Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling
Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling
复制标题
多粒子建模中数据稀疏张量积近似的概念
DOI：
10.1142/9789812836021_0020
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
R. Schneider
中科院分区：
文献类型：
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作者：
H. Flad;W. Hackbusch;B. Khoromskij;R. Schneider
摘要提出了量子化学多粒子模型中函数和算子的数据稀疏张量逼近的概念。我们的方法是基于系统地使用结构化张量积表示，其中低维分量以层次或基于小波的矩阵格式表示。讨论了高维张量积近似的现代方法，重点讨论了基于解析的方法。我们给出了数值实例，证实了张量分解技术在电子结构计算中的有效性。关键词：Schr¨odinger方程，Hartree-Fock方法，密度泛函数理论，张量积近似1介绍当今科学计算中最具挑战性的问题之一是高维问题，例如，多粒子相互作用，d上的积分或微分方程以及d≥3时的相关数值算子演算。许多标准方法的计算复杂度在维度上呈指数增长，因此由于众所周知的“维度诅咒”而失败。为了摆脱这种指数增长的复杂性，可以在解决过程的所有阶段使用张量积结构的思想（参见[85]）。因此，我们对张量积格式中感兴趣的量进行近似，并对剩余的低维分量使用其他近似方法。根据问题的具体性质，这些低维分量是1
AbstractWe present concepts of data-sparse tensor approximations to the functions and operatorsarising in many-particle models of quantum chemistry. Our approach is based on thesystematic use of structured tensor-product representations where the low-dimensionalcomponentsare representedin hierarchicalor waveletbased matrix formats. The modernmethods of tensor-product approximation in higher dimensions are discussed with thefocus on analytically based approaches. We give numerical illustrations which confirmthe efficiency of tensor decomposition techniques in electronic structure calculations. AMS Subject Classification: 65F30, 65F50, 65N35, 65F10Key words: Schr¨odinger equation, Hartree-Fock method, density functional theory, tensor-product approximation 1 Introduction Among the most challenging problems of scientific computing nowadays are those of high di-mensions, for instance, multi-particle interactions, integral or differential equations on [0,1] d and the related numerical operator calculus for d≥ 3. Many standard approaches have acomputational complexity that grows exponentially in the dimension dand thus fail becauseof the well known “curse of dimensionality”. To get rid of this exponential growth in thecomplexity one can use the idea of tensor-product constructions (cf. [85]) on all stages ofthe solution process. Hereby we approximate the quantity of interest in tensor-product for-mats and use other approximation methods for the remaining low-dimensional components.Depending on the specific properties of the problem, these low-dimensional components are1