Model Reduction of Linear Dynamical Systems via Balancing for Bayesian Inference

Abstract

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.

Appendices

Appendix A Proof of Proposition 4.1

Proof

The Eq. (4.1) can be solved by integrating factors in the standard way because the stochastic product rule applied to $\mathrm {d}(\mathrm {e}^{\varvec{A}t}x)$ produces no corrections:

$$\begin{aligned} \varvec{x}(t)=\int _{-\infty }^t \mathrm {e}^{\varvec{A}(t-s)} \varvec{B}\, \mathrm {d}\varvec{W}(s) \end{aligned}$$

With the Ito isometry, which is equivalent to the formal rule $\mathbb {E}[\mathrm {d}\varvec{W}(s) \mathrm {d}\varvec{W}(s')] = I \delta (s-s') \, \mathrm {d}s \, \mathrm {d}s'$, we compute:

$$\begin{aligned} \mathbb {E}[\varvec{x}(t)\varvec{x}^\top (t)]&= \int _{-\infty }^t \mathrm {e}^{\varvec{A}(t-s)} \varvec{B}(\mathrm {e}^{\varvec{A}(t-s)}\varvec{B})^\top \, \mathrm {d}s = \int _{-\infty }^t \mathrm {e}^{\varvec{A}(t-s)} \varvec{B}\varvec{B}^\top \mathrm {e}^{\varvec{A}^\top (t-s)} \, \mathrm {d}s \nonumber \\&= \int _0^\infty \mathrm {e}^{\varvec{A}s} \varvec{B}\varvec{B}^\top \mathrm {e}^{\varvec{A}^\top s} \, \mathrm {d}s, \end{aligned}$$

(6.1)

which also shows that $\mathrm {d}\mathbb {E}[\varvec{x}\varvec{x}^\top ] = 0$. The method of moments gives:

$$\begin{aligned} d(\varvec{x}\varvec{x}^\top )&= (\varvec{A}\varvec{x}\,\mathrm {d}t + \varvec{B}\, \mathrm {d}\varvec{W}(t))x^\top + \varvec{x}( \varvec{x}^\top \varvec{A}^\top + \mathrm {d}\varvec{W}^\top (t) \varvec{B}^\top ) + \varvec{B}\,\mathrm {d}\varvec{W}(t) \mathrm {d}\varvec{W}(t)^\top \varvec{B}^\top \nonumber \\&= \varvec{A}\varvec{x}\varvec{x}^\top \, \mathrm {d}t + \varvec{x}\varvec{x}^\top \varvec{A}^\top \, \mathrm {d}t + \varvec{B}\,\mathrm {d}\varvec{W}(t) \varvec{x}^\top + \varvec{x}\,\mathrm {d}\varvec{W}(t)^\top \varvec{B}^\top + \varvec{B}\varvec{B}^\top \, \mathrm {d}t \end{aligned}$$

(6.2)

Taking expectations and using $\mathbb {E}[\mathrm {d}\varvec{W}x]=0$, we recover the desired Lyapunov Eq. (4.3):

$$\begin{aligned} \mathrm {d}\mathbb {E}[\varvec{x}\varvec{x}^\top ] = 0 = (\varvec{A}\mathbb {E}[\varvec{x}\varvec{x}^\top ] + \mathbb {E}[\varvec{x}\varvec{x}^\top ] \varvec{A}^\top + \varvec{B}\varvec{B}^\top ) \, \mathrm {d}t. \end{aligned}$$

$\square$

Appendix B Optimality Property of Mahalanobis Distance for Gaussian Distributions

Proposition B.1

The neighborhood $M\equiv \{\varvec{z}\in \mathbb {R}^d: D\left( \varvec{z}, \mathcal {N}( \varvec{\mu },\varSigma )\right) < \delta \}$ defined by the Mahalanobis distance (3.1) satisfies the following optimality condition:

$$\begin{aligned} M= \mathop {\mathrm{argmax}}\limits _{S \in \mathcal {O}: \lambda (S)=\lambda (M)} {\mathbb {P}}(S). \end{aligned}$$

where ${\mathbb {P}}$ denotes the probability measure on $\mathbb {R}^d$ corresponding to a multivariate Gaussian distribution with mean $\varvec{\mu }\in \mathbb {R}^d$ and covariance matrix $\varSigma \in \mathbb {R}^{d\times d}$, $\mathcal {O}$ is the collection of open sets in $\mathbb {R}^d$ and $\lambda (S)$ denotes the Lebesgue measure of set S.

