Model Reduction of Linear Dynamical Systems via Balancing for Bayesian Inference

by   Elizabeth Qian, et al.

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 reduction offers a variety of computational tools that seek to reduce this computational burden. In particular, balanced truncation is 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.


page 1

page 2

page 3

page 4


Time-limited Balanced Truncation for Data Assimilation Problems

Balanced truncation is a well-established model order reduction method i...

Gramians, Energy Functionals and Balanced Truncation for Linear Dynamical Systems with Quadratic Outputs

Model order reduction is a technique that is used to construct low-order...

Gradient-based data and parameter dimension reduction for Bayesian models: an information theoretic perspective

We consider the problem of reducing the dimensions of parameters and dat...

A time domain a posteriori error bound for balancing-related model order reduction

The aim in model order reduction is to approximate an input-output map d...

Full state approximation by Galerkin projection reduced order models for stochastic and bilinear systems

In this paper, the problem of full state approximation by model reductio...

Deep Learning Aided Laplace Based Bayesian Inference for Epidemiological Systems

Parameter estimation and associated uncertainty quantification is an imp...

Efficient Learning of a Linear Dynamical System with Stability Guarantees

We propose a principled method for projecting an arbitrary square matrix...

Please sign up or login with your details

Forgot password? Click here to reset