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Model reduction from partial observations
Author(s) -
Herzet C.,
Héas P.,
Drémeau A.
Publication year - 2017
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5623
Subject(s) - subspace topology , linear subspace , partial differential equation , reduction (mathematics) , manifold (fluid mechanics) , operator (biology) , mathematics , dimensionality reduction , mathematical optimization , term (time) , computer science , algorithm , mathematical analysis , pure mathematics , artificial intelligence , physics , mechanical engineering , biochemistry , chemistry , geometry , repressor , transcription factor , engineering , gene , quantum mechanics
Summary This paper deals with model‐order reduction of parametric partial differential equations (PPDEs). 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 2 sources of information are available: ( a ) a “rough” prior knowledge taking the form of a manifold containing the target solution manifold and ( b ) partial linear measurements of the solutions of the PPDE (the term partial refers to the fact that observation operators cannot be inverted). We provide and study several tools to derive good approximation subspaces from these 2 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.