Bayesian Probabilistic Numerical Methods in Time-Dependent State Estimation for Industrial Hydrocyclone Equipment

Abstract

The use of high-power industrial equipment, such as large-scale mixing equipment or a hydrocyclone for separation of particles in liquid suspension, demands careful monitoring to ensure correct operation. The fundamental task of state-estimation for the liquid suspension can be posed as a time-evolving inverse problem and solved with Bayesian statistical methods. In this article, we extend Bayesian methods to incorporate statistical models for the error that is incurred in the numerical solution of the physical governing equations. This enables full uncertainty quantification within a principled computation-precision trade-off, in contrast to the over-confident inferences that are obtained when all sources of numerical error are ignored. The method is cast within a sequential Monte Carlo framework and an optimized implementation is provided in Python.

Keywords:

• Electrical tomography
• Inverse problems
• Partial differential equations
• Probabilistic meshless methods
• Sequential Monte Carlo

Acknowledgments

The authors are grateful for detailed suggestions from the associate editor and two anonymous reviewers. For the numerical results reported in Section 3, we thank T. J. Sullivan for the use of computing facilities at the Freie Universität Berlin, funded by the Excellence Initiative of the German Research Foundation.

Funding

Additional information

Funding

This research was supported by the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers and by the Key Technology Partnership program at the University of Technology Sydney. CJO and MG were supported by the Lloyd’s Register Foundation programme on data-centric engineering at the Alan Turing Institute, UK. MG was supported by the EPSRC grants [EP/K034154/1, EP/R018413/1, EP/P020720/1, EP/L014165/1], an EPSRC Established Career Fellowship [EP/J016934/1] and a Royal Academy of Engineering Research Chair in Data Centric Engineering. The collection of tomographic data was supported by an EPSRC grant [GR/R22148/01]. This material was based upon work partially supported by the National Science Foundation under Grant DMS-1127914 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.