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

Thermalization of rate-independent processes by entropic regularization

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.
Citation: T. J. Sullivan, M. Koslowski, F. Theil, Michael Ortiz. Thermalization of rate-independent processes by entropic regularization. Discrete and 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.

[2]

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

[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.

[4]

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

[5]

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

[6]

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

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[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.

[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.

[15]

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

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.

[2]

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

[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.

[4]

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

[5]

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

[6]

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

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[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.

[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.

[15]

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

[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 and 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 and 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 and Applied Analysis, 2020, 19 (11) : 5033-5057. doi: 10.3934/cpaa.2020221

[5]

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

[6]

Yacine Chitour, Zhenyu Liao, Romain Couillet. A geometric approach of gradient descent algorithms in linear neural networks. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022021

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (0)

[Back to Top]