American Institute of Mathematical Sciences

Advanced
February  2013, 6(1): 215-233. doi: 10.3934/dcdss.2013.6.215

Thermalization of rate-independent processes by entropic regularization

T. J. Sullivan 1, , M. Koslowski 2, , F. Theil 3, and  Michael Ortiz 4,

1. 

Applied & Computational Mathematics and Graduate Aerospace Laboratories, California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125-9400, United States
2. 

School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907-2088, United States
3. 

Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
4. 

Graduate Aerospace Laboratories, California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, United States

Received  May 2011 Revised  July 2011 Published  October 2012

We consider the effective behaviour of a rate-independent process when it is placed in contact with a heat bath. The method used to ``thermalize'' the process is an interior-point entropic regularization of the Moreau--Yosida incremental formulation of the unperturbed process. It is shown that the heat bath destroys the rate independence in a controlled and deterministic way, and that the effective dynamics are those of a non-linear gradient descent in the original energetic potential with respect to a different and non-trivial effective dissipation potential.
Keywords: non-linear evolution equations, Gradient descent, thermodynamics..
Mathematics Subject Classification: Primary: 47J35, 82C3.
Citation: T. J. Sullivan, M. Koslowski, F. Theil, Michael Ortiz. Thermalization of rate-independent processes by entropic regularization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 215-233. doi: 10.3934/dcdss.2013.6.215
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, second ed., 2008.  Google Scholar

[2]

S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004.  Google Scholar

[3]

R. Jordan and D. Kinderlehrer, An extended variational principle, Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., Dekker, New York, 177 (1996), 187-200.  Google Scholar

[4]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  Google Scholar

[5]

M. Koslowski, "A Phase-Field Model of Dislocations in Ductile Single Crystals," Ph. D. thesis, California Institute of Technology, Pasadena, California, USA, 2003. Google Scholar

[6]

A. Mielke, Evolution of rate-independent systems, Evolutionary Equations. Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559.  Google Scholar

[7]

\bysame, Modeling and analysis of rate-independent processes, January 2007, Lipschitz Lecture held in Bonn: http://www.wias-berlin.de/people/mielke/papers/Lipschitz07Mielke.pdf. Google Scholar

[8]

A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), 585-615.  Google Scholar

[9]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.  Google Scholar

[10]

J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299.  Google Scholar

[11]

J. P. Penot and M. Théra, Semicontinuous mappings in general topology, Arch. Math. (Basel), 38 (1982), 158-166.  Google Scholar

[12]

D. Preiss, Geometry of measures in $\mathbbR^n$distribution, rectifiability, and densities: , Ann. of Math., 125 (1987), 537-643.  Google Scholar

[13]

T. J. Sullivan, "Analysis of Gradient Descents in Random Energies and Heat Baths," Ph. D. thesis, Mathematitics Institute, University of Warwick, Coventry, UK, 2009. Google Scholar

[14]

T. J. Sullivan, M. Koslowski, F. Theil and M. Ortiz, On the behavior of dissipative systems in contact with a heat bath: application to Andrade creep, J. Mech. Phys. Solids, 57 (2009), 1058-1077.  Google Scholar

[15]

K. Yosida, "Functional Analysis," Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York, 1965.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, second ed., 2008.  Google Scholar

[2]

S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004.  Google Scholar

[3]

R. Jordan and D. Kinderlehrer, An extended variational principle, Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., Dekker, New York, 177 (1996), 187-200.  Google Scholar

[4]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  Google Scholar

[5]

M. Koslowski, "A Phase-Field Model of Dislocations in Ductile Single Crystals," Ph. D. thesis, California Institute of Technology, Pasadena, California, USA, 2003. Google Scholar

[6]

A. Mielke, Evolution of rate-independent systems, Evolutionary Equations. Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559.  Google Scholar

[7]

\bysame, Modeling and analysis of rate-independent processes, January 2007, Lipschitz Lecture held in Bonn: http://www.wias-berlin.de/people/mielke/papers/Lipschitz07Mielke.pdf. Google Scholar

[8]

A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), 585-615.  Google Scholar

[9]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.  Google Scholar

[10]

J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299.  Google Scholar

[11]

J. P. Penot and M. Théra, Semicontinuous mappings in general topology, Arch. Math. (Basel), 38 (1982), 158-166.  Google Scholar

[12]

D. Preiss, Geometry of measures in $\mathbbR^n$distribution, rectifiability, and densities: , Ann. of Math., 125 (1987), 537-643.  Google Scholar

[13]

T. J. Sullivan, "Analysis of Gradient Descents in Random Energies and Heat Baths," Ph. D. thesis, Mathematitics Institute, University of Warwick, Coventry, UK, 2009. Google Scholar

[14]

T. J. Sullivan, M. Koslowski, F. Theil and M. Ortiz, On the behavior of dissipative systems in contact with a heat bath: application to Andrade creep, J. Mech. Phys. Solids, 57 (2009), 1058-1077.  Google Scholar

[15]

K. Yosida, "Functional Analysis," Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York, 1965.  Google Scholar

[1]

G. A. Athanassoulis, K. A. Belibassakis. New evolution equations for non-linear water waves in general bathymetry with application to steady travelling solutions in constant, but arbitrary, depth. Conference Publications, 2007, 2007 (Special) : 75-84. doi: 10.3934/proc.2007.2007.75
[2]

Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems & Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004
[3]

Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841
[4]

Tarek Saanouni. Non-linear bi-harmonic Choquard equations. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5033-5057. doi: 10.3934/cpaa.2020221
[5]

Christoph Walker. Age-dependent equations with non-linear diffusion. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 691-712. doi: 10.3934/dcds.2010.26.691
[6]

Simona Fornaro, Ugo Gianazza. Local properties of non-negative solutions to some doubly non-linear degenerate parabolic equations. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 481-492. doi: 10.3934/dcds.2010.26.481
[7]

Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : i-iv. doi: 10.3934/dcdss.201702i
[8]

Dmitry Dolgopyat. Bouncing balls in non-linear potentials. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 165-182. doi: 10.3934/dcds.2008.22.165
[9]

Dorin Ervin Dutkay and Palle E. T. Jorgensen. Wavelet constructions in non-linear dynamics. Electronic Research Announcements, 2005, 11: 21-33.

[10]

Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems & Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675
[11]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435
[12]

Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239
[13]

Anugu Sumith Reddy, Amit Apte. Stability of non-linear filter for deterministic dynamics. Foundations of Data Science, 2021, 3 (3) : 647-675. doi: 10.3934/fods.2021025
[14]

Ahmad El Hajj, Aya Oussaily. Continuous solution for a non-linear eikonal system. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3795-3823. doi: 10.3934/cpaa.2021131
[15]

Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517
[16]

Sanjay Khattri. Another note on some quadrature based three-step iterative methods for non-linear equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 549-555. doi: 10.3934/naco.2013.3.549
[17]

Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic & Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713
[18]

Radhia Ghanmi, Tarek Saanouni. Well-posedness issues for some critical coupled non-linear Klein-Gordon equations. Communications on Pure & Applied Analysis, 2019, 18 (2) : 603-623. doi: 10.3934/cpaa.2019030
[19]

Michela Procesi. Quasi-periodic solutions for completely resonant non-linear wave equations in 1D and 2D. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 541-552. doi: 10.3934/dcds.2005.13.541
[20]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy