Time-limited Balanced Truncation for Data Assimilation Problems

TL;DR

This paper discusses the application of balanced truncation to linear Gaussian Bayesian inference, and, in particular, the 4D-Var method, thereby strengthening the link between systems theory and data assimilation further.

Abstract

Balanced truncation is a well-established model order reduction method which has been applied to a variety of problems. Recently, a connection between linear Gaussian Bayesian inference problems and the system-theoretic concept of balanced truncation has been drawn. Although this connection is new, the application of balanced truncation to data assimilation is not a novel idea: it has already been used in four-dimensional variational data assimilation (4D-Var). This paper discusses the application of balanced truncation to linear Gaussian Bayesian inference, and, in particular, the 4D-Var method, thereby strengthening the link between systems theory and data assimilation further. Similarities between both types of data assimilation problems enable a generalisation of the state-of-the-art approach to the use of arbitrary prior covariances as reachability Gramians. Furthermore, we propose an enhanced approach using time-limited balanced truncation that allows to balance Bayesian inference for unstable systems and in addition improves the numerical results for short observation periods.

Figures (5)

• Figure 1: Comparison of a given non-compatible prior (NC) and a modified compatible prior (C) in balanced truncation (BT) Qian2021Balancing and the optimal low-rank approach (OLR) Spantini2015Optimal of LG Bayesian inference for the heat equation. The first letters denote the prior used for the approx. posterior computation and the second the one used for the reference. Measurements are spaced $h=0.005$ apart inside $\mathcal{T} = [0,10]$
• Figure 2: Comparison of time-limited balanced truncation (TLBT) with standard balanced truncation (BT) Qian2021Balancing and the optimal low-rank approach (OLR) Spantini2015Optimal of LG Bayesian inference for the heat equation model. Measurements are spaced $h=0.005$ apart inside $\mathcal{T} = [0,t_e]$ for three different end times $t_e = 1,3,10$. In the left panel, the normalised square roots $\delta_i$ and $\delta^{TL}_i$ of the generalised eigenvalues of the matrix pencils $(\pmb{Q}_{\infty}^{\text{LG}}, \pmb{\Gamma}_{\mathrm{pr}}^{-1})$ and $(\pmb{Q}_{\mathcal{T}}^{\text{LG}}, \pmb{\Gamma}_{\mathrm{pr}}^{-1})$ are plotted, corresponding to the Hankel singular values
• Figure 3: Quantities plotted as in Figure \ref{['3x3 heat']}; here for the ISS1R model. Measurements are equispaced with $h=0.1$ inside $\mathcal{T} = [0,t_e]$ for three different end times $t_e = 1,3,10$
• Figure 4: Comparison of time-limited balanced truncation (TLBT) and balanced truncation with Fisher matrix $\pmb{H}$ as observability Gramian (BT-H) Qian2021Balancing and the optimal low-rank approach (OLR) Spantini2015Optimal of LG Bayesian inference for the unstable discretised advection-diffusion equation; $d = 200$. Measurements are spaced $h=0.001$ apart inside $\mathcal{T} = [0,t_e]$ for three different end times $t_e = 0.1,0.5,1$
• Figure 5: Model and plotted quantities as in Figure \ref{['3x3 AdDiff']}; here for $d = 1200$. Measurements are equispaced with $h=0.001$ inside $\mathcal{T} = [0,t_e]$ for three different end times $t_e = 0.1,0.5,1$

Theorems & Definitions (8)

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