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

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。解的数值近似受到维数灾难的影响，即计算复杂度与空间的维数呈指数关系。因此，它们的数值处理总是涉及到张量空间上的数据稀疏表示的近似方法。最近在张量积近似方面有了显著的进展。新引入的多尺度张量化技术提供了新的张量格式。我们的目标是继续和加强这些发展，并使用它的设计和严格的评估算法的新张量格式治疗高维谱问题。一个特别强调的是对这些方法可能与各种众所周知的算法用于治疗的电子薛定谔方程的潜力进行系统的检查。特别是，我们的目标是联合收割机结合这些新技术的要求和现有的完善的电子薛定谔方程的算法，特别是与单粒子模型的Hartree-Fock和Kohn-Sham方法。在这方面，该提案的目的是继续我们的项目在SPP 1324，有关的发展新的，数学上健全的张量积方法的数值处理高维特征值问题。

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.