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Accelerated construction of projection-based reduced-order models via incremental approaches

Eki Agouzal, Tommaso Taddei

Abstract

We present an accelerated greedy strategy for training of projection-based reduced-order models for parametric steady and unsteady partial differential equations. Our approach exploits hierarchical approximate proper orthogonal decomposition to speed up the construction of the empirical test space for least-square Petrov-Galerkin formulations, a progressive construction of the empirical quadrature rule based on a warm start of the non-negative least-square algorithm, and a two-fidelity sampling strategy to reduce the number of expensive greedy iterations. We illustrate the performance of our method for two test cases: a two-dimensional compressible inviscid flow past a LS89 blade at moderate Mach number, and a three-dimensional nonlinear mechanics problem to predict the long-time structural response of the standard section of a nuclear containment building under external loading.

Accelerated construction of projection-based reduced-order models via incremental approaches

Abstract

We present an accelerated greedy strategy for training of projection-based reduced-order models for parametric steady and unsteady partial differential equations. Our approach exploits hierarchical approximate proper orthogonal decomposition to speed up the construction of the empirical test space for least-square Petrov-Galerkin formulations, a progressive construction of the empirical quadrature rule based on a warm start of the non-negative least-square algorithm, and a two-fidelity sampling strategy to reduce the number of expensive greedy iterations. We illustrate the performance of our method for two test cases: a two-dimensional compressible inviscid flow past a LS89 blade at moderate Mach number, and a three-dimensional nonlinear mechanics problem to predict the long-time structural response of the standard section of a nuclear containment building under external loading.
Paper Structure (26 sections, 40 equations, 11 figures, 3 tables, 7 algorithms)

This paper contains 26 sections, 40 equations, 11 figures, 3 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Accelerated construction of least-square Petrov-Galerkin ROMs for steady problems
  3. Preliminary definitions and tools
  4. Least-square Petrov-Galerkin formulation
  5. Construction of the empirical test space
  6. Construction of the empirical quadrature rule
  7. A posteriori error indicator
  8. Overview of the computational procedure
  9. Progressive construction of empirical test space and quadrature
  10. Empirical test space
  11. Empirical quadrature
  12. Summary of the incremental weak-greedy algorithm
  13. Multi-fidelity sampling
  14. Accelerated construction of Galerkin ROMs for unsteady problems
  15. Vanilla POD-greedy for Galerkin time-marching ROMs
  16. ...and 11 more sections

Figures (11)

  • Figure 1: inviscid flow past an array of LS89 turbine blades. (a) partition associated with the geometric map. (b)-(c) behavior of the Mach number for two parameter values.
  • Figure 2: compressible flow past a LS89 blade. (a)-(b)-(c) three computational meshes. (d) behavior of the average and maximum error \ref{['eq:grid_error']}.
  • Figure 3: compressible flow past a LS89 blade. Progressive construction of quadrature rule and test space, coarse mesh ($N_{\rm e} = 1827$).
  • Figure 4: compressible flow past a LS89 blade. Progressive construction of quadrature rule and test space; fine mesh ($N_{\rm e} = 16353$).
  • Figure 5: compressible flow past a LS89 blade; sampling. (a) behavior of $E_{n}^{{\rm proj},(i)}$\ref{['eq:proj_error_sampling']} for six different choices of the coarse mesh, and for random samples. (b)-(c) parameters $\{ \mu^{\star,j} \}_j$ selected by Algorithm \ref{['alg:multifidelity_greedy']} for two different coarse meshes.
  • ...and 6 more figures