SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 3, Page 1013-1033, January 2021.
We study numerical approximation for one-dimensional stochastic elliptic equations with integral fractional Laplacian and the additive Gaussian noise of power-law: $1/f^\beta$ noise and fractional Brownian noise. We present an optimal convergence of our method using spectral expansions of noises. We first establish the well-posedness of a corresponding deterministic problem and show the stability of solutions for the rough data via negative norms in weighted Sobolev spaces. We also analyze the regularity of the noise and approximation properties of their finite truncations. Next, we show the optimal error estimates of our method for a wide range of parameters in the order of fractional operator and the fractional Gaussian noise. Finally, we present several numerical examples to illustrate the mean-square convergence orders and verify our optimal convergence rates.

中文翻译：

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具有加性分数高斯噪声的分数椭圆方程最优收敛的数值逼近

SIAM/ASA 不确定性量化杂志，第 9 卷，第 3 期，第 1013-1033 页，2021 年 1 月。
我们研究了具有积分分数拉普拉斯算子和幂律加性高斯噪声的一维随机椭圆方程的数值近似：$1/f^\beta$ 噪声和分数布朗噪声。我们使用噪声的频谱扩展来呈现我们方法的最佳收敛。我们首先建立相应确定性问题的适定性，并通过加权 Sobolev 空间中的负范数显示粗数据解的稳定性。我们还分析了其有限截断的噪声和近似特性的规律性。接下来，我们按照分数运算符和分数高斯噪声的顺序展示了我们的方法对各种参数的最佳误差估计。最后，