论文标题
$ L_P $正规化合奏Kalman倒置
$l_p$ regularization for ensemble Kalman inversion
论文作者
论文摘要
Ensemble Kalman倒置（EKI）是一种无衍生的优化方法，位于确定性和反向问题的概率方法之间。 Eki迭代了基于集合的Kalman过滤器的Kalman更新，其集合将其收敛到目标函数的最小化器。 EKI通过将合奏限制为初始集合的线性跨度或通过早期停止迭代正则化来规则规范性问题。 EKI Tikhonov Eki的另一种正规化方法使用$ L_2 $罚款项惩罚了目标函数，从而阻止了标准EKI的过度拟合。本文提出了一种实现$ L_P的策略，0 <P \ leq 1，EKI恢复解决方案中稀疏结构的正则化。该策略将$ L_P $问题转换为$ L_2 $问题，然后由Tikhonov Eki解决。转换是明确的，因此所提出的方法具有与Tikhonov Eki相当的计算成本。我们通过一系列数值实验（包括压缩传感和地下流动反向问题）来验证所提出的方法的有效性和鲁棒性。
Ensemble Kalman inversion (EKI) is a derivative-free optimization method that lies between the deterministic and the probabilistic approaches for inverse problems. EKI iterates the Kalman update of ensemble-based Kalman filters, whose ensemble converges to a minimizer of an objective function. EKI regularizes ill-posed problems by restricting the ensemble to the linear span of the initial ensemble, or by iterating regularization with early stopping. Another regularization approach for EKI, Tikhonov EKI, penalizes the objective function using the $l_2$ penalty term, preventing overfitting in the standard EKI. This paper proposes a strategy to implement $l_p, 0<p\leq 1,$ regularization for EKI to recover sparse structures in the solution. The strategy transforms a $l_p$ problem into a $l_2$ problem, which is then solved by Tikhonov EKI. The transformation is explicit, and thus the proposed approach has a computational cost comparable to Tikhonov EKI. We validate the proposed approach's effectiveness and robustness through a suite of numerical experiments, including compressive sensing and subsurface flow inverse problems.
