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Construction of local reduced spaces for Friedrichs' systems via randomized training

Christian Engwer, Mario Ohlberger, Lukas Renelt

TL;DR

This work develops and analyzes localized training for Friedrichs' systems by introducing transfer operators that map boundary data on oversampling domains to interior solutions and proving their compactness via a Caccioppoli-type energy decay and Maxwell-type embedding results. The authors formulate local problems, derive the associated transfer operators, and show that the resulting local spaces are quasi-optimal through a randomized range-finder approach, enabling practical construction of near-optimal reduced spaces. The methodology is then applied to a convection-diffusion-reaction problem in first-order form, where the Friedrichs framework provides the necessary smoothing and compactness properties. Numerical experiments with heterogeneous diffusion channels demonstrate exponential convergence of the localized spaces, validating the approach and highlighting the influence of oversampling geometry on performance, with implications for efficient parametric and multiscale PDE solvers.

Abstract

This contribution extends the localized training approach, traditionally employed for multiscale problems and parameterized partial differential equations (PDEs) featuring locally heterogeneous coefficients, to the class of linear, positive symmetric operators, known as Friedrichs' operators. Considering a local subdomain with corresponding oversampling domain we prove the compactness of the transfer operator which maps boundary data to solutions on the interior domain. While a Caccioppoli-inequality quantifying the energy decay to the interior holds true for all Friedrichs' systems, showing a compactness result for the graph-spaces hosting the solution is additionally necessary. We discuss the mixed formulation of a convection-diffusion-reaction problem where the necessary compactness result is obtained by the Picard-Weck-Weber theorem. Our numerical results, focusing on a scenario involving heterogeneous diffusion fields with multiple high-conductivity channels, demonstrate the effectiveness of the proposed method.

Construction of local reduced spaces for Friedrichs' systems via randomized training

TL;DR

This work develops and analyzes localized training for Friedrichs' systems by introducing transfer operators that map boundary data on oversampling domains to interior solutions and proving their compactness via a Caccioppoli-type energy decay and Maxwell-type embedding results. The authors formulate local problems, derive the associated transfer operators, and show that the resulting local spaces are quasi-optimal through a randomized range-finder approach, enabling practical construction of near-optimal reduced spaces. The methodology is then applied to a convection-diffusion-reaction problem in first-order form, where the Friedrichs framework provides the necessary smoothing and compactness properties. Numerical experiments with heterogeneous diffusion channels demonstrate exponential convergence of the localized spaces, validating the approach and highlighting the influence of oversampling geometry on performance, with implications for efficient parametric and multiscale PDE solvers.

Abstract

This contribution extends the localized training approach, traditionally employed for multiscale problems and parameterized partial differential equations (PDEs) featuring locally heterogeneous coefficients, to the class of linear, positive symmetric operators, known as Friedrichs' operators. Considering a local subdomain with corresponding oversampling domain we prove the compactness of the transfer operator which maps boundary data to solutions on the interior domain. While a Caccioppoli-inequality quantifying the energy decay to the interior holds true for all Friedrichs' systems, showing a compactness result for the graph-spaces hosting the solution is additionally necessary. We discuss the mixed formulation of a convection-diffusion-reaction problem where the necessary compactness result is obtained by the Picard-Weck-Weber theorem. Our numerical results, focusing on a scenario involving heterogeneous diffusion fields with multiple high-conductivity channels, demonstrate the effectiveness of the proposed method.
Paper Structure (12 sections, 8 theorems, 28 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 12 sections, 8 theorems, 28 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Linear, positive symmetric PDE-operators
  3. Optimal local approximation spaces
  4. Derivation of the local problem formulation
  5. Compactness of the transfer operator
  6. Quasi-optimal approximation spaces via localized training
  7. Application to diffusion problems in mixed form
  8. Weak formulation
  9. Numerical experiments
  10. Different oversampling sizes
  11. Results for the full CDR problem
  12. Conclusion

Key Result

Theorem 2.1

For any $f\in L^2(\Omega)^m$, the problem is well-posed. Its solution $u$ is the unique minimizer of the residual energy

Figures (7)

  • Figure 1: Influence of different oversampling sizes $\delta$ in a simple diffusion test case. The same grid width $h=1/30$ was used in all tests.
  • Figure 2: Computational domain.
  • Figure 3: Diagonal entries of the diffusion coefficient $D_{ii}$ exhibiting high-conductivity channels.
  • Figure 4: Exemplary solution to the local problem with channels in $x$-direction.
  • Figure 5: Error in the range approximation for non intersecting parallel high-conductivity channels.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 2.1: Well-posedness ernGuermond2007
  • Proposition 3.1: Caccioppoli inequality
  • Theorem 3.2: Compactness of $T_i$
  • proof
  • Proposition 4.1: buhr2018randomized
  • Proposition 5.1
  • Lemma 5.2
  • proof
  • Theorem 5.3: Picard-Weber-Weck picard1984maxwellweber1980maxwellweck1974maxwell
  • Theorem 5.4: Rellich compactness theorem rellich1930