This paper presents a novel method for the accurate functional approximation of possibly highly concentrated probability densities. It is based on the combination of several modern techniques such as transport maps and low-rank approximations via a nonintrusive tensor train reconstruction. The central idea is to carry out computations for statistical quantities of interest such as moments based on a convenient representation of a reference density for which accurate numerical methods can be employed. Since the transport from target to reference can usually not be determined exactly, one has to cope with a perturbed reference density due to a numerically approximated transport map. By the introduction of a layered approximation and appropriate coordinate transformations, the problem is split into a set of independent approximations in seperately chosen orthonormal basis functions, combining the notions h- and p-refinement (i.e. “mesh size” and polynomial degree). An efficient low-rank representation of the perturbed reference density is achieved via the Variational Monte Carlo method. This nonintrusive regression technique reconstructs the map in the tensor train format. An a priori convergence analysis with respect to the error terms introduced by the different (deterministic and statistical) approximations in the Hellinger distance and the Kullback–Leibler divergence is derived. Important applications are presented and in particular the context of Bayesian inverse problems is illuminated which is a main motivation for the developed approach. Several numerical examples illustrate the efficacy with densities of different complexity and degrees of perturbation of the transport to the reference density. The (superior) convergence is demonstrated in comparison to Monte Carlo and Markov Chain Monte Carlo methods.

本文提出了一种对可能高度集中的概率密度进行精确泛函逼近的新方法。它基于几种现代技术的组合，例如传输图和通过非侵入式张量列车重建的低秩近似。中心思想是基于参考密度的方便表示来计算感兴趣的统计量，例如矩量，可以使用精确的数值方法。由于从目标到参考的传输通常无法准确确定，因此必须处理由于数字近似传输图而导致的扰动参考密度。通过引入分层近似和适当的坐标变换，问题被分解为一组独立的近似，分别选择正交基函数，结合概念 h 和 p 细化（即“网格大小”和多项式次数）。通过变分蒙特卡罗方法实现了扰动参考密度的有效低秩表示。这种非侵入式回归技术以张量训练格式重建地图。导出了关于由 Hellinger 距离和 Kullback-Leibler 散度中的不同（确定性和统计）近似引入的误差项的先验收敛性分析。介绍了重要的应用，特别是阐明了贝叶斯逆问题的背景，这是开发方法的主要动机。几个数值示例说明了不同复杂度的密度和传输到参考密度的扰动程度的功效。与蒙特卡罗和马尔可夫链蒙特卡罗方法相比，证明了（优越的）收敛性。