Summary
We provide an example of a stochastic approach to relaxation of the variational integrals with non-attainable infima in one dimension. We provide an approximation for the coefficients of the Laplace transformation of the Probability Density Function. This approaximation yields the relaxing microstructures.
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.C. Antoulas and D.C. Sorensen. Approximation of large-scale dynamical systems: An overview. Special Issue in Numerical Analysis and System Theory, Edited by S.L. Campbell, International J. of Applied Mathematics and Computational Science, 11:1093–1121, 2001.
A.C. Antoulas and D.C. Sorensen. The sylvester equation and approximate balanced reduction. Special Issue in Numerical Analysis and System Theory; Edited by V. Blondel, D. Hinrichsen, J. Rosenthal, and P.M. van Dooren, Linear Algebra and It’s Applications, 2002, to appear.
J. M. Ball. Singularities and computation of minimizers for variational problems. Oxford FoCM, Lecture Notes, 1999.
C. Carstensen. Numerical analysis of non-convex minimization problems alllowing microstructures. Zeitschrift für Angewandte mathematik and Mechanik, 76(S2):437–438, 1996.
D. Cox, P. Klouček, and D. R. Reynolds. The non-local relaxation of nonattainable differential inclusions using a subgrid projection method: One dimensional theory and computations. Technical Report 10, École Polytechnique Fédérale de Lausanne, July 2001. to appear in: SIAM J. Sci. Comp., (2003).
D. Cox, P. Klouček, and D. R. Reynolds. A subgrid projection mathod for nonattainable differential inclusions. pages 575–584, 2001. in proceedings: ENUMATH 2001, Numerical Mathematics and Advanced Applications, F. Brezzi, A. Buffa, S. Corsaro, A. Murli (Eds).
D. Cox, P. Klouček, and D. R. Reynolds. On the asymptotically stochastic computational modeling of microstructures. Future Generation Computer Systems, 1080:1–16, 2003.
B. Dacorogna and P. Marcellini. Implicit Partial Differential Equations. Birkhäuser, 2000.
R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29:1–17, 1998.
H. Risken. The Fokker-Planck equation: methods of solution and applications. Springer-Verlag, Berlin-New York, 1984. Springer series in synergetics: 18.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cox, D.D., Klouček, P., Reynolds, D.R., Šolín, P. (2004). Stochastic Relaxation of Variational Integrals with Non-attainable Infima. In: Feistauer, M., Dolejší, V., Knobloch, P., Najzar, K. (eds) Numerical Mathematics and Advanced Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18775-9_21
Download citation
DOI: https://doi.org/10.1007/978-3-642-18775-9_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-62288-5
Online ISBN: 978-3-642-18775-9
eBook Packages: Springer Book Archive