论文标题
贝叶斯反问题的复制交换预处理曲柄nicolson langevin动态MCMC方法
A replica exchange preconditioned Crank-Nicolson Langevin dynamic MCMC method for Bayesian inverse problems
论文作者
论文摘要
本文提出了由曲柄 - 尼科尔森方案（REPCNLD）离散的复制式交换朗格文扩散，以处理高维和多模式分布问题。从高维和多模式分布中抽样是一个具有挑战性的问题。随着参数尺寸的增加，许多标准MCMC链的性能会恶化，并且如果能量函数不是凸，许多MCMC算法将无法捕获所有模式。所提出的REPCNLD可以加速单链PCNLD的收敛性，并可以捕获多模式分布的所有模式。我们提出了曲柄 - 尼科尔森离散化，这是强大的。此外，离散误差相对于时间步长线性增长。我们将RETCNLD扩展到多变量设置，以进一步加速收敛并节省计算成本。此外，我们得出了多变量依赖方法交换率的公正估计器，为选择第二链中使用的低保真模型提供了指南。我们使用高维高斯混合模型和高维非线性PDE逆问题测试我们的方法。特别是，我们采用离散的伴随方法来有效计算非线性PDE逆问题的梯度。
This paper proposes a replica exchange preconditioned Langevin diffusion discretized by the Crank-Nicolson scheme (repCNLD) to handle high-dimensional and multi-modal distribution problems. Sampling from high-dimensional and multi-modal distributions is a challenging question. The performance of many standard MCMC chains deteriorates as the dimension of parameters increases, and many MCMC algorithms cannot capture all modes if the energy function is not convex. The proposed repCNLD can accelerate the convergence of the single-chain pCNLD, and can capture all modes of the multi-modal distributions. We proposed the Crank-Nicolson discretization, which is robust. Moreover, the discretization error grows linearly with respect to the time step size. We extend repCNLD to the multi-variance setting to further accelerate the convergence and save computation costs. Additionally, we derive an unbiased estimator of the swapping rate for the multi-variance repCNLD method, providing a guide for the choice of the low-fidelity model used in the second chain. We test our methods with high-dimensional Gaussian mixture models and high-dimensional nonlinear PDE inverse problems. Particularly, we employ the discrete adjoint method to efficiently calculate gradients for nonlinear PDE inverse problems.
