Numerical analysis of the MLMC ensemble scheme for transient heat equations with uncertain inputs

Tingfu Yao; Changlun Ye; Xianbing Luo; Tingfu Yao; Changlun Ye; Xianbing Luo

doi:10.3934/math.2025884

AIMS Mathematics

2025, Volume 10, Issue 8: 19816-19844. doi: 10.3934/math.2025884

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Research article Topical Sections

Numerical analysis of the MLMC ensemble scheme for transient heat equations with uncertain inputs

Tingfu Yao ^{1
,
,},
Changlun Ye ²,
Xianbing Luo ^{3
,
,}

1.
College of Science, Guiyang University, Guiyang 550005, China
2.
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China
3.
School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China

Received: 02 June 2025 Revised: 18 August 2025 Accepted: 22 August 2025 Published: 28 August 2025
MSC : 65C05, 65M60, 65N12

A multilevel Monte Carlo ensemble (MLMCE) coupled with the finite element method is applied to address numerically transient heat equations characterized by random diffusion and Robin coefficients. By incorporating two ensemble averages for the Robin boundary and diffusion coefficients, we present an extended Monte Carlo ensemble scheme tailored for the uncertain transient heat equation. The suggested MLMCE approach resolves a single linear system that entails multiple right-hand-side vectors for a group during each time step, thereby decreasing both the storage requirements and the computational expenses associated with the solution process. Stability analysis and error estimates of the method are derived under some conditions involving two ratios between fluctuations in the thermal conductivity and a random Robin coefficient corresponding to their mean. Numerical experiments are presented to confirm the theoretical results and verify the feasibility and effectiveness of the proposed approach.
- multilevel Monte Carlo method,
- ensemble,
- finite element,
- transient heat equation,
- Robin coefficient,
- diffusion coefficient,
- uncertain inputs
Citation: Tingfu Yao, Changlun Ye, Xianbing Luo. Numerical analysis of the MLMC ensemble scheme for transient heat equations with uncertain inputs[J]. AIMS Mathematics, 2025, 10(8): 19816-19844. doi: 10.3934/math.2025884

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Abstract

A multilevel Monte Carlo ensemble (MLMCE) coupled with the finite element method is applied to address numerically transient heat equations characterized by random diffusion and Robin coefficients. By incorporating two ensemble averages for the Robin boundary and diffusion coefficients, we present an extended Monte Carlo ensemble scheme tailored for the uncertain transient heat equation. The suggested MLMCE approach resolves a single linear system that entails multiple right-hand-side vectors for a group during each time step, thereby decreasing both the storage requirements and the computational expenses associated with the solution process. Stability analysis and error estimates of the method are derived under some conditions involving two ratios between fluctuations in the thermal conductivity and a random Robin coefficient corresponding to their mean. Numerical experiments are presented to confirm the theoretical results and verify the feasibility and effectiveness of the proposed approach.

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