Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling

2010 ◽  
pp. 313-347 ◽  
Author(s):  
Heinz-Jürgen Flad ◽  
Wolfgang Hackbusch ◽  
Boris N. Khoromskij ◽  
Reinhold Schneider
Keyword(s):  
Tensor Product ◽  
Tensor Product Approximation

scholarly journals On tensor product approximation of analytic functions

2016 ◽  
Vol 207 ◽  
pp. 348-379 ◽  
Author(s):  
Michael Griebel ◽  
Jens Oettershagen
Keyword(s):  
Tensor Product ◽  
Analytic Functions ◽  
Tensor Product Approximation

scholarly journals On optimal tensor product approximation

1976 ◽  
Vol 18 (2) ◽  
pp. 99-107 ◽  
Author(s):  
F.-J Delvos ◽  
H Posdorf
Keyword(s):  
Tensor Product ◽  
Tensor Product Approximation

Optimized Tensor-Product Approximation Spaces

2000 ◽  
Vol 16 (4) ◽  
pp. 525-540 ◽  
Author(s):  
M. Griebel ◽  
S. Knapek
Keyword(s):  
Tensor Product ◽  
Approximation Spaces ◽  
Tensor Product Approximation

Sparse Tensor Product Approximation for Radiative Transfer

2007 ◽  
Author(s):  
Gisela Widmer ◽  
Ralf Hiptmair
Keyword(s):  
Radiative Transfer ◽  
Tensor Product ◽  
Degrees Of Freedom ◽  
Convergence Result ◽  
Superior Performance ◽  
Discrete Ordinates ◽  
Strong Regularity ◽  
Scattering Media ◽  
Translation Invariant ◽  
Tensor Product Approximation

The stationary monochromatic radiative transfer equation is stated in five dimensions, with the intensity depending on both a position in a three-dimensional domain as well as a direction. In order to overcome the high dimensionality of the problem, we propose and analyse a new multiscale Galerkin Finite Element discretizaton that, under strong regularity assumptions on the solution, reduces the complexity of the problem to the number of degrees of freedom in space only (up to logarithmic terms). The sparse tensor product approximation adapts the idea of so-called ‘Sparse Grids’ for the product space of functions on the physical domain and the unit sphere. We present some details of the sparse tensor product construction including a convergence result that shows that, assuming strong regularity of the solution, the method converges with essentially optimal asymptotic rates while its complexity grows essentially only as that for a linear transport problem. Numerical experiments in a translation invariant setting in non-scattering media agree with predictions of theory and demonstrate the superior performance of the sparse tensor product method compared to the discrete ordinates method.

scholarly journals Counterexample to a Problem on Tensor Product Approximation

1998 ◽  
Vol 94 (2) ◽  
pp. 300-305
Author(s):  
Vilmos Totik
Keyword(s):  
Tensor Product ◽  
Tensor Product Approximation

scholarly journals Hermitian tensor product approximation of complex matrices and separability

2006 ◽  
Vol 57 (2) ◽  
pp. 271-288 ◽  
Author(s):  
Shao-Ming Fei ◽  
Naihuan Jing ◽  
Bao-Zhi Suna
Keyword(s):  
Tensor Product ◽  
Complex Matrices ◽  
Tensor Product Approximation

scholarly journals Tensor product approximation with optimal rank in quantum chemistry

2007 ◽  
Vol 127 (8) ◽  
pp. 084110 ◽  
Author(s):  
Sambasiva Rao Chinnamsetty ◽  
Mike Espig ◽  
Boris N. Khoromskij ◽  
Wolfgang Hackbusch ◽  
Heinz-Jürgen Flad
Keyword(s):  
Quantum Chemistry ◽  
Tensor Product ◽  
Tensor Product Approximation

scholarly journals Tensor-product approximation to operators and functions in high dimensions

2007 ◽  
Vol 23 (4-6) ◽  
pp. 697-714 ◽  
Author(s):  
Wolfgang Hackbusch ◽  
Boris N. Khoromskij
Keyword(s):  
Tensor Product ◽  
High Dimensions ◽  
Tensor Product Approximation
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