Nonlinear compressive reduced basis approximation for PDE’s
Linear model reduction techniques design offline low-dimensional subspaces that are tailored to the approximation of solutions to a parameterized partial differential equation, for the purpose of fast online numerical simulations. These methods, such as the Proper Orthogonal Decomposition (POD) or R...
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-09-01
|
| Series: | Comptes Rendus. Mécanique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.191/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1825206005508603904 |
|---|---|
| author | Cohen, Albert Farhat, Charbel Maday, Yvon Somacal, Agustin |
| author_facet | Cohen, Albert Farhat, Charbel Maday, Yvon Somacal, Agustin |
| author_sort | Cohen, Albert |
| collection | DOAJ |
| description | Linear model reduction techniques design offline low-dimensional subspaces that are tailored to the approximation of solutions to a parameterized partial differential equation, for the purpose of fast online numerical simulations. These methods, such as the Proper Orthogonal Decomposition (POD) or Reduced Basis (RB) methods, are very effective when the family of solutions has fast-decaying Karhunen–Loève eigenvalues or Kolmogorov widths, reflecting the approximability by finite-dimensional linear spaces. On the other hand, they become ineffective when these quantities have a slow decay, in particular for families of solutions to hyperbolic transport equations with parameter-dependent shock positions. The objective of this work is to explore the ability of nonlinear model reduction to circumvent this particular situation. To this end, we first describe particular notions of nonlinear widths that have a substantially faster decay for the aforementioned families. Then, we discuss a systematic approach for achieving better performance via a nonlinear reconstruction from the first coordinates of a linear reduced model approximation, thus allowing us to stay in the same “classical” framework of projection-based model reduction. We analyze the approach and report on its performance for a simple and yet instructive univariate test case. |
| format | Article |
| id | doaj-art-6570fc798d8b4344891e36e1116c8d58 |
| institution | Kabale University |
| issn | 1873-7234 |
| language | English |
| publishDate | 2023-09-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mécanique |
| spelling | doaj-art-6570fc798d8b4344891e36e1116c8d582025-02-07T13:46:20ZengAcadémie des sciencesComptes Rendus. Mécanique1873-72342023-09-01351S135737410.5802/crmeca.19110.5802/crmeca.191Nonlinear compressive reduced basis approximation for PDE’sCohen, Albert0Farhat, Charbel1Maday, Yvon2Somacal, Agustin3Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, FranceDepartment of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA; Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA; Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305, USASorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, FranceSorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, FranceLinear model reduction techniques design offline low-dimensional subspaces that are tailored to the approximation of solutions to a parameterized partial differential equation, for the purpose of fast online numerical simulations. These methods, such as the Proper Orthogonal Decomposition (POD) or Reduced Basis (RB) methods, are very effective when the family of solutions has fast-decaying Karhunen–Loève eigenvalues or Kolmogorov widths, reflecting the approximability by finite-dimensional linear spaces. On the other hand, they become ineffective when these quantities have a slow decay, in particular for families of solutions to hyperbolic transport equations with parameter-dependent shock positions. The objective of this work is to explore the ability of nonlinear model reduction to circumvent this particular situation. To this end, we first describe particular notions of nonlinear widths that have a substantially faster decay for the aforementioned families. Then, we discuss a systematic approach for achieving better performance via a nonlinear reconstruction from the first coordinates of a linear reduced model approximation, thus allowing us to stay in the same “classical” framework of projection-based model reduction. We analyze the approach and report on its performance for a simple and yet instructive univariate test case.https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.191/non linear reduced basiscompressed sensingsolution manifoldmachine learning$m$-width |
| spellingShingle | Cohen, Albert Farhat, Charbel Maday, Yvon Somacal, Agustin Nonlinear compressive reduced basis approximation for PDE’s Comptes Rendus. Mécanique non linear reduced basis compressed sensing solution manifold machine learning $m$-width |
| title | Nonlinear compressive reduced basis approximation for PDE’s |
| title_full | Nonlinear compressive reduced basis approximation for PDE’s |
| title_fullStr | Nonlinear compressive reduced basis approximation for PDE’s |
| title_full_unstemmed | Nonlinear compressive reduced basis approximation for PDE’s |
| title_short | Nonlinear compressive reduced basis approximation for PDE’s |
| title_sort | nonlinear compressive reduced basis approximation for pde s |
| topic | non linear reduced basis compressed sensing solution manifold machine learning $m$-width |
| url | https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.191/ |
| work_keys_str_mv | AT cohenalbert nonlinearcompressivereducedbasisapproximationforpdes AT farhatcharbel nonlinearcompressivereducedbasisapproximationforpdes AT madayyvon nonlinearcompressivereducedbasisapproximationforpdes AT somacalagustin nonlinearcompressivereducedbasisapproximationforpdes |