scholarly journals Improving a constant in high-dimensional discrepancy estimates

2020 ◽  
Vol 107 (121) ◽  
pp. 67-74
Author(s):  
Hendrik Pasing ◽  
Christian Weiß
Keyword(s):  
High Dimensional ◽  
Star Discrepancy

For all s ? 1 and N ? 1 there exist sequences (z1,..., zN) in [0, 1]s such that the star-discrepancy of these points can be bounded by D* N(z1,..., zN) ? c ?s/?N . In practice it is desirable to obtain low values of c. The best known value for the constant is c = 10 as has been calculated by Aistleitner. In this paper we improve the bound to c = 9.

Preface to the special issue, CMAM 2011, no. 3.

2011 ◽  
Vol 11 (3) ◽  
pp. 272
Author(s):  
Ivan Gavrilyuk ◽  
Boris Khoromskij ◽  
Eugene Tyrtyshnikov
Keyword(s):  
Separation Of Variables ◽  
Numerical Algorithms ◽  
Multilevel Methods ◽  
High Dimensional ◽  
Special Issue ◽  
High Dimensions ◽  
Guiding Principle ◽  
Tensor Methods ◽  
Closed World ◽  
The One

Abstract In the recent years, multidimensional numerical simulations with tensor-structured data formats have been recognized as the basic concept for breaking the "curse of dimensionality". Modern applications of tensor methods include the challenging high-dimensional problems of material sciences, bio-science, stochastic modeling, signal processing, machine learning, and data mining, financial mathematics, etc. The guiding principle of the tensor methods is an approximation of multivariate functions and operators with some separation of variables to keep the computational process in a low parametric tensor-structured manifold. Tensors structures had been wildly used as models of data and discussed in the contexts of differential geometry, mechanics, algebraic geometry, data analysis etc. before tensor methods recently have penetrated into numerical computations. On the one hand, the existing tensor representation formats remained to be of a limited use in many high-dimensional problems because of lack of sufficiently reliable and fast software. On the other hand, for moderate dimensional problems (e.g. in "ab-initio" quantum chemistry) as well as for selected model problems of very high dimensions, the application of traditional canonical and Tucker formats in combination with the ideas of multilevel methods has led to the new efficient algorithms. The recent progress in tensor numerical methods is achieved with new representation formats now known as "tensor-train representations" and "hierarchical Tucker representations". Note that the formats themselves could have been picked up earlier in the literature on the modeling of quantum systems. Until 2009 they lived in a closed world of those quantum theory publications and never trespassed the territory of numerical analysis. The tremendous progress during the very recent years shows the new tensor tools in various applications and in the development of these tools and study of their approximation and algebraic properties. This special issue treats tensors as a base for efficient numerical algorithms in various modern applications and with special emphases on the new representation formats.

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