论文标题
基于优化的非线性分层统计反问题的MCMC方法
Optimization-Based MCMC Methods for Nonlinear Hierarchical Statistical Inverse Problems
论文作者
论文摘要
在许多层次逆问题中，我们不仅需要在参数到观察的图中估算高或无限维模型参数，而且我们还必须估算代表统计和数学模型过程中重要假设的超级参数。作为高差异性，非线性依赖性和联合后后验分布中的非符号结构的关节作用，与模型参数和超参数相比，解决层次贝叶斯环境中的逆问题构成了重大的计算挑战。在这项工作中，我们旨在开发基于可扩展优化的马尔可夫链蒙特卡洛（MCMC）方法，用于解决分层贝叶斯逆问题，具有非线性参数到可观察的图和更广泛的超级标准。我们的算法开发基于最近开发的可扩展的随机化 - 优化（RTO）方法[4]，用于探索高或无限维模型参数空间。通过将RTO用作更新的大都市中的提案分布，或者将pseudo-marginal mcmc的偏置分布作为偏置分布[2]，我们能够为层次结构贝叶斯倒置设计有效的采样工具。特别是，RTO和伪核心MCMC的整合具有对模型参数尺寸的采样性能鲁棒。我们还将方法扩展到泊松分布的测量值的非线性反问题。 PDE约束的反问题和正电子发射断层扫描（PET）中的数值示例用于证明我们方法的性能。
In many hierarchical inverse problems, not only do we want to estimate high- or infinite-dimensional model parameters in the parameter-to-observable maps, but we also have to estimate hyperparameters that represent critical assumptions in the statistical and mathematical modeling processes. As a joint effect of high-dimensionality, nonlinear dependence, and non-concave structures in the joint posterior posterior distribution over model parameters and hyperparameters, solving inverse problems in the hierarchical Bayesian setting poses a significant computational challenge. In this work, we aim to develop scalable optimization-based Markov chain Monte Carlo (MCMC) methods for solving hierarchical Bayesian inverse problems with nonlinear parameter-to-observable maps and a broader class of hyperparameters. Our algorithmic development is based on the recently developed scalable randomize-then-optimize (RTO) method [4] for exploring the high- or infinite-dimensional model parameter space. By using RTO either as a proposal distribution in a Metropolis-within-Gibbs update or as a biasing distribution in the pseudo-marginal MCMC [2], we are able to design efficient sampling tools for hierarchical Bayesian inversion. In particular, the integration of RTO and the pseudo-marginal MCMC has sampling performance robust to model parameter dimensions. We also extend our methods to nonlinear inverse problems with Poisson-distributed measurements. Numerical examples in PDE-constrained inverse problems and positron emission tomography (PET) are used to demonstrate the performance of our methods.
