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A Goal-Oriented Adaptive Discrete Empirical Interpolation Method

R. Stefanescu, A. Sandu

TL;DR

The paper addresses the computational cost of nonlinear reduced order models by deriving an a-posteriori, goal-oriented error Estimation for quantities of interest (QoI) within POD/DEIM frameworks. It provides an adjoint-based gradient and a first-order QoI error approximation, enabling efficient estimation using a single reduced forward and adjoint pass plus high-fidelity residuals, with explicit and implicit Euler variants. A novel adaptive DEIM strategy relocates interpolation points using dual weighted residuals and nonlinear-term bases to improve QoI accuracy, demonstrated on 1D Burgers and Shallow Water Equations. Numerical results show strong agreement between estimated and true QoI errors and demonstrate improved QoI accuracy through adaptive point placement, highlighting potential for robust parametric ROMs in nonlinear PDE contexts.

Abstract

In this study we propose a-posteriori error estimation results to approximate the precision loss in quantities of interests computed using reduced order models. To generate the surrogate models we employ Proper Orthogonal Decomposition and Discrete Empirical Interpolation Method. First order expansions of the components of the quantity of interest obtained as the product between the components gradient and model residuals are summed up to generate the error estimation result. Efficient versions are derived for explicit and implicit Euler schemes and require only one reduced forward and adjoint models and high-fidelity model residuals estimation. Then we derive an adaptive DEIM algorithm to enhance the accuracy of these quantities of interests. The adaptive DEIM algorithm uses dual weighted residuals singular vectors in combination with the non-linear term basis. Both the a-posteriori error estimation results and the adaptive DEIM algorithm were assessed using the 1D-Burgers and Shallow Water Equation models and the numerical experiments shows very good agreement with the theoretical results.

A Goal-Oriented Adaptive Discrete Empirical Interpolation Method

TL;DR

The paper addresses the computational cost of nonlinear reduced order models by deriving an a-posteriori, goal-oriented error Estimation for quantities of interest (QoI) within POD/DEIM frameworks. It provides an adjoint-based gradient and a first-order QoI error approximation, enabling efficient estimation using a single reduced forward and adjoint pass plus high-fidelity residuals, with explicit and implicit Euler variants. A novel adaptive DEIM strategy relocates interpolation points using dual weighted residuals and nonlinear-term bases to improve QoI accuracy, demonstrated on 1D Burgers and Shallow Water Equations. Numerical results show strong agreement between estimated and true QoI errors and demonstrate improved QoI accuracy through adaptive point placement, highlighting potential for robust parametric ROMs in nonlinear PDE contexts.

Abstract

In this study we propose a-posteriori error estimation results to approximate the precision loss in quantities of interests computed using reduced order models. To generate the surrogate models we employ Proper Orthogonal Decomposition and Discrete Empirical Interpolation Method. First order expansions of the components of the quantity of interest obtained as the product between the components gradient and model residuals are summed up to generate the error estimation result. Efficient versions are derived for explicit and implicit Euler schemes and require only one reduced forward and adjoint models and high-fidelity model residuals estimation. Then we derive an adaptive DEIM algorithm to enhance the accuracy of these quantities of interests. The adaptive DEIM algorithm uses dual weighted residuals singular vectors in combination with the non-linear term basis. Both the a-posteriori error estimation results and the adaptive DEIM algorithm were assessed using the 1D-Burgers and Shallow Water Equation models and the numerical experiments shows very good agreement with the theoretical results.

Paper Structure

This paper contains 19 sections, 2 theorems, 58 equations, 12 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Reduced order modeling
  3. Proper Orthogonal Decomposition
  4. Discrete Empirical Interpolation Method
  5. Problem formulation
  6. A-posteriori error estimates
  7. Gradient of the quantity of interest
  8. Error estimates
  9. Efficient computation of the error estimates
  10. Explicit Euler scheme
  11. Implicit Euler scheme
  12. Adaptive location of DEIM interpolation points
  13. Numerical experiments
  14. One-dimensional Burgers model
  15. Numerical experiments with the one-dimensional Burgers model
  16. ...and 4 more sections

Key Result

Theorem 1

Assume that the model operators $\mathcal{M}_{i,i+1} : \mathbb{R}^{N_{\rm{state}} \times \mathcal{\tilde{P}}} \to \mathbb{R}^{N_{\rm{state}}},~i=0,..,N_t-1,$ are of class $C^1$, and the scalar functions $r_i:\mathbb{R}^{N_{\rm{state}} \times \mathcal{\tilde{P}}} \to \mathbb{R},~i=0,..,N_t,$ belong t where ${\mathbf M}_{i+1,i}^*$ is the Jacobian matrix of $\mathcal{M}_{i,i+1}$.

Figures (12)

  • Figure 1: Geometrical interpretation of the a-posteriori error estimate \ref{['eqn::aposteriori1']}.
  • Figure 2: Initial conditions of Burgers model for $\mu = 0.1$ (left panel). The Burgers model solution at final time (right panel). The green color depicts the model trajectory used to compute the quantity of interest.
  • Figure 3: Singular values of state variable and non-linear term of the 1D-Burgers model.
  • Figure 4: A-posteriori error estimates for the same parametric configuration - $\mu = 0.1$.
  • Figure 5: A-posteriori error estimates for different parametric configuration - $\mu = 0.07$.
  • ...and 7 more figures

Theorems & Definitions (7)

  • Theorem 1: Minimization of the QoI
  • Proof 1
  • Theorem 2: A-posteriori error estimation
  • Proof 2
  • Remark 1
  • Remark 2
  • Remark 3