Affine Approximation of Parametrized Kernels and Model Order Reduction for Nonlocal and Fractional Laplace Models
Affine Approximation of Parametrized Kernels and Model Order Reduction for Nonlocal and Fractional Laplace Models
复制标题
非局部和分数拉普拉斯模型参数化核的仿射逼近和模型降阶
DOI：
10.1137/19m124321x
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
M. Gunzburger
中科院分区：
文献类型：
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作者：
O. Burkovska;M. Gunzburger
我们考虑由具有参数依赖核的空间非局部积分算子驱动的参数化问题。特别是，内核与不同的非局部相互作用半径$\delta > 0$和分数拉普拉斯内核，参数化的分数功率$s\in（0，1）$，进行了研究。为了提供一个有效的和可靠的近似的解决方案，为不同的参数值，我们开发的参数模型降阶方法的减少基础的方法。主要的困难出现，因为内核不是仿射的参数，奇异的，和不连续的。此外，空间的正则性的解决方案取决于不同的分数功率$s$。为了解决这个问题，我们得到的规律性和可微性的结果相对于$\delta$和$s$，这是独立的其他应用程序，如优化和参数识别的兴趣。然后，我们使用这些结果来构建仿射近似的内核的局部多项式。最后，我们通过提供可靠的后验误差估计来验证该方法，该估计考虑了所有近似误差，并通过数值实验支持了理论结果。
We consider parametrized problems driven by spatially nonlocal integral operators with parameter-dependent kernels. In particular, kernels with varying nonlocal interaction radius $\delta > 0$ and fractional Laplace kernels, parametrized by the fractional power $s\in(0,1)$, are studied. In order to provide an efficient and reliable approximation of the solution for different values of the parameters, we develop the reduced basis method as a parametric model order reduction approach. Major difficulties arise since the kernels are not affine in the parameters, singular, and discontinuous. Moreover, the spatial regularity of the solutions depends on the varying fractional power $s$. To address this, we derive regularity and differentiability results with respect to $\delta$ and $s$, which are of independent interest for other applications such as optimization and parameter identification. We then use these results to construct affine approximations of the kernels by local polynomials. Finally, we certify the method by providing reliable a posteriori error estimators, which account for all approximation errors, and support the theoretical findings by numerical experiments.
DOI：
10.1137/18m1204802
发表时间：
2018-08
期刊：
SIAM J. Sci. Comput.
影响因子：
--
作者：
Harbir Antil;Yanlai Chen;A. Narayan
通讯作者：
Harbir Antil;Yanlai Chen;A. Narayan