论文标题

在Langevin动力学中与正向模型的相互作用较少

Less interaction with forward models in Langevin dynamics

论文作者

Eigel, Martin, Gruhlke, Robert, Sommer, David

论文摘要

合奏方法已无处不在,用于解决贝叶斯推理问题的解决方案。最先进的Langevin采样器,例如Ensemble Kalman采样器(EKS),仿射不变的Langevin Dynamics(ALDI)或使用加权协方差估算的扩展,取决于对远期模型或其梯度的连续评估。因此,这些方法的主要缺点是它们大量所需的远程调用以及在涉及更多后验措施(例如多模态分布)的情况下可能缺乏融合。本文的目的是解决这些挑战。首先,讨论了几种可能的自适应集合富集策略,这些策略依次扩大了基础Langevin动力学中的粒子数量,从而导致正向调用总数的大幅减少。其次,为线性向前模型提供了集合富集方法的分析一致性保证。第三,为了解决更多涉及的目标分布,该方法通过基于同质性形式主义应用改编的langevin动力学来扩展该方法,该动力学证明了收敛性。最后,对几个基准问题的数值研究说明了该方法的可能增益,将其与最先进的Langevin采样器进行了比较。

Ensemble methods have become ubiquitous for the solution of Bayesian inference problems. State-of-the-art Langevin samplers such as the Ensemble Kalman Sampler (EKS), Affine Invariant Langevin Dynamics (ALDI) or its extension using weighted covariance estimates rely on successive evaluations of the forward model or its gradient. A main drawback of these methods hence is their vast number of required forward calls as well as their possible lack of convergence in the case of more involved posterior measures such as multimodal distributions. The goal of this paper is to address these challenges to some extend. First, several possible adaptive ensemble enrichment strategies that successively enlarge the number of particles in the underlying Langevin dynamics are discusses that in turn lead to a significant reduction of the total number of forward calls. Second, analytical consistency guarantees of the ensemble enrichment method are provided for linear forward models. Third, to address more involved target distributions, the method is extended by applying adapted Langevin dynamics based on a homotopy formalism for which convergence is proved. Finally, numerical investigations of several benchmark problems illustrates the possible gain of the proposed method, comparing it to state-of-the-art Langevin samplers.

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