The ensemble Kalman inversion (EKI), as a derivative-free methodology, has been widely used in the parameter estimation of inverse problems. Unfortunately, its cost may become moderately large for systems described by high dimensional nonlinear PDEs, as EKI requires a relatively large ensemble size to guarantee its performance. In this paper, we propose an adaptive multi-fidelity polynomial chaos (PC) based EKI technique to address this challenge. Our new strategy combines a large number of low-order PC surrogate model evaluations and a small number of high-fidelity forward model evaluations, yielding a multi-fidelity approach. Especially, we present a new approach that adaptively constructs and refines a multi-fidelity PC surrogate during the EKI simulation. Since the forward model evaluations are only required for updating the low-order multi-fidelity PC model, whose number can be much smaller than the total ensemble size of the classic EKI, the entire computational costs are thus significantly reduced. The new algorithm was tested through the two-dimensional time fractional inverse diffusion problems and demonstrated great effectiveness in comparison with PC based EKI and classic EKI.

中文翻译：

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用于逆问题的自适应多保真基于 PC 的集成卡尔曼反演

集成卡尔曼反演（EKI）作为一种无导数方法，已广泛用于反问题的参数估计。不幸的是，对于由高维非线性偏微分方程描述的系统，它的成本可能会相当大，因为 EKI 需要相对较大的集合大小来保证其性能。在本文中，我们提出了一种基于自适应多保真多项式混沌 (PC) 的 EKI 技术来应对这一挑战。我们的新策略结合了大量低阶 PC 代理模型评估和少量高保真前向模型评估，产生了多保真方法。特别是，我们提出了一种新方法，可以在 EKI 模拟过程中自适应地构建和完善多保真 PC 代理。由于前向模型评估只需要更新低阶多保真 PC 模型，其数量可以远小于经典 EKI 的总集合大小，因此整个计算成本显着降低。新算法通过二维时间分数逆扩散问题进行了测试，与基于 PC 的 EKI 和经典 EKI 相比显示出极大的有效性。