Model reduction for fractional elliptic problems using Kato's formula
Model reduction for fractional elliptic problems using Kato's formula
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DOI：
10.3934/mcrf.2021004
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发表时间：
2019-04
影响因子：
1.2
通讯作者：
H. Dinh;Harbir Antil;Yanlai Chen;E. Cherkaev;A. Narayan
中科院分区：
文献类型：
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作者：
H. Dinh;Harbir Antil;Yanlai Chen;E. Cherkaev;A. Narayan
我们提出了一种新的数值算法，利用模型约简来计算涉及谱分数阶拉普拉斯方程的平稳偏微分方程的解。我们的方法利用了已知的解的表征，即经典椭圆型问题的解的积分。我们把这个积分重新表述成一个表达式，它的连续和离散表达式都是稳定的；离散公式是稳定的，与所有离散参数无关。然后应用降基方法完成被积函数的模型阶约简。我们在积分离散化中选择的正交是一个全局高斯正交规则，我们观察到它比以前提出的正交规则更有效。最后，模型约简方法使人们能够以比传统求解器低数量级的成本计算多查询分数阶拉普拉斯问题的解。
We propose a novel numerical algorithm utilizing model reduction for computing solutions to stationary partial differential equations involving the spectral fractional Laplacian. Our approach utilizes a known characterization of the solution in terms of an integral of solutions to classical elliptic problems. We reformulate this integral into an expression whose continuous and discrete formulations are stable; the discrete formulations are stable independent of all discretization parameters. We subsequently apply the reduced basis method to accomplish model order reduction for the integrand. Our choice of quadrature in discretization of the integral is a global Gaussian quadrature rule that we observe is more efficient than previously proposed quadrature rules. Finally, the model reduction approach enables one to compute solutions to multi-query fractional Laplace problems with order of magnitude less cost than a traditional solver.