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Evolution of Staircase Structures in Diffusive Convection
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Multilevel Monte Carlo Methods for Stochastic Convection-Diffusion Eigenvalue Problems.

Tiangang Cui1, Hans De Sterck2, Alexander D Gilbert3

  • 1School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006 Australia.

Journal of Scientific Computing
|May 6, 2024
PubMed
Summary
This summary is machine-generated.

New multilevel Monte Carlo (MLMC) methods efficiently estimate the smallest eigenvalue of stochastic operators. These methods offer superior complexity compared to plain Monte Carlo, enhancing computational efficiency for complex problems.

Keywords:
Convection–diffusion eigenvalue problemsHomotopyMultilevel Monte CarloUncertainty quantification

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Scientific computing

Background:

  • Estimating eigenvalues of stochastic operators is crucial in various scientific fields.
  • Traditional Monte Carlo methods can be computationally expensive for such problems.
  • Stochastic convection-diffusion operators with random coefficients present unique challenges.

Purpose of the Study:

  • To develop and analyze new multilevel Monte Carlo (MLMC) methods for estimating the smallest eigenvalue of stochastic convection-diffusion operators.
  • To improve the computational efficiency and stability of eigenvalue estimation for stochastic problems.
  • To compare the performance of MLMC with plain Monte Carlo and quasi-Monte Carlo (QMC) methods.

Main Methods:

  • Development of MLMC methods using a hierarchy of finite element (FE) discretizations.
  • Application of Rayleigh quotient (RQ) iteration and implicitly restarted Arnoldi (IRA) for algebraic eigenproblems.
  • Adaptation of FE error bounds to stochastic settings and analysis of variance on each level.
  • Implementation of mesh hierarchy exploitation and streamline upwind Petrov-Galerkin formulation for improved efficiency and stability.
  • Introduction of a multilevel quasi-Monte Carlo (QMC) method for enhanced complexity.

Main Results:

  • The developed MLMC estimator demonstrates superior complexity bounds compared to plain Monte Carlo.
  • Strategies for exploiting mesh hierarchy and employing streamline upwind Petrov-Galerkin formulations enhance MLMC efficiency and stability.
  • The multilevel QMC method further improves overall complexity due to faster convergence.
  • Numerical results validate the theoretical analysis and demonstrate the practical feasibility and superiority of MLMC methods.

Conclusions:

  • MLMC methods provide a computationally efficient and stable approach for estimating eigenvalues of stochastic convection-diffusion operators.
  • The proposed enhancements, including QMC integration, significantly outperform traditional Monte Carlo techniques.
  • These methods offer a robust framework for tackling complex eigenvalue problems in scientific and engineering applications.