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<p>In this paper, we propose a Monte Carlo method for solving high-dimensional PDEs involving the variable-order fractional Laplacian operator (VoFL) on bounded domains. By utilizing Green function and Poisson kernel associated with the VoFL, we construct the Feynman-Kac formula for the variable-order fractional Poisson equation on the unit ball in arbitrary dimensions. Together with the idea of walk-on-sphere method, we then derive a conditional trajectory sampling algorithm for solving the variable-order fractional Poisson equation in irregular domains. The proposed method achieves remarkable efficiency in solving high-dimensional Poisson equations with VoFL, as it only requires the evaluation of expectation integrals over maximally inscribed balls within smaller balls, thereby overcoming the curse of dimensionality. Additionally, we prove that the proposed method is unbiased and provide the error bounds. Furthermore, we extend the method to parabolic equations with VoFL. Extensive numerical results, including cases up to 100 dimensions, not only validate our theoretical findings and demonstrate the robustness and accuracy of the proposed approach, but also illustrate that, compared to constant-order fractional PDEs, variable-order fractional PDEs and the proposed numerical methods offer better simulations of more complex physical phenomena, such as the ion diffusion between homogeneous and heterogeneous media, the coexistence of anomalous and normal diffusion, and many others.</p>

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On Probabilistic Methods for Linear PDEs Involving Variable-Order fractional Laplacian in High Dimensions

  • 期刊论文

  • 发表日期：

    2026年3月18日

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  • Zhiyuan Hui,
  • Huan Liu,
  • Changtao Sheng,
  • Chenglong Xu

摘要

In this paper, we propose a Monte Carlo method for solving high-dimensional PDEs involving the variable-order fractional Laplacian operator (VoFL) on bounded domains. By utilizing Green function and Poisson kernel associated with the VoFL, we construct the Feynman-Kac formula for the variable-order fractional Poisson equation on the unit ball in arbitrary dimensions. Together with the idea of walk-on-sphere method, we then derive a conditional trajectory sampling algorithm for solving the variable-order fractional Poisson equation in irregular domains. The proposed method achieves remarkable efficiency in solving high-dimensional Poisson equations with VoFL, as it only requires the evaluation of expectation integrals over maximally inscribed balls within smaller balls, thereby overcoming the curse of dimensionality. Additionally, we prove that the proposed method is unbiased and provide the error bounds. Furthermore, we extend the method to parabolic equations with VoFL. Extensive numerical results, including cases up to 100 dimensions, not only validate our theoretical findings and demonstrate the robustness and accuracy of the proposed approach, but also illustrate that, compared to constant-order fractional PDEs, variable-order fractional PDEs and the proposed numerical methods offer better simulations of more complex physical phenomena, such as the ion diffusion between homogeneous and heterogeneous media, the coexistence of anomalous and normal diffusion, and many others.