhonov regularization in the ensemble inversion context.

中文翻译：

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集成卡尔曼反演中的 Tikhonov 正则化

集成卡尔曼反演是一种用于解决逆或参数估计问题的可并行化方法。虽然它基于卡尔曼滤波的思想，但它可以被视为一种无导数的优化方法。在其最基本的形式中，它通过子空间属性对不适定的逆问题进行正则化：找到的解决方案在所采用的初始集合的线性跨度中。在这项工作中，我们展示了如何施加进一步的正则化，结合有关潜在未知的先验信息。我们特别研究了如何施加类似 Tikhonov 的 Sobolev 惩罚。在介绍这种改进的集合卡尔曼反演方法的同时，我们还研究了它的连续时间限制，证明了集合崩溃；在多智能体优化的语言中，这可以被视为达成共识。