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Reduced-order modelling numerical homogenization.

A Abdulle1, Y Bai2

  • 1ANMC, Mathematics Section, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland assyr.adulle@epfl.ch.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
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PubMed
Summary
This summary is machine-generated.

This study introduces a framework combining numerical homogenization and reduced-order modeling for multiscale partial differential equations (PDEs). This approach significantly reduces computational cost for complex problems by optimizing microproblem calculations.

Keywords:
multiscale methodoscillatory partial differential equationsreduced basis

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Area of Science:

  • Computational Mathematics
  • Applied Mathematics
  • Scientific Computing

Background:

  • Multiscale partial differential equations (PDEs) present significant computational challenges.
  • Traditional numerical homogenization methods require solving numerous microproblems, leading to high computational expense, especially for high-dimensional, time-dependent, or nonlinear problems.

Purpose of the Study:

  • To develop a general framework combining numerical homogenization and reduced-order modeling (ROM) techniques for multiscale PDEs.
  • To significantly reduce the computational cost associated with classical numerical homogenization.

Main Methods:

  • Integration of numerical homogenization with reduced-order modeling techniques.
  • Implementation of an offline stage for accurate computation of effective data at selected grid points.
  • Development of an online stage for efficient 'interpolation' of effective data, achieving online costs comparable to single-scale solvers.

Main Results:

  • The proposed methodology effectively reduces the computational burden of numerical homogenization for multiscale PDEs.
  • The framework is applicable to a wide range of PDEs, including elliptic, parabolic, wave, and nonlinear problems.
  • Numerical examples demonstrate the method's efficacy in wave propagation and solute transport simulations.

Conclusions:

  • The combination of numerical homogenization and reduced-order modeling offers a computationally efficient solution for multiscale PDEs.
  • This integrated approach makes complex multiscale simulations more tractable and cost-effective.