We investigate solution methods for large-scale inverse problems governed by partial differential equations (PDEs) via Bayesian inference. The Bayesian framework provides a statistical setting to infer uncertain parameters from noisy measurements. To quantify posterior uncertainty, we adopt Markov Chain Monte Carlo (MCMC) approaches for generating samples. To increase the efficiency of these approaches in high-dimension, we make use of local information about gradient and Hessian of the target potential, also via Hamiltonian Monte Carlo (HMC). Our target application is inferring the field of soil permeability processing observations of pore pressure, using a nonlinear PDE poromechanics model for predicting pressure from permeability. We compare the performance of different sampling approaches in this and other settings. We also investigate the effect of dimensionality and non-gaussianity of distributions on the performance of different sampling methods.

翻译：我们通过贝叶斯推断研究受偏微分方程（PDE）控制的大规模反问题的求解方法。贝叶斯框架提供了从含噪测量中推断不确定参数的统计建模方法。为量化后验不确定性，我们采用马尔可夫链蒙特卡洛（MCMC）方法生成样本。为提升该方法在高维问题中的效率，我们利用目标势函数的梯度与海森矩阵的局部信息，并引入哈密顿蒙特卡洛（HMC）方法。目标应用是通过非线性PDE孔隙力学模型（用于根据渗透率预测压力）处理孔隙压力观测数据，推断土壤渗透率场。我们比较了不同采样方法在该场景及其他场景中的性能，并研究了分布的高维性与非高斯性对不同采样方法性能的影响。