Dictionary-based model reduction for state estimation

TL;DR

This work tackles state estimation from few linear measurements for states lying on a parameter-dependent solution manifold. It extends the PBDW framework by introducing a dictionary-based, multi-space approach and a randomized offline-online decomposition to enable scalable, stable recovery even when the manifold cannot be well approximated by a single low-dimensional space. Key contributions include a randomized selection criterion $\mathcal{S}^{\Theta}$, a dictionary-based multi-space construction from snapshot dictionaries $\mathcal{D}_K$, and an efficient, provably stable offline-online scheme for parameter-dependent PDEs with affine parametrizations. Numerical experiments on thermal-block and advection-diffusion problems confirm improved mean recovery accuracy with large dictionaries and demonstrate the practicality of randomized sketches in high-dimensional settings. The approach offers a flexible framework for robust inverse problems with reduced computational cost, and lays groundwork for extensions to noisy data and joint observation-background dictionary strategies.

Abstract

We consider the problem of state estimation from a few linear measurements, where the state to recover is an element of the manifold $\mathcal{M}$ of solutions of a parameter-dependent equation. The state is estimated using prior knowledge on $\mathcal{M}$ coming from model order reduction. Variational approaches based on linear approximation of $\mathcal{M}$, such as PBDW, yields a recovery error limited by the Kolmogorov width of $\mathcal{M}$. To overcome this issue, piecewise-affine approximations of $\mathcal{M}$ have also been considered, that consist in using a library of linear spaces among which one is selected by minimizing some distance to $\mathcal{M}$. In this paper, we propose a state estimation method relying on dictionary-based model reduction, where a space is selected from a library generated by a dictionary of snapshots, using a distance to the manifold. The selection is performed among a set of candidate spaces obtained from a set of $\ell_1$-regularized least-squares problems. Then, in the framework of parameter-dependent operator equations (or PDEs) with affine parametrizations, we provide an efficient offline-online decomposition based on randomized linear algebra, that ensures efficient and stable computations while preserving theoretical guarantees.

Figures (7)

• Figure 1: Geometric interpretation of the PBDW recovery for one observation $w=P_W u$, with dim$(W)=2$ and dim$(V)=1$.
• Figure 2: The thermal block problem. On the left, the geometry of the problem, with sensors locations (crosses). On the middle, the (normalized) Riesz representer of a sensor. On the right, an example of snapshot.
• Figure 3: Evolution of the constants in the error bound \ref{['equ:pbdw error bound']}, on $500$ test snapshots, with increasing number of POD modes and $m=64$ measurements.
• Figure 4: Evolution of the recovery errors in $U$-norm, on $500$ test snapshots, with growing dictionary sizes and different values of $m$. We compare the PBDW recovery based on the best adaptive POD truncation $A^{(1)}$, in red, defined in \ref{['equ:best adaptive pod']}, the dictionary-based recovery with randomized selection criterion $A^{(2)} = A^{\text{dic}}_{\mathcal{S}^{\Theta}}$, in blue, as well as the best one produced by the LARS algorithm $A^{(3)}$, in cyan, defined in \ref{['equ:best lars recovery']} . The full line is the mean relative error and the dotted line is the maximal relative error.
• Figure 5: The advection diffusion problem. On the left, the geometry of the problem, with sensors locations (crosses) and advection fields (circular arrows). On the middle, the (normalized) Riesz representer of the central sensor. On the right, an example of snapshot.

Theorems & Definitions (17)

• Proposition 3.1
• proof
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Definition 4.1
• Definition 4.2
• Remark 4.1
• Proposition 4.1
• proof