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.

中文翻译：

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非局部和分数拉普拉斯模型的参数化核的仿射近似和模型降阶

我们考虑由具有参数相关内核的空间非局部积分运算符驱动的参数化问题。特别地，研究了具有不同非局部相互作用半径 $\delta > 0$ 的内核和由分数幂 $s\in(0,1)$ 参数化的分数拉普拉斯内核。为了为参数的不同值提供有效且可靠的解决方案近似值，我们开发了简化基方法作为参数模型降阶方法。由于内核在参数、奇异和不连续方面不仿射，因此出现了主要困难。此外，解的空间规律性取决于变化的分数幂 $s$。为了解决这个问题，我们推导出关于 $\delta$ 和 $s$ 的规律性和可微性结果，它们对其他应用程序（例如优化和参数识别）具有独立的兴趣。然后我们使用这些结果通过局部多项式构建内核的仿射近似。最后，我们通过提供可靠的后验误差估计器来证明该方法，该估计器解释了所有近似误差，并通过数值实验支持理论发现。