We develop a multilevel Monte Carlo (MLMC)-FEM algorithm for linear, elliptic diffusion problems in polytopal domain $\mathcal D\subset \mathbb R^d$, with Besov-tree random coefficients. This is to say that the logarithms of the diffusion coefficients are sampled from so-called Besov-tree priors, which have recently been proposed to model data for fractal phenomena in science and engineering. Numerical analysis of the fully discrete FEM for the elliptic PDE includes quadrature approximation and must account for a) nonuniform pathwise upper and lower coefficient bounds, and for b) low path-regularity of the Besov-tree coefficients. Admissible non-parametric random coefficients correspond to random functions exhibiting singularities on random fractals with tunable fractal dimension, but involve no a-priori specification of the fractal geometry of singular supports of sample paths. Optimal complexity and convergence rate estimates for quantities of interest and for their second moments are proved. A convergence analysis for MLMC-FEM is performed which yields choices of the algorithmic steering parameters for efficient implementation. A complexity (``error vs work'') analysis of the MLMC-FEM approximations is provided.

翻译：针对多面体域$\mathcal D\subset \mathbb R^d$中具有Besov树随机系数的线性椭圆扩散问题，我们发展了一种多水平蒙特卡洛（MLMC）-有限元算法。具体而言，扩散系数的对数从所谓的Besov树先验中采样，该先验最近被提出用于模拟科学与工程中分形现象的数据。椭圆型偏微分方程全离散有限元法的数值分析需考虑求积近似，并须处理：a）非均匀路径上下系数界；b）Besov树系数的低路径正则性。可容许的非参数随机系数对应在具有可调分形维数的随机分形上呈现奇异性的随机函数，但无需预先指定样本路径奇异支撑的分形几何结构。我们证明了目标量及其二阶矩的最优复杂度与收敛率估计。通过收敛性分析确定了MLMC-FEM算法的高效实现参数选择，并给出了MLMC-FEM近似的复杂度（"误差与工作量"）分析。