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2017 | Supplement | Buchkapitel

16. Adaptive Sampling for Nonlinear Dimensionality Reduction Based on Manifold Learning

verfasst von : Thomas Franz, Ralf Zimmermann, Stefan Görtz

Erschienen in: Model Reduction of Parametrized Systems

Verlag: Springer International Publishing

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Abstract

We make use of the non-intrusive dimensionality reduction method Isomap in order to emulate nonlinear parametric flow problems that are governed by the Reynolds-averaged Navier-Stokes equations. Isomap is a manifold learning approach that provides a low-dimensional embedding space that is approximately isometric to the manifold that is assumed to be formed by the high-fidelity Navier-Stokes flow solutions under smooth variations of the inflow conditions. The focus of the work at hand is the adaptive construction and refinement of the Isomap emulator: We exploit the non-Euclidean Isomap metric to detect and fill up gaps in the sampling in the embedding space. The performance of the proposed manifold filling method will be illustrated by numerical experiments, where we consider nonlinear parameter-dependent steady-state Navier-Stokes flows in the transonic regime.

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Vorheriges Kapitel Hi-POD Solution of Parametrized Fluid Dynamics Problems: Preliminary Results

Nächstes Kapitel Cross-Gramian-Based Model Reduction: A Comparison

i.e., a bijective both-ways differentiable mapping.

The number of nearest neighbors used for the embedding was chosen automatically in each iteration according to [7, Sect. 4.3.3].

All computations were conducted sequentially on the same standard desktop computer endowed with an Intel^®; Xeon^®; E3-1270 v3 Processor (8M Cache, 3.50 GHz) and 32 GB RAM.

For applications where the dimension of the manifold is unknown, there exist various methods to estimate the intrinsic dimensionality of the data, e.g. by looking for the “elbow” [17].

Error quantification is with respect to the surface C _p distributions and is based on 2500 uniformly distributed TAU reference CFD solutions.

Table 16.1

The mean relative error, its standard deviation and the maximum relative error after a full sampling process of various sampling strategies/designs for the NACA 64A010 test case

Method	Mean rel. error	STD. deviation	Max. rel. error
MDE	2. 3347 ⋅ 10⁻²	1. 6616 ⋅ 10⁻²	9. 2956 ⋅ 10⁻²
HYE	2. 1903 ⋅ 10⁻²	1. 0320 ⋅ 10⁻²	5. 3337 ⋅ 10⁻²
Halton	2. 6670 ⋅ 10⁻²	2. 7398 ⋅ 10⁻²	2. 3016 ⋅ 10⁻¹
LHS	3. 1262 ⋅ 10⁻²	2. 6257 ⋅ 10⁻²	1. 8009 ⋅ 10⁻¹

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Titel: Adaptive Sampling for Nonlinear Dimensionality Reduction Based on Manifold Learning
verfasst von: Thomas Franz
Ralf Zimmermann
Stefan Görtz
Verlag: Springer International Publishing
Buch: Model Reduction of Parametrized Systems
Print ISBN: 978-3-319-58785-1

Electronic ISBN: 978-3-319-58786-8

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-58786-8_16

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