Nonlinear dimensionality reduction for parametric problems: A kernel proper orthogonal decomposition

Autor: Alba Muixí, Alberto García-González, Pedro Díez, Sergio Zlotnik
Přispěvatelé: Universitat Politècnica de Catalunya. Departament d'Enginyeria Civil i Ambiental, Universitat Politècnica de Catalunya. LACÀN - Mètodes Numèrics en Ciències Aplicades i Enginyeria
Rok vydání: 2021
Předmět:
Reduced-order models
Computer science
Matemàtiques i estadística::Matemàtica aplicada a les ciències [Àrees temàtiques de la UPC]
Computing Methodologies
Kernel principal component analysis
Kernel (linear algebra)
Dimension (vector space)
Sistemes de control
Informàtica
Tangent space
68 Computer science::68U Computing methodologies and applications [Classificació AMS]
System theory
93 Systems Theory
Control::93B Controllability
observability
and system structure [Classificació AMS]
Parametric statistics
93 Systems Theory [Classificació AMS]
Numerical Analysis
Basis (linear algebra)
Nonlinear multidimensionality reduction
Applied Mathematics
General Engineering
Nonlinear dimensionality reduction
Linear subspace
93B Controllability
observability
and system structure [Control]
kPCA
Matemàtiques i estadística::Investigació operativa::Simulació [Àrees temàtiques de la UPC]
Parametric problems
Algorithm
Zdroj: UPCommons. Portal del coneixement obert de la UPC
Universitat Politècnica de Catalunya (UPC)
ISSN: 1097-0207
0029-5981
DOI: 10.1002/nme.6831
Popis: This is the peer reviewed version of the following article: Diez, P. [et al.]. Nonlinear dimensionality reduction for parametric problems: a kernel proper orthogonal decomposition. "International journal for numerical methods in engineering", 30 Desembre 2021, vol. 122, núm. 24, p. 7306-7327, which has been published in final form at DOI: 10.1002/nme.6831. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Reduced-order models are essential tools to deal with parametric problems in the context of optimization, uncertainty quantification, or control and inverse problems. The set of parametric solutions lies in a low-dimensional manifold (with dimension equal to the number of independent parameters) embedded in a large-dimensional space (dimension equal to the number of degrees of freedom of the full-order discrete model). A posteriori model reduction is based on constructing a basis from a family of snapshots (solutions of the full-order model computed offline), and then use this new basis to solve the subsequent instances online. Proper orthogonal decomposition (POD) reduces the problem into a linear subspace of lower dimension, eliminating redundancies in the family of snapshots. The strategy proposed here is to use a nonlinear dimensionality reduction technique, namely, the kernel principal component analysis (kPCA), in order to find a nonlinear manifold, with an expected much lower dimension, and to solve the problem in this low-dimensional manifold. Guided by this paradigm, the methodology devised here introduces different novel ideas, namely, 1) characterizing the nonlinear manifold using local tangent spaces, where the reduced-order problem is linear and based on the neighboring snapshots, 2) the approximation space is enriched with the cross-products of the snapshots, introducing a quadratic description, 3) the kernel for kPCA is defined ad hoc, based on physical considerations, and 4) the iterations in the reduced-dimensional space are performed using an algorithm based on a Delaunay tessellation of the cloud of snapshots in the reduced space. The resulting computational strategy is performing outstandingly in the numerical tests, alleviating many of the problems associated with POD and improving the numerical accuracy. Generalitat de Catalunya, 2017-SGR-1278; Ministerio de Ciencia e Innovación, CEX2018-000797-S; PID2020-113463RB-C32; PID2020-113463RB-C33
Databáze: OpenAIRE
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