In this paper, we consider a Bayesian inverse problem modeled by elliptic partial differential equations (PDEs). Specifically, we propose a data-driven and model-based approach to accelerate the Hamiltonian Monte Carlo (HMC) method in solving large-scale Bayesian inverse problems. The key idea is to exploit (model-based) and construct (data-based) the intrinsic approximate low-dimensional structure of the underlying problem which consists of two components - a training component that computes a set of data-driven basis to achieve significant dimension reduction in the solution space, and a fast solving component that computes the solution and its derivatives for a newly sampled elliptic PDE with the constructed data-driven basis. Hence we achieve an effective data and model-based approach for the Bayesian inverse problem and overcome the typical computational bottleneck of HMC - repeated evaluation of the Hamiltonian involving the solution (and its derivatives) modeled by a complex system, a multiscale elliptic PDE in our case. We present numerical examples to demonstrate the accuracy and efficiency of the proposed method.
翻译:在本文中,我们考虑了以椭圆部分差异方程式(PDEs)为模型模型的巴伊西亚反面问题。具体地说,我们建议采用以数据驱动和模型为基础的方法加速汉密尔顿-蒙特卡洛(HMC)解决大规模巴伊西亚反面问题的方法。关键的想法是利用(基于模型的)和构建(基于数据的)由两个组成部分组成的根本问题的内在近似低维结构,其中两个组成部分是:一个培训部分,它计算出一套数据驱动的基础,以实现解决方案空间的显著缩小,一个快速解决部分,用构建的数据驱动基础,将解决方案及其衍生物计算成新样本的埃利奥利普蒂PDE的解决方案及其衍生物。因此,我们为巴伊西的反面问题找到一个有效的数据和模型基方法,克服HMC典型的计算瓶颈,即对汉密尔密尔顿人(及其衍生物)的反复评价,其中以复杂的系统为模型,一个多尺度的椭PDE。我们举了一些数字例子,以证明拟议方法的准确性和效率。