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Reduced basis solvers for unfitted methods on parameterized domains

Nicholas Mueller, Santiago Badia, Yiran Zhao

TL;DR

This paper addresses reduced-order modeling for parametrized PDEs on domains that change with parameters. It fuses unfitted finite elements with both classical RB and tensor-train RB (TT-RB), using a deformation map to a fixed reference configuration and a localization strategy to build local reduced subspaces with MDEIM-based hyper-reduction. It extends the framework to saddle-point problems via supremizer enrichment defined on the reference configuration, and demonstrates the approach on Poisson, linear elasticity, Stokes, and Navier–Stokes problems, achieving high accuracy with meaningful speedups. The work advances model reduction on general geometries by improving compressibility through deformation-based representations and locality, offering a scalable pathway for efficient simulations on complex, parameterized domains.

Abstract

In this paper, we present a unified framework for reduced basis approximations of parametrized partial differential equations defined on parameter-dependent domains. Our approach combines unfitted finite element methods with both classical and tensor-based reduced basis techniques -- particularly the tensor-train reduced basis method -- to enable efficient and accurate model reduction on general geometries. To address the challenge of reconciling geometric variability with fixed-dimensional snapshot representations, we adopt a deformation-based strategy that maps a reference configuration to each parameterized domain. Furthermore, we introduce a localization procedure to construct dictionaries of reduced subspaces and hyper-reduction approximations, which are obtained via matrix discrete empirical interpolation in our work. We extend the proposed framework to saddle-point problems by adapting the supremizer enrichment strategy to unfitted methods and deformed configurations, demonstrating that the supremizer operator can be defined on the reference configuration without loss of stability. Numerical experiments on two- and three-dimensional problems -- including Poisson, linear elasticity, incompressible Stokes and Navier-Stokes equations -- demonstrate the flexibility, accuracy and efficiency of the proposed methodology.

Reduced basis solvers for unfitted methods on parameterized domains

TL;DR

This paper addresses reduced-order modeling for parametrized PDEs on domains that change with parameters. It fuses unfitted finite elements with both classical RB and tensor-train RB (TT-RB), using a deformation map to a fixed reference configuration and a localization strategy to build local reduced subspaces with MDEIM-based hyper-reduction. It extends the framework to saddle-point problems via supremizer enrichment defined on the reference configuration, and demonstrates the approach on Poisson, linear elasticity, Stokes, and Navier–Stokes problems, achieving high accuracy with meaningful speedups. The work advances model reduction on general geometries by improving compressibility through deformation-based representations and locality, offering a scalable pathway for efficient simulations on complex, parameterized domains.

Abstract

In this paper, we present a unified framework for reduced basis approximations of parametrized partial differential equations defined on parameter-dependent domains. Our approach combines unfitted finite element methods with both classical and tensor-based reduced basis techniques -- particularly the tensor-train reduced basis method -- to enable efficient and accurate model reduction on general geometries. To address the challenge of reconciling geometric variability with fixed-dimensional snapshot representations, we adopt a deformation-based strategy that maps a reference configuration to each parameterized domain. Furthermore, we introduce a localization procedure to construct dictionaries of reduced subspaces and hyper-reduction approximations, which are obtained via matrix discrete empirical interpolation in our work. We extend the proposed framework to saddle-point problems by adapting the supremizer enrichment strategy to unfitted methods and deformed configurations, demonstrating that the supremizer operator can be defined on the reference configuration without loss of stability. Numerical experiments on two- and three-dimensional problems -- including Poisson, linear elasticity, incompressible Stokes and Navier-Stokes equations -- demonstrate the flexibility, accuracy and efficiency of the proposed methodology.

Paper Structure

This paper contains 27 sections, 3 theorems, 92 equations, 7 figures, 5 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Full order model
  3. Mesh deformation techniques
  4. Aggregated cell method
  5. Full order model
  6. Nodal value extension
  7. Induced norms
  8. Reduced basis method
  9. Global subspace generation
  10. Reduced order equations
  11. Local subspaces method
  12. Tensor-train reduced basis method
  13. Tensor-train decomposition
  14. Global TT-RB subspace generation
  15. Notes on reduced order equations, hyper-reduction and localized strategy for TT-RB
  16. ...and 12 more sections

Key Result

Theorem 1

Let $(\mathcal{V}_h,\mathcal{Q}_h)$ be the aggregated fe spaces defined above. Moreover, we define where $\underline{\widetilde{u}}_h \doteq \underline{u}_h \circ \underline{\psi}^{-1}$, $\underline{\widetilde{v}}_h \doteq \underline{v}_h \circ \underline{\psi}^{-1}$ and $\widetilde{q}_h \doteq q_h \circ \underline{\psi}^{-1}$. Assuming $\eta$ is large enough Neiva2021, there exists a positive co

Figures (7)

  • Figure 1: Reference and deformed mesh obtained respectively for $\widetilde{\bm{\mu}} = (0.5,0.12)$ and $\bm{\mu} = (0.6,0.15)$.
  • Figure 2: Cells and nodes in the aggregated cell method in the case of a circular hole cut from a background grid.
  • Figure 3: Harmonic extension into a circular hole cut from the background grid.
  • Figure 4: Results for the Poisson equation benchmark, obtained with ttrb. fe solution (left), ttrb solution obtained with a tolerance $\varepsilon = 10^{-4}$ (center-left), and point-wise error (centre-right, with a tolerance $\varepsilon = 10^{-3}$, and right, with a tolerance $\varepsilon = 10^{-4}$). Value of the test parameter: $\bm{\mu} = (0.43, 0.26)^T$.
  • Figure 5: Results for the linear elasticity equation, obtained with ttrb. Longitudinal mid-section view of the fe solution (left), ttrb solution obtained with a tolerance $\varepsilon = 10^{-4}$ (center-left), and point-wise error magnitude (centre-right, with a tolerance $\varepsilon = 10^{-3}$, and right, with a tolerance $\varepsilon = 10^{-4}$). Value of the test parameter: $\bm{\mu} = (1.09, 0.52)^T$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof