We propose a multilevel Monte Carlo-FEM algorithm to solve elliptic Bayesian inverse problems with "Besov random tree prior". These priors are given by a wavelet series with stochastic coefficients, and certain terms in the expansion vanishing at random, according to the law of so-called Galton-Watson trees. This allows to incorporate random fractal structures and large deviations in the log-diffusion, which occur naturally in many applications from geophysics or medical imaging. This framework entails two main difficulties: First, the associated diffusion coefficient does not satisfy a uniform ellipticity condition, which leads to non-integrable terms and thus divergence of standard multilevel estimators. Secondly, the associated space of parameters is Polish, but not a normed linear space. We address the first point by introducing cut-off functions in the estimator to compensate for the non-integrable terms, while the second issue is resolved by employing an independence Metropolis-Hastings sampler. The resulting algorithm converges in the mean-square sense with essentially optimal asymptotic complexity, and dimension-independent acceptance probabilities.

翻译：我们提出了一种多层蒙特卡洛-有限元算法，用于求解具有"Besov随机树先验"的椭圆贝叶斯反问题。此类先验由带有随机系数的子波级数定义，其中部分展开项根据所谓高尔顿-沃森树的规律随机消失。这使得在扩散对数项中能够纳入随机分形结构和大偏差特性，而这类特性在地球物理或医学成像等许多应用中自然出现。该框架存在两大难点：首先，相应的扩散系数不满足一致椭圆条件，导致存在不可积项，从而使标准多层估计量发散；其次，相应的参数空间是波兰空间，但并非赋范线性空间。针对第一个问题，我们在估计量中引入截断函数以补偿不可积项；针对第二个问题，则采用独立梅特罗波利斯-黑斯廷斯采样器。最终算法均方收敛，具有本质上最优的渐近复杂度，且接受概率与维数无关。