Numerically accurate computational techniques for optimal estimator analyses of multi-parameter models

Lukas Berger*, Konstantin Kleinheinz, Antonio Attili, Fabrizio Bisetti, Heinz Pitsch, Michael E. Mueller

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

Modelling unclosed terms in partial differential equations typically involves two steps: First, a set of known quantities needs to be specified as input parameters for a model, and second, a specific functional form needs to be defined to model the unclosed terms by the input parameters. Both steps involve a certain modelling error, with the former known as the irreducible error and the latter referred to as the functional error. Typically, only the total modelling error, which is the sum of functional and irreducible error, is assessed, but the concept of the optimal estimator enables the separate analysis of the total and the irreducible errors, yielding a systematic modelling error decomposition. In this work, attention is paid to the techniques themselves required for the practical computation of irreducible errors. Typically, histograms are used for optimal estimator analyses, but this technique is found to add a non-negligible spurious contribution to the irreducible error if models with multiple input parameters are assessed. Thus, the error decomposition of an optimal estimator analysis becomes inaccurate, and misleading conclusions concerning modelling errors may be drawn. In this work, numerically accurate techniques for optimal estimator analyses are identified and a suitable evaluation of irreducible errors is presented. Four different computational techniques are considered: a histogram technique, artificial neural networks, multivariate adaptive regression splines, and an additive model based on a kernel method. For multiple input parameter models, only artificial neural networks and multivariate adaptive regression splines are found to yield satisfactorily accurate results. Beyond a certain number of input parameters, the assessment of models in an optimal estimator analysis even becomes practically infeasible if histograms are used. The optimal estimator analysis in this paper is applied to modelling the filtered soot intermittency in large eddy simulations using a dataset of a direct numerical simulation of a non-premixed sooting turbulent flame.

Original languageEnglish (US)
Pages (from-to)480-504
Number of pages25
JournalCombustion Theory and Modelling
Volume22
Issue number3
DOIs
StatePublished - May 4 2018

Bibliographical note

Funding Information:
Generous support of the Deutsche Forschungsgemeinschaft (DFG) and the Research Association for Combustion Engines (FVV) under grant numbers PI 368/6-2 (DFG) and 6012390 (FVV) for L. B., K. K., and H. P. is gratefully acknowledged. A. A. and H. P. gratefully acknowledge generous support of the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 695747).

Funding Information:
The analyses presented in this article were performed in part on computational resources supported by the Princeton Institute for Computational Science and Engineering (PICSciE) and the Office of Information Technology’s Research Computing department at Princeton University and in part by the Gauss Centre for Supercomputing e.V. on the GCS Supercomputer SuperMuc at Leibniz Supercomputing Centre in Munich. We acknowledge the KAUST Supercomputing Laboratory (KSL) for providing the computational time on the supercomputer “Shaheen” used to perform the DNS.

Publisher Copyright:
© 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.

Keywords

  • DNS
  • artificial neural networks
  • multivariate adaptive regression splines
  • optimal estimator
  • soot

ASJC Scopus subject areas

  • Chemistry(all)
  • Chemical Engineering(all)
  • Modeling and Simulation
  • Fuel Technology
  • Energy Engineering and Power Technology
  • Physics and Astronomy(all)

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