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Model order reduction of an ultraweak and optimally stable variational formulation for parametrized reactive transport problems

Christian Engwer, Mario Ohlberger, Lukas Renelt

TL;DR

The paper develops a model order reduction framework for parametrized advection-reaction problems using an ultraweak variational formulation with an $L^2$-like trial space and an optimal test space, achieving unit condition number and a symmetric normal-equation reformulation. Exponential decay of the Kolmogorov $N$-width is established under parameter-separability and norm-equivalence assumptions, enabling efficient offline-online reduction via a greedy procedure on the test-space; the reduced dual solution drives the online computation and primal solutions can be reconstructed or evaluated through dual functionals. Numerical experiments on a catalytic-filter reactive transport benchmark demonstrate exponential convergence of the reduced spaces and orders-of-magnitude speedups over the full-order model, validating the approach and its potential for real-time parametric simulations. The framework is positioned to extend to Friedrichs'-systems and time-dependent problems, with future work focusing on scalable solution of the normal equations and broader applications.

Abstract

This contribution introduces a model order reduction approach for an advection-reaction problem with a parametrized reaction function. The underlying discretization uses an ultraweak formulation with an $L^2$-like trial space and an 'optimal' test space as introduced by Demkowicz et al. This ensures the stability of the discretization and in addition allows for a symmetric reformulation of the problem in terms of a dual solution which can also be interpreted as the normal equations of an adjoint least-squares problem. Classic model order reduction techniques can then be applied to the space of dual solutions which also immediately gives a reduced primal space. We show that the necessary computations do not require the reconstruction of any primal solutions and can instead be performed entirely on the space of dual solutions. We prove exponential convergence of the Kolmogorov $N$-width and show that a greedy algorithm produces quasi-optimal approximation spaces for both the primal and the dual solution space. Numerical experiments based on the benchmark problem of a catalytic filter confirm the applicability of the proposed method.

Model order reduction of an ultraweak and optimally stable variational formulation for parametrized reactive transport problems

TL;DR

The paper develops a model order reduction framework for parametrized advection-reaction problems using an ultraweak variational formulation with an -like trial space and an optimal test space, achieving unit condition number and a symmetric normal-equation reformulation. Exponential decay of the Kolmogorov -width is established under parameter-separability and norm-equivalence assumptions, enabling efficient offline-online reduction via a greedy procedure on the test-space; the reduced dual solution drives the online computation and primal solutions can be reconstructed or evaluated through dual functionals. Numerical experiments on a catalytic-filter reactive transport benchmark demonstrate exponential convergence of the reduced spaces and orders-of-magnitude speedups over the full-order model, validating the approach and its potential for real-time parametric simulations. The framework is positioned to extend to Friedrichs'-systems and time-dependent problems, with future work focusing on scalable solution of the normal equations and broader applications.

Abstract

This contribution introduces a model order reduction approach for an advection-reaction problem with a parametrized reaction function. The underlying discretization uses an ultraweak formulation with an -like trial space and an 'optimal' test space as introduced by Demkowicz et al. This ensures the stability of the discretization and in addition allows for a symmetric reformulation of the problem in terms of a dual solution which can also be interpreted as the normal equations of an adjoint least-squares problem. Classic model order reduction techniques can then be applied to the space of dual solutions which also immediately gives a reduced primal space. We show that the necessary computations do not require the reconstruction of any primal solutions and can instead be performed entirely on the space of dual solutions. We prove exponential convergence of the Kolmogorov -width and show that a greedy algorithm produces quasi-optimal approximation spaces for both the primal and the dual solution space. Numerical experiments based on the benchmark problem of a catalytic filter confirm the applicability of the proposed method.
Paper Structure (18 sections, 16 theorems, 100 equations, 5 figures, 4 tables, 1 algorithm)

This paper contains 18 sections, 16 theorems, 100 equations, 5 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Ultraweak Petrov-Galerkin formulation of reactive transport
  3. Derivation and characterization of the trial space X
  4. The optimal test space Y
  5. Optimally stable ultraweak formulation and normal equation
  6. Discretization
  7. A-priori error analysis
  8. Approximability of the parametrized problem
  9. The parametrized problem
  10. Approximation theory
  11. Test space based model order reduction
  12. Basis generation
  13. Error estimator
  14. Numerical experiments
  15. The parametrized problem
  16. ...and 3 more sections

Key Result

Corollary 2.3

There exists a linear and continuous trace operatorWe will abbreviate different trace operators by the same symbol $\gamma$ and only use subscripts if necessary. as well as a linear and continuous projection operator fulfilling $\gamma_{\mathcal{X}}(u) = u\raisebox{-.5ex}{$|$}_{{\Gamma_{out}}}$ and $\operatorname{pr}_{L^2}(u) = u$ for all $u\in \mathcal{X}_{\circ} \subset \mathcal{X}$.

Figures (5)

  • Figure 1: Geometric setup and flow profile for the Poiseuille model \ref{['eq:poiseuilleFlow']}
  • Figure 2: Geometric setup and flow field for the Darcy model \ref{['eq:darcyFlow']}
  • Figure 3: Convergence of the $L^2(\Omega)$-error under $h$-refinement. Top: Poiseuille model \ref{['eq:poiseuilleFlow']}, Bottom: Darcy model \ref{['eq:darcyFlow']}
  • Figure 4: Solutions $u_\mu$ to the parametrized problem \ref{['eq:parametrizedStrongProblem']} for different values of the parameter $\mu =(c_w,c_c,g_0)$.
  • Figure 5: Evaluation of the reduced models on a test set of $n_{test}=500$ additional randomly chosen parameters. The reduced model was obtained using the $H^1(\vec{b}, \Omega)$-norm.

Theorems & Definitions (39)

  • Remark 2.1
  • Remark 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5: Characterization of $\mathcal{X}$
  • proof
  • Corollary 2.6
  • Definition 2.7: $\Omega$-filling flow
  • ...and 29 more