Tensor methods in multi-dimensional spectral problems with particular application in electronic structure calculations

多维谱问题中的张量方法，特别适用于电子结构计算

• 批准号: 79050120
• 负责人: Professor Dr. Wolfgang Hackbusch
• 金额: --
• 依托单位: Fachgebiet Modellierung, Simulation und Optimierung in Natur- und Ingenieurwissenschaften
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2008
• 资助国家: 德国
• 起止时间: 2007-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/79050120?language=en
• 关键词: Tensor; methods; multi; dimensional; spectral

Many problems from modern physics and other fields are posed on high-dimensional tensor spaces in a natural way. Numerical approximation of solutions suffers from the curse of dimensionality, i.e. the computational complexity scales exponentially with the dimension of the space. Their numerical treatment therefore always involves methods of approximation by data-sparse representation ontensor spaces. There has very recently been remarkable progress in tensor product approximations. Newly introduced multiscale tensorisation techniques are provided by novel tensor formats. We aim to continue and enhance these developments and to use it for the design and critical assessment of algorithms for the new tensor formats treating high dimensional spectral problems. A particular emphasis is on the systematic examination of the potential which these methods may have in connection with various well-known algorithms used for the treatment of the electronic Schrödinger equation. In particular, our goal is to combine those novel techniques with the requirements and existing wellestablished algorithms for the electronic Schrödinger equation, in particular with one-particle models as the Hartree-Fock and Kohn-Sham method. In this respect, the proposal aims at the continuation of our project within the SPP 1324, concerned with the development of novel, mathematically sound tensor product methods for the numerical treatment of high-dimensional eigenvalue problems.

现代物理学和其他领域的许多问题都是以自然的方式在高维张量空间上提出的。解的数值近似会受到维数灾难的影响，即计算复杂性随着空间维数呈指数级增长。因此，它们的数值处理总是涉及通过张量空间上的数据稀疏表示来近似的方法。最近在张量积近似方面取得了显着的进展。新引入的多尺度张量化技术由新颖的张量格式提供。我们的目标是继续并加强这些开发，并将其用于处理高维谱问题的新张量格式的算法的设计和关键评估。特别强调的是对这些方法可能与用于处理电子薛定谔方程的各种众所周知的算法相关的潜力进行系统检查。特别是，我们的目标是将这些新技术与电子薛定谔方程的要求和现有的完善算法相结合，特别是与 Hartree-Fock 和 Kohn-Sham 方法等单粒子模型相结合。在这方面，该提案旨在继续我们在 SPP 1324 内的项目，涉及开发新颖的、数学上合理的张量积方法，用于高维特征值问题的数值处理。