PBDW method for state estimation: error analysis for noisy data and nonlinear formulation

TL;DR

This work analyzes the Parameterized-Background Data-Weak (PBDW) framework for state estimation under noisy measurements, introducing a constrained background approach that yields nonlinear PBDW and improves robustness. It provides a rigorous a priori error analysis for linear PBDW, a stability bound for the nonlinear case, and optimality results that connect PBDW to classical recovery theory. The authors introduce and study constants that quantify sensitivity to noise, model bias, and data misfit, and they demonstrate how box-constrained backgrounds and SGreedy sensor placement affect performance. Through 2D and 3D numerical experiments, the paper shows that nonlinear PBDW with background constraints substantially enhances stability and accuracy in the presence of noise, offering practical guidelines for parameter and sensor configuration.

Abstract

We present an error analysis and further numerical investigations of the Parameterized-Background Data-Weak (PBDW) formulation to variational Data Assimilation (state estimation), proposed in [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933-965]. The PBDW algorithm is a state estimation method involving reduced models. It aims at approximating an unknown function $u^{\rm true}$ living in a high-dimensional Hilbert space from $M$ measurement observations given in the form $y_m = \ell_m(u^{\rm true}),\, m=1,\dots,M$, where $\ell_m$ are linear functionals. The method approximates $u^{\rm true}$ with $\hat{u} = \hat{z} + \hatη$. The \emph{background} $\hat{z}$ belongs to an $N$-dimensional linear space $\mathcal{Z}_N$ built from reduced modelling of a parameterized mathematical model, and the \emph{update} $\hatη$ belongs to the space $\mathcal{U}_M$ spanned by the Riesz representers of $(\ell_1,\dots, \ell_M)$. When the measurements are noisy {--- i.e., $y_m = \ell_m(u^{\rm true})+ε_m$ with $ε_m$ being a noise term --- } the classical PBDW formulation is not robust in the sense that, if $N$ increases, the reconstruction accuracy degrades. In this paper, we propose to address this issue with an extension of the classical formulation, {which consists in} searching for the background $\hat{z}$ either on the whole $\mathcal{Z}_N$ in the noise-free case, or on a well-chosen subset $\mathcal{K}_N \subset \mathcal{Z}_N$ in presence of noise. The restriction to $\mathcal{K}_N$ makes the reconstruction be nonlinear and is the key to make the algorithm significantly more robust against noise. We {further} present an \emph{a priori} error and stability analysis, and we illustrate the efficiency of the approach on several numerical examples.

Figures (11)

• Figure 1: two-dimensional problem. Behavior of $\Lambda_2$, $\Lambda_{\mathcal{U}}$ and $\Lambda_{\mathcal{U}}^{\rm bias}$ for several choices of $N$ and $M$.
• Figure 2: two-dimensional problem. Behavior of $E_{\rm avg}$ for several values of $\xi$ and three choices of $N,M$, for linear PBDW ($\Phi_N = \mathbb{R}^N$).
• Figure 3: two-dimensional problem. Figures (a)-(d): location of the observation centers selected by SGreedy. Figures (b)-(e): behavior of $\Lambda_{\mathcal{U}}$ with $M$ for different choices of the observation centers. Figures (c)-(f): behavior of $\Lambda_{2}$ with $M$ for different choices of the observation centers.
• Figure 4: two-dimensional problem. Behavior of $E_{\rm avg}$ with $N$ for several values of $M$, for linear PBDW ($\Phi_N = \mathbb{R}^N$).
• Figure 5: two-dimensional problem. Behavior of $E_{\rm avg}$ with $N$ for several values of $M$, for nonlinear PBDW ($\Phi_N \subsetneq \mathbb{R}^N$).

Theorems & Definitions (20)

• Proposition 2.1
• Corollary 2.2
• proof
• Remark 2.1
• Lemma 3.1
• proof
• Proposition 3.1
• proof
• Remark 3.1
• proof