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High-order empirical interpolation methods for real time solution of parametrized nonlinear PDEs

Ngoc Cuong Nguyen

TL;DR

This work proposes a class of high-order empirical interpolation methods to derive basis functions and interpolation points by using high-order partial derivatives of the nonlinear terms and develops effective a posteriori estimator to quantify the interpolation errors.

Abstract

We present novel model reduction methods for rapid solution of parametrized nonlinear partial differential equations (PDEs) in real-time or many-query contexts. Our approach combines reduced basis (RB) space for rapidly convergent approximation of the parametric solution manifold, Galerkin projection of the underlying PDEs onto the RB space for dimensionality reduction, and high-order empirical interpolation for efficient treatment of the nonlinear terms. We propose a class of high-order empirical interpolation methods to derive basis functions and interpolation points by using high-order partial derivatives of the nonlinear terms. As these methods can generate high-quality basis functions and interpolation points from a snapshot set of full-order model (FOM) solutions, they significantly improve the approximation accuracy. We develop effective a posteriori estimator to quantify the interpolation errors and construct a parameter sample via greedy sampling. Furthermore, we implement two hyperreduction schemes to construct efficient reduced-order models: one that applies the empirical interpolation before Newton's method and another after. The latter scheme shows flexibility in controlling hyperreduction errors. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed methods.

High-order empirical interpolation methods for real time solution of parametrized nonlinear PDEs

TL;DR

This work proposes a class of high-order empirical interpolation methods to derive basis functions and interpolation points by using high-order partial derivatives of the nonlinear terms and develops effective a posteriori estimator to quantify the interpolation errors.

Abstract

We present novel model reduction methods for rapid solution of parametrized nonlinear partial differential equations (PDEs) in real-time or many-query contexts. Our approach combines reduced basis (RB) space for rapidly convergent approximation of the parametric solution manifold, Galerkin projection of the underlying PDEs onto the RB space for dimensionality reduction, and high-order empirical interpolation for efficient treatment of the nonlinear terms. We propose a class of high-order empirical interpolation methods to derive basis functions and interpolation points by using high-order partial derivatives of the nonlinear terms. As these methods can generate high-quality basis functions and interpolation points from a snapshot set of full-order model (FOM) solutions, they significantly improve the approximation accuracy. We develop effective a posteriori estimator to quantify the interpolation errors and construct a parameter sample via greedy sampling. Furthermore, we implement two hyperreduction schemes to construct efficient reduced-order models: one that applies the empirical interpolation before Newton's method and another after. The latter scheme shows flexibility in controlling hyperreduction errors. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed methods.
Paper Structure (19 sections, 5 theorems, 76 equations, 8 figures, 7 tables, 5 algorithms)

This paper contains 19 sections, 5 theorems, 76 equations, 8 figures, 7 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Model Reduction Methods for Parametrized Nonlinear PDEs
  3. Finite element approximation
  4. Galerkin-Newton method
  5. Hyperreduction--Galerkin-Newton method
  6. Newton-Gakerin--Hypereduction method
  7. High-Order Empirical Interpolation Methods
  8. Interpolation procedure
  9. Basis functions
  10. Interpolation points
  11. Stability of the high-order empirical interpolation
  12. Convergence of the high-order empirical interpolation
  13. Error estimate of the high-order empirical interpolation
  14. Greedy sampling
  15. A simple test case
  16. ...and 4 more sections

Key Result

Lemma 1

Assume that $\Psi_{M}$ is of dimension $M$ and that $\bm B_{M}$ is invertible, then we have $\mathcal{I}(\Psi_M, T_M)[v] = v$ for any $v \in \Psi_{M}$. In other words, the interpolation is exact for all v in $\Psi_{M}$.

Figures (8)

  • Figure 1: (a) Plots of $g(u(x,\mu))$ as a function of $x$ for different values of $\mu$, and (b) the maximum error estimate $\hat{\varepsilon}_{M,P} (\mu_{N+1})$ in the greedy sampling.
  • Figure 2: Comparison of accuracy between EIM, FOEIM, and SOEIM: (a) the maximum test error and (b) the mean test error as a function of $N$.
  • Figure 3: Convergence of the mean error $\varepsilon_M^{\rm mean}$ for SOEIM with $M=6N$ as a function of $N$ for Greedy Sampling, Extended Chebyshev, and Uniform Distribution.
  • Figure 4: (a) distribution of the parameter sample points selected using the greedy sampling, and (b) distribution of the interpolation points for the SOEIM method for $N=15$.
  • Figure 5: Comparison of accuracy between EIM-GN, FOEIM-GN, SOEIM-GN, GN-SOEIM, and GN methods.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Proposition 1
  • Theorem 3
  • proof