Model Reduction from Partial Observations

TL;DR

This paper develops methods for model-order reduction of parametric PDEs using partial observations and prior knowledge, providing theoretical bounds and procedures for optimal approximation subspaces.

Contribution

It introduces new tools to derive approximation subspaces from partial measurements and prior knowledge, along with performance bounds and analysis of the observation operator's impact.

Findings

Identified best worst-case approximation performance.

Proposed procedures for near-optimal subspace construction.

Analyzed the influence of observation operator and prior knowledge.

Abstract

This paper deals with model-order reduction of parametric partial differential equations (PPDE). More specifically, we consider the problem of finding a good approximation subspace of the solution manifold of the PPDE when only partial information on the latter is available. We assume that two sources of information are available: i) a "rough" prior knowledge, taking the form of a manifold containing the target solution manifold, ii) partial linear measurements of the solutions of the PPDE (the term partial refers to the fact that observation operator cannot be inverted). We provide and study several tools to derive good approximation subspaces from these two sources of information. We first identify the best worst-case performance achievable in this setup and propose simple procedures to approximate the corresponding optimal approximation subspace. We then provide, in a simplified setup, a theoretical analysis relating the achievable reduction performance to the choice of the observation operator and the prior knowledge available on the solution manifold.