Convex optimization on Landau Theory Models
Within the theoretical framework of phase transformation physics, one of the most studied and applied strategies to comprehend the universal behavior of many different physical systems is given by the Landau theory, which provides a phenomenological theory for the system's second order phase tr...
- Autores:
-
Afanador Rodríguez, Daniel Felipe
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2024
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/75214
- Acceso en línea:
- https://hdl.handle.net/1992/75214
- Palabra clave:
- Convex Optimization
Landau Theory
Matemáticas
- Rights
- openAccess
- License
- Attribution-NonCommercial-NoDerivatives 4.0 International
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Convex optimization on Landau Theory Models |
| title |
Convex optimization on Landau Theory Models |
| spellingShingle |
Convex optimization on Landau Theory Models Convex Optimization Landau Theory Matemáticas |
| title_short |
Convex optimization on Landau Theory Models |
| title_full |
Convex optimization on Landau Theory Models |
| title_fullStr |
Convex optimization on Landau Theory Models |
| title_full_unstemmed |
Convex optimization on Landau Theory Models |
| title_sort |
Convex optimization on Landau Theory Models |
| dc.creator.fl_str_mv |
Afanador Rodríguez, Daniel Felipe |
| dc.contributor.advisor.none.fl_str_mv |
Meziat Vélez, René Joaquín |
| dc.contributor.author.none.fl_str_mv |
Afanador Rodríguez, Daniel Felipe |
| dc.contributor.jury.none.fl_str_mv |
Giniatoulline, Andrei |
| dc.subject.keyword.eng.fl_str_mv |
Convex Optimization Landau Theory |
| topic |
Convex Optimization Landau Theory Matemáticas |
| dc.subject.themes.none.fl_str_mv |
Matemáticas |
| description |
Within the theoretical framework of phase transformation physics, one of the most studied and applied strategies to comprehend the universal behavior of many different physical systems is given by the Landau theory, which provides a phenomenological theory for the system's second order phase transformations, where the second derivative of the free energy with respect to temperature is discontinuous at the transition temperature. Furthermore, in many study cases, it is of utter importance to find global minima of the free energy density for the identification of such transition temperatures. In accordance with this problematic, due to the possible high complexity of the system's free energy polynomial, this work proposes an approach to locate its global minima by means of its convex envelope, considering the optimization problem given by the Method of Moments. The development of this approach utilizes semidefinite programming techniques to construct a self-contained algorithm that is applicable under the scope of applied mathematics, physics and mechanical engineering. |
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2024-12-02T13:27:16Z |
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2024-01-15 |
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Trabajo de grado - Pregrado |
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https://hdl.handle.net/1992/75214 |
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eng |
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eng |
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[Akh65] N.I. Akhiezer, The classical moment problem: And some related questions in analysis, University mathematical monographs