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

层次张量多态不确定性的自适应可靠数值处理

• 批准号: 312863472
• 负责人: Professor Dr. Lars Grasedyck
• 金额: --
• 依托单位: Institut für Geometrie und Praktische Mathematik (IGPM)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/312863472?language=en
• 关键词: Self; Adaptive; Reliable; Numerical; Treatment

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.

该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。这个想法是使用与张量压缩交织在一起的模型简化技术，以生成所考虑的高分辨率模型的参数表示。自适应是必要的，因为压缩模型应该被不专门研究张量的研究人员用作黑匣子。模型缩减部分负责从 PDE 模型的离散化中降低高分辨率。张量压缩部分可以处理来自模型不确定性的许多参数或相当高的维度。两个部分组合在一起提供了一种工具，可以以复杂性生成压缩表示，该复杂性与参数数量呈线性关系，并且与偏微分方程离散化的未知数数量大小呈线性关系。我们考虑几个实际模型问题，涉及模型参数、外力和初始数据产生的不确定性的混合。我们将这个问题转化为多参数和高维问题，其中参数可能来自不同的不确定性来源。参数模型的简化产生了压缩的分层低秩张量表示，可以针对任何给定的参数选择立即对其进行评估。