A Hierarchical A-Posteriori Error Estimatorfor the Reduced Basis Method Article Swipe

MAKE IMPACT

Stefan Hain , Mario Ohlberger , Mladjan Radic , Karsten Urban · YOU? · · 2018 · Open Access · · DOI: https://doi.org/10.48550/arxiv.1802.03298

In this contribution we are concerned with tight a posteriori error estimation for projection based model order reduction of $\inf$-$\sup$ stable parameterized variational problems. In particular, we consider the Reduced Basis Method in a Petrov-Galerkin framework, where the reduced approximation spaces are constructed by the (weak) Greedy algorithm. We propose and analyze a hierarchical a posteriori error estimator which evaluates the difference of two reduced approximations of different accuracy. Based on the a priori error analysis of the (weak) Greedy algorithm, it is expected that the hierarchical error estimator is sharp with efficiency index close to one, if the Kolmogorov N-with decays fast for the underlying problem and if a suitable saturation assumption for the reduced approximation is satisfied. We investigate the tightness of the hierarchical a posteriori estimator both from a theoretical and numerical perspective. For the respective approximation with higher accuracy we study and compare basis enrichment of Lagrange- and Taylor-type reduced bases. Numerical experiments indicate the efficiency for both, the construction of a reduced basis using the hierarchical error estimator in a weak Greedy algorithm, and for tight online certification of reduced approximations. This is particularly relevant in cases where the $\inf$-$\sup$ constant may become small depending on the parameter. In such cases a standard residual-based error estimator -- complemented by the successive constrained method to compute a lower bound of the parameter dependent $\inf$-$\sup$ constant -- may become infeasible.

Type

preprint

Language

en

Landing Page

http://arxiv.org/abs/1802.03298

PDF

https://arxiv.org/pdf/1802.03298

OA Status

green

Cited By

1

References

30

Related Works

10

OpenAlex ID

https://openalex.org/W2786593157

Raw OpenAlex JSON

OpenAlex ID

https://openalex.org/W2786593157

Canonical identifier for this work in OpenAlex

DOI

https://doi.org/10.48550/arxiv.1802.03298

Digital Object Identifier

Title

A Hierarchical A-Posteriori Error Estimatorfor the Reduced Basis Method

Work title

Type

preprint

OpenAlex work type

Language

en

Primary language

Publication year

2018

Year of publication

Publication date

2018-02-09

Full publication date if available

Authors

Stefan Hain, Mario Ohlberger, Mladjan Radic, Karsten Urban

List of authors in order

Landing page

https://arxiv.org/abs/1802.03298

Publisher landing page

PDF URL

https://arxiv.org/pdf/1802.03298

Direct link to full text PDF

Open access

Yes

Whether a free full text is available

OA status

green

Open access status per OpenAlex

OA URL

https://arxiv.org/pdf/1802.03298

Direct OA link when available

Concepts

Basis (linear algebra), A priori and a posteriori, Computer science, Statistics, Algorithm, Mathematics, Epistemology, Philosophy, Geometry

Top concepts (fields/topics) attached by OpenAlex

Cited by

1

Total citation count in OpenAlex

Citations by year (recent)

2018: 1

Per-year citation counts (last 5 years)

References (count)

30

Number of works referenced by this work

Related works (count)

10

Other works algorithmically related by OpenAlex

Full payload