Geometric MCMC for infinite-dimensional inverse problems
Geometric MCMC for infinite-dimensional inverse problems
复制标题
DOI：
10.1016/j.jcp.2016.12.041
复制
发表时间：
2017-04-15
影响因子：
4.1
通讯作者：
Stuart, Andrew M.
中科院分区：
文献类型：
--
作者：
Beskos, Alexandros;Girolami, Mark;Stuart, Andrew M.
贝叶斯逆问题通常涉及在无限尺寸函数空间上取样后验分布。传统的马尔可夫链蒙特卡洛（MCMC）算法的特征是当有限的尺寸近似值变得更加准确时，在网状再填充时混合时间恶化。这种方法通常被迫随着离散化的效果而降低步进尺寸，因此随着维度的函数而变得昂贵。最近，出现了一种新的具有与网格无关的收敛时间的MCMC方法。但是，很少有人考虑到数据告知的后验的几何形状。同时，已发现最近开发的几何MCMC算法在探索复杂的分布方面具有强大的功能，这些分布显着偏离了椭圆形的高斯法律，但对于在无限维度定义的模型中通常在计算上是可悲的。在这项工作中，我们将有限维子空间上的几何方法与网状独立的无限二维方法相结合。我们的目标是加快MCMC混合时间，而不会显着增加每个步骤的计算成本（例如，与香草预处理的曲柄尼古尔森（PCN）方法相比）。这是通过使用几何MCMC的想法来探测内在有限维子空间的复杂结构来实现的，其中大多数数据信息集中在同时，同时随着尺寸通过在互补子空间中使用PCN样方法而增长的稳健混合时间。在地下流动，热传导和不可压缩的流量控制中产生的三个具有挑战性的反问题的背景下，证明了所得算法。与PCN方法相比，该算法在采样效率方面最多提高了两个数量级。 （c）2017年作者。由Elsevier Inc.出版
Bayesian inverse problems often involve sampling posterior distributions on infinite dimensional function spaces. Traditional Markov chain Monte Carlo (MCMC) algorithms are characterized by deteriorating mixing times upon mesh-refinement, when the finite dimensional approximations become more accurate. Such methods are typically forced to reduce step-sizes as the discretization gets finer, and thus are expensive as a function of dimension. Recently, a new class of MCMC methods with mesh-independent convergence times has emerged. However, few of them take into account the geometry of the posterior informed by the data. At the same time, recently developed geometric MCMC algorithms have been found to be powerful in exploring complicated distributions that deviate significantly from elliptic Gaussian laws, but are in general computationally intractable for models defined in infinite dimensions. In this work, we combine geometric methods on a finite-dimensional subspace with mesh-independent infinite-dimensional approaches. Our objective is to speed up MCMC mixing times, without significantly increasing the computational cost per step (for instance, in comparison with the vanilla preconditioned Crank Nicolson (pCN) method). This is achieved by using ideas from geometric MCMC to probe the complex structure of an intrinsic finite-dimensional subspace where most data information concentrates, while retaining robust mixing times as the dimension grows by using pCN-like methods in the complementary subspace. The resulting algorithms are demonstrated in the context of three challenging inverse problems arising in subsurface flow, heat conduction and incompressible flow control. The algorithms exhibit up to two orders of magnitude improvement in sampling efficiency when compared with the pCN method. (C) 2017 The Authors. Published by Elsevier Inc.