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Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models.

A A Shah1, W W Xing1, V Triantafyllidis1

  • 1School of Engineering, University of Warwick, Coventry CV4 7AL, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences
|May 10, 2017
PubMed
Summary

This study introduces efficient reduced-order models for dynamic equations using proper orthogonal decomposition (POD). The method accurately approximates solutions for new parameters, even in nonlinear cases.

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Area of Science:

  • Computational Science and Engineering
  • Applied Mathematics
  • Numerical Analysis

Background:

  • Reduced-order modeling (ROM) is crucial for simulating complex dynamic systems.
  • Parameter-dependent partial differential equations (PDEs) pose challenges in computational efficiency.
  • Proper Orthogonal Decomposition (POD) is a common technique for dimensionality reduction.

Purpose of the Study:

  • To develop accurate and efficient reduced-order models for dynamic, parameter-dependent linear and nonlinear PDEs.
  • To address challenges in approximating POD bases for new parameter values.
  • To efficiently handle nonlinear terms within the reduced-order modeling framework.

Main Methods:

  • Utilized Proper Orthogonal Decomposition (POD) for dimensionality reduction.
  • Employed Bayesian nonlinear regression to learn solution snapshots and nonlinearities for new parameters.
  • Integrated manifold learning for low-dimensional emulation to ensure computational efficiency.

Main Results:

  • Demonstrated the accuracy of the proposed method on both linear and nonlinear examples.
  • Showcased efficient approximation of POD bases for unseen parameter values.
  • Validated the effective handling of nonlinear terms in dynamic systems.

Conclusions:

  • The developed Bayesian POD-based ROM offers an accurate and computationally efficient approach for parameter-dependent PDEs.
  • The method provides a robust framework for simulating complex dynamic systems across various parameters.
  • This technique advances the field of reduced-order modeling for scientific and engineering applications.