球面上对数正态扩散方程的数值分析
Numerical analysis of lognormal diffusions on the sphere
AI总结：
本文研究了单位球面上静止扩散方程的数值解，通过获得各向同性高斯随机场的Hölder正则性，将其与驱动的对数正态系数的正则性联系起来，从而在Sobolev空间中获得解的L^p正则性。随后推导了多级蒙特卡洛有限元和谱元离散化方法的收敛率估计，并通过数值实验验证了理论结果。
摘要：
考虑了单位球面上具有各向同性对数正态扩散系数的静止扩散方程的数值解。获得了各向同性高斯随机场在L^p意义下的Hölder正则性，并将其与驱动的对数正态系数的正则性相关联。这导致了扩散问题解在Sobolev空间中的L^p正则性。随后推导了这些问题的多级蒙特卡洛有限元和谱元离散化方法的收敛率估计。具体而言，提供了收敛分析，给出了解的收敛率估计，以蒙特卡洛样本数和空间离散化总自由度数为变量，并给出了有限元离散化情况下算法总工作量的界限。所获得的收敛率仅与扩散系数（对数）的角功率谱衰减有关。数值实验验证了所提出的理论。
英文摘要：
Numerical solutions of stationary diffusion equations on the unit sphere with isotropic lognormal diffusion coefficients are considered. Hölder regularity in $L^p$ sense for isotropic Gaussian random fields is obtained and related to the regularity of the driving lognormal coefficients. This yields regularity in $L^p$ sense of the solution to the diffusion problem in Sobolev spaces. Convergence rate estimates of multilevel Monte Carlo Finite and Spectral Element discretizations of these problems are then deduced. Specifically, a convergence analysis is provided with convergence rate estimates in terms of the number of Monte Carlo samples of the solution to the considered diffusion equation and in terms of the total number of degrees of freedom of the spatial discretization, and with bounds for the total work required by the algorithm in the case of Finite Element discretizations. The obtained convergence rates are solely in terms of the decay of the angular power spectrum of the (logarithm) of the diffusion coefficient. Numerical examples confirm the presented theory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
