Self-Adaptive Reliable Numerical Treatment of Polymorphic Uncertainty by Hierarchical Tensors

本项目的目的是开发一种快速可靠的自适应仿真工具，可用于多态不确定性量化。我们的想法是使用与张量压缩交织在一起的模型简化技术，以产生所考虑的高分辨率模型的参数表示。自适应性是必要的，因为压缩模型应该被非张量专业的研究人员用作黑盒。模型简化部分负责从PDE模型的离散化中降低高分辨率。张量压缩部分可以科普来自模型中的不确定性的许多参数或相当高的维度。组合的两个部分提供了一种工具，该工具以在参数的数量上是线性的并且在用于PDE离散化的未知数的数量的大小上是线性的复杂度产生压缩表示。我们考虑几个实际的模型问题，涉及的混合物中的模型，外力和初始数据中的参数所产生的不确定性。我们把这个问题转化为一个多参数和高维的参数可能来自不同来源的不确定性。参数模型的减少产生了一个压缩的层次低秩张量表示，可以立即评估任何给定的参数选择。

The aim of this project is to develop a fast and reliable self-adaptive simulation tool that can be used for polymorphic uncertainty quantification. The idea is to use model reduction techniques intertwined with tensor compression in order to produce a parametric representation of the high resolution model under consideration. The self-adaptivity is necessary since the compressed model should be used as a black box by researchers that are not specialized in tensors. The model reduction part is responsible for the reduction of the high resolution from the discretisation of the PDE model. The tensor compression part can cope with the many parameters or equivalently high dimensionality from the uncertainty in the model. Both parts combined provide a tool that produces the compressed representation in a complexity that is linear in the number of parameters and linear in the size of the number of unknowns for the PDE discretisation. We consider several practical model problems involving a mixture of uncertainties that arise from parameters in the model, external forces and initial data. We transform this problem into a multiparametric and high-dimensional one where parameters may come from different sources of uncertainty. The reduction of the parametric model gives rise to a compressed hierarchical low rank tensor representation which can be evaluated instantly for any given choice of parameters.