Complete Deterministic Dynamics and Spectral Decomposition of the Linear Ensemble Kalman Inversion

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2023-03-15 DOI:10.1137/21m1429461
Leon Bungert, Philipp Wacker
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引用次数: 1

Abstract

The ensemble Kalman inversion (EKI) for the solution of Bayesian inverse problems of type , with being an unknown parameter, a given datum, and measurement noise, is a powerful tool usually derived from a sequential Monte Carlo point of view. It describes the dynamics of an ensemble of particles , whose initial empirical measure is sampled from the prior, evolving over an artificial time toward an approximate solution of the inverse problem, with emulating the posterior, and corresponding to the underregularized minimum-norm solution of the inverse problem. Using spectral techniques, we provide a complete description of the deterministic dynamics of EKI and its asymptotic behavior in parameter space. In particular, we analyze the dynamics of naive EKI and mean-field EKI with a special focus on their time asymptotic behavior. Furthermore, we show that—even in the deterministic case—residuals in parameter space do not decrease monotonously in the Euclidean norm and suggest a problem-adapted norm, where monotonicity can be proved. Finally, we derive a system of ordinary differential equations governing the spectrum and eigenvectors of the covariance matrix. While the analysis is aimed at the EKI, we believe that it can be applied to understand more general particle-based dynamical systems.
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线性集合卡尔曼反演的完全确定性动力学和谱分解
集合卡尔曼反演(EKI)用于求解具有未知参数、给定基准和测量噪声的贝叶斯反问题,通常是从顺序蒙特卡罗的角度衍生出来的强大工具。它描述了粒子集合的动力学,其初始经验测量从先验中采样,在人工时间内向反问题的近似解演化,模拟后验,并对应于反问题的未正则化最小范数解。利用谱技术,我们给出了EKI的确定性动力学及其在参数空间中的渐近行为的完整描述。特别地,我们分析了朴素EKI和平均场EKI的动力学,特别关注它们的时间渐近行为。此外,我们表明,即使在确定性情况下,参数空间的残差在欧几里得范数中也不会单调减少,并提出了一个问题适应范数,其中单调性可以证明。最后,我们导出了一个控制协方差矩阵的谱和特征向量的常微分方程组。虽然分析的目标是EKI,但我们相信它可以应用于理解更一般的基于粒子的动力系统。
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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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