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:我们建议一个多层次的蒙特卡洛-FEM 算法, 以解决“ Besov 随机树 ” 的流体巴耶斯反向问题。 这些前缀是由一个带有随机系数的波盘序列和一些条件随机消失的扩展条件给出的。 根据所谓的Galton- Watson 树的法律, 这允许将随机的分形结构和大量偏差纳入日志扩散中, 这自然出现在来自地球物理学或医学成像的许多应用中。 这个框架包含两个主要难题 : 首先, 相关的传播系数不能满足统一的流体状态, 从而导致无法加固的术语, 从而导致标准多级别估量器的差异 。 其次, 相关的参数空间是波兰的, 但不是规范的线性空间 。 我们通过在顶点引入断函数来补偿非可加固的术语, 而第二个问题则通过使用独立的Metropolis- Hastings 样板来解决 。 由此产生的算法在平均值上会趋同准, 其基本为最优的接受性复杂度和维度。