Abstract
The fine description of the mechanics and structure of materials at nanometric scale introduces some specific challenges related to the impressive number of degrees of freedom required due to the highly dimensional spaces in which those models are defined. This is the case of quantum mechanics models, in which the wavefunction is defined in a space of dimension 3×N p , being N p the number of particles involved, that leads to the terrific curse of dimensionality. Despite the fact that spectacular progresses have been accomplished in the context of computational mechanics in the last decade, the treatment of those models, as we describe in the present work, needs further developments.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Ammar, B. Mokda, F. Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, Journal of Non-Newtonian Fluid Mechanics, 139, 2006, 153–176
A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Part II: transient simulation using space-time separated representation, Journal of Non-Newtonian Fluid Mechanics, 144, 2007, 98–121.
G. Beylkin, M.J. Mohlenkamp, Algorithm for numerical analysis in high dimensions, SIAM J. Sci. Comput., 26/6, 2005, 2133–2159.
H.J. Bungartz, M. Griebel, Sparse grids, Acta Numer., 13, 2004, pp. 1–123.
E. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris, Y. Maday, Computational Quantum Chemistry: a primer, handbook of Numerical Analysis, Vol. X, Elsevier, 2003, pp. 3–270.
J.S. Chen, W. Hu, M. Puso, Orbital HP-clouds for solving Schrdinger equation in quantum mechanics, Comput. Methods Appl. Mech. Engrg., 196/37-40, 2007, pp. 3693–3705.
V. Gavini, K. Bhattacharya, M. Ortiz, Quasi-continuum orbital-free density-functional theory: A route to multi-million atom non-periodic DFT calculation, Journal of the Mechanics and Physics of Solids, 55/4, 2007, pp. 697–718.
Handbook of Numerical Analysis, Vol. X: Computational Chemistry C Le Bris editor, Elsevier, 2003.
R.B. Laughlin The theory of everything, Proceeding of the U.S.A. National Academy of Sciences, 2000
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ammar, A., Chinesta, F. (2008). Circumventing Curse of Dimensionality in the Solution of Highly Multidimensional Models Encountered in Quantum Mechanics Using Meshfree Finite Sums Decomposition. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations IV. Lecture Notes in Computational Science and Engineering, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79994-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-540-79994-8_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79993-1
Online ISBN: 978-3-540-79994-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)