Reduced Basis Methods for Fractional Laplace Equations via Extension
Reduced Basis Methods for Fractional Laplace Equations via Extension
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DOI：
10.1137/18m1204802
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发表时间：
2018-08
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通讯作者：
Harbir Antil;Yanlai Chen;A. Narayan
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文献类型：
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作者：
Harbir Antil;Yanlai Chen;A. Narayan
分数阶拉普拉斯方程正成为数学建模和预测的重要工具。近年来，在开发精确和鲁棒的算法来数值求解此类问题方面取得了很大进展，但大多数分数问题的求解器在计算上是昂贵的。实践者通常对选择数学模型的分数指数以匹配实验和/或观测数据感兴趣;这需要对进入模型的指数和其他参数的多个值的分数方程的计算解，这是计算上昂贵的多查询问题。为了解决这个困难，我们提出了一个模型降阶策略的分数拉普拉斯问题，利用减少的基础方法（RBM）。我们的RBM算法的分数阶偏微分方程（PDE），使我们能够实现显着的加速相比，传统的PDE求解器，同时保持精度。我们的数值结果表明，我们的RBM算法的精度和效率的分数拉普拉斯问题在两个空间维。
Fractional Laplace equations are becoming important tools for mathematical modeling and prediction. Recent years have shown much progress in developing accurate and robust algorithms to numerically solve such problems, yet most solvers for fractional problems are computationally expensive. Practitioners are often interested in choosing the fractional exponent of the mathematical model to match experimental and/or observational data; this requires the computational solution to the fractional equation for several values of the both exponent and other parameters that enter the model, which is a computationally expensive many-query problem. To address this difficulty, we present a model order reduction strategy for fractional Laplace problems utilizing the reduced basis method (RBM). Our RBM algorithm for this fractional partial differential equation (PDE) allows us to accomplish significant acceleration compared to a traditional PDE solver while maintaining accuracy. Our numerical results demonstrate this accuracy and efficiency of our RBM algorithm on fractional Laplace problems in two spatial dimensions.