Proof

M can be verified by definition to correspond to an ellipsoidal level set of the multivariate Gaussian density, with the probability density strictly greater on the interior of M than on the complement. Consequently, for any $S \in \mathcal {O}$, the probability density on $S \setminus M$ is strictly greater than the probability density on $M \setminus S$. If M and S have the same Lebesgue measure, then so do $S \setminus M$ and $M \setminus S$. Thus $P(S \setminus M) \le P(M\setminus S)$, with equality holding only if $P(S \setminus M)=0$, which can only happen if $S=M$ since both sets are open. $\square$

Appendix C Stochastic Forcing Continuing Through Positive Times

We consider here how the connection between balanced truncation and the optimal low-rank posterior update procedure described in [54] is affected if we were to consider the more natural case of a stochastic linear dynamical system driving consistently for both positive and negative times:

$$\begin{aligned} \mathrm {d}\varvec{x}= \varvec{A}\varvec{x}\, \mathrm {d}t + \varvec{B}\,\mathrm {d}\varvec{W}(t), \qquad -\infty< t < \infty . \end{aligned}$$

(6.3)

Now the observations involve noise not only from the measurement process, but from the dynamical noise at previous times. We can write in place of Eq. (2.2):

$$\begin{aligned} \varvec{m}_i&= \varvec{C}\mathrm {e}^{\varvec{A}t_i} \varvec{x}_0+ \varvec{\var** }_i, \end{aligned}$$

where

$$\begin{aligned} \varvec{\var** }_i= \int _0^{t_i} \varvec{C}\mathrm {e}^{\varvec{A}(t_i-s)} \varvec{B}\, d \varvec{W}(s) + \varvec{\epsilon }_i. \end{aligned}$$

In the expression of the abstract linear Bayesian inference problem:

$$\begin{aligned} \varvec{m}= \varvec{G}\varvec{x}_0+ \varvec{\var** }, \end{aligned}$$

the noisy component $\varvec{\var** }\in \mathbb {R}^{d_{\mathrm{obs}}}$ of the measurements is now correlated across observation times. Thus, $\varvec{\var** }\sim \mathcal {N}(0,\varvec{\varGamma }_{\mathrm {tot}})$ where $\varvec{\varGamma }_{\mathrm {tot}}\in \mathbb {R}^{d_{\mathrm{obs}}\times d_{\mathrm{obs}}}$ has block structure:

$$\begin{aligned} \varvec{\varGamma }_{\!\mathrm {obs}}= \begin{bmatrix} \varvec{\varGamma }_{\varvec{\epsilon }_1,\varvec{\epsilon }_1} &{} \varvec{\varGamma }_{\varvec{\epsilon }_1,\varvec{\epsilon }_2} &{} \cdots &{} \varvec{\varGamma }_{\varvec{\epsilon }_1,\varvec{\epsilon }_{n}} \\ \varvec{\varGamma }_{\varvec{\epsilon }_2,\varvec{\epsilon }_1} &{} \varvec{\varGamma }_{\varvec{\epsilon }_2,\varvec{\epsilon }_2} &{} \cdots &{} \varvec{\varGamma }_{\varvec{\epsilon }_2,\varvec{\epsilon }_{n}} \\ \vdots &{} \vdots &{} \cdots &{} \vdots \\ \varvec{\varGamma }_{\varvec{\epsilon }_{n},\varvec{\epsilon }_1} &{} \varvec{\varGamma }_{\varvec{\epsilon }_{n},\varvec{\epsilon }_2} &{} \cdots &{} \varvec{\varGamma }_{\varvec{\epsilon }_{n},\varvec{\epsilon }_{n}} \end{bmatrix}. \end{aligned}$$

(6.4)

with covariances of measurements at different times given explicitly by;

$$\begin{aligned} \varvec{\varGamma }_{\varvec{\epsilon }_i,\varvec{\epsilon }_j} = \varvec{\varGamma }_{\varvec{\epsilon }_i}\delta _{ij}+ \int _{|t_i-t_j|}^{\max (t_i,t_j)} \varvec{C}\mathrm {e}^{\varvec{A}s} {\varvec{BB}}^\top \mathrm {e}^{\varvec{A}^\top s}\varvec{C}^\top \, \mathrm {d}s \end{aligned}$$

with Kronecker delta function $\delta _{ij}$. The reason for the correlations is the influence of the common past driving noise. These correlations are indeed mitigated if the observation times are spaced sufficiently far apart so the minimal eigenvalue of $-A |t_i-t_j|$ becomes large, but this is not a terribly interesting case as it would imply also weak dependence of the observations on the state $\varvec{x}_0$. The correlation in the effective measurement noise would lead to a very different expression for the Fisher information matrix in Eq. (2.8)), as $\varvec{\varGamma }_{\!\mathrm {obs}}^{-1}$ would no longer be block diagonal, and we would thereby lose interpretability of the Fisher information matrix as a discretized version of an observability Gramian.

Appendix D Modification of Covariance to Induce Prior-Compatibility

This appendix complements the discussion in Sect. 4.1.2: We provide below a Matlab routine that, given a system matrix A and a prior covariance Gamma0, returns the upper-triangular Cholesky factor R_Gam of a modified prior that is compatible with the state dynamics of A. Our matlab function listed below utilizes the lyapchol routine from the Control System Toolbox to solve (4.5) for the Cholesky factor E of the prior modification $\varDelta$.