Model Reduction from Partial Observations
For researchers in model reduction of parametric PDEs, this work provides a framework and theoretical guarantees for handling partial observations, though it is incremental as it builds on existing manifold-based reduction concepts.
The paper addresses model-order reduction for parametric PDEs when only partial observations of solutions are available, alongside a rough prior manifold. It identifies optimal worst-case performance and proposes procedures to approximate optimal subspaces, with theoretical analysis linking performance to observation operator and prior knowledge.
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.