Model Reduction of Linear Dynamical Systems via Balancing for Bayesian Inference
Model Reduction of Linear Dynamical Systems via Balancing for Bayesian Inference
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DOI：
10.1007/s10915-022-01798-8
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发表时间：
2021-11
影响因子：
2.5
通讯作者：
E. Qian;Jemima M. Tabeart;C. Beattie;S. Gugercin;Jiahua Jiang;P. Kramer;A. Narayan
中科院分区：
文献类型：
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作者：
E. Qian;Jemima M. Tabeart;C. Beattie;S. Gugercin;Jiahua Jiang;P. Kramer;A. Narayan
我们认为贝叶斯方法的线性高斯推断问题的初始条件的线性动力系统的噪声输出测量后的初始时间。在实际应用中，动力系统状态的维数很大，这给计算精确的后验分布带来了计算上的困难，模型降阶提供了各种计算工具来减少这一计算负担。特别是平衡截断是一种系统理论的方法来降低模型，获得一个有效的降维动力系统，通过投影系统运营商的状态方向，权衡可达性和可观测性的状态方向表示通过相关的Gramian。我们引入Gramian定义相关的推理设置，并提出了一个平衡的截断方法，这些推理Gramian的基础上，产生一个减少的动力系统，可以用来便宜地近似后验均值和协方差。我们的定义利用了（i）可达性Gramian和先验协方差以及（ii）可观测性Gramian和Fisher信息之间的自然联系。由此产生的减少模型，然后继承系统理论的考虑，稳定性和误差界，并在某些设置产生一个最佳的后验协方差近似。对模型降阶中的两个基准问题的数值实验表明，该方法可以得到近似最优的后验协方差近似，并具有数量级的状态降维。
We consider the Bayesian approach to the linear Gaussian inference problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribution.Model reductionoffers a variety of computational tools that seek to reduce this computational burden. In particular,balanced truncationis a system-theoretic approach to model reduction which obtains an efficient reduced-dimension dynamical system by projecting the system operators onto state directions which trade off the reachability and observability of state directions as expressed through the associated Gramians. We introduce Gramian definitions relevant to the inference setting and propose a balanced truncation approach based on these inference Gramians that yield a reduced dynamical system that can be used to cheaply approximate the posterior mean and covariance. Our definitions exploit natural connections between (i) the reachability Gramian and the prior covariance and (ii) the observability Gramian and the Fisher information. The resulting reduced model then inherits stability properties and error bounds from system theoretic considerations, and in some settings yields an optimal posterior covariance approximation. Numerical demonstrations on two benchmark problems in model reduction show that our method can yield near-optimal posterior covariance approximations with order-of-magnitude state dimension reduction.