Summary
If a mathematical model contains many different scales the computational cost for its numerical solution is very large. The smallest scale must be resolved over the distance of the largest scale. A huge number of unknowns are required and until recently many such problems could not be treated computationally. We will discuss a new set of numerical techniques that couples models for different scales in the same simulation in order to handle many realistic multi-scale problems. In most of this presentation we shall survey existing methods but we shall also give some new observations.
In honor of Lennart Carleson on his 75th birthday
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
V.I. Arnold: Mathematical methods in classical mechanics, 2nd edn (Springer Verlag 1989)
A. Bensoussan, J.-L. Lions, J. Papanicolaou: Asymptotic analysis for periodic structures (North-Holland Publ. 1978)
G. Beylkin, R. Coifman, V. Rokhlin: Comm. Pure. Appl. Math. 44, 141 (1991)
A. Brandt: Springer Lecture Notes in Comp. Sci. and Eng. 20, 3 (2002)
L. Carleson: After the ‘Golden Age’: What Next? In: Mathematics Unlimited-2001 and Beyond, (Springer Verlag 2001) pp 455–463
W. E, B. Engquist: Comm. Math. Sci. 1, 87 (2003)
W. E, B. Engquist: Notices American Math. Soc. 50, 1062 (2003)
W. E, D. Liu, E. Vanden-Eijnden: Analysis for stochastic differential equations, Comm. Pure Appl. Math, to appear.
B. Engquist, G. Golub: From numerical analysis to computational science. In: Mathematics Unlimited-2001 and Beyond (Springer Verlag 2001) pp 433–448
G.S. Fishman: Monte Carlo concepts, algorithms and applications (Springer Verlag 1996)
A.C. Gilbert, S. Guha, P. Indyk et al: ACM STOC 2002, 152 (2002)
I.G. Kevrekides, C.W. Gear, J.M. Hyman et al: Comm. Math. Sci. 1, 715 (2003)
C.E. Shannon: Bell System Technical Journal 27, 379, 623 (1948)
E.B. Tadmor, M. Ortiz, R. Phillips: Philos. Mag. A73, 1529 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Engquist, B. (2005). Multi-scale Modeling. In: Benedicks, M., Jones, P.W., Smirnov, S., Winckler, B. (eds) Perspectives in Analysis. Mathematical Physics Studies, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-30434-7_5
Download citation
DOI: https://doi.org/10.1007/3-540-30434-7_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30432-6
Online ISBN: 978-3-540-30434-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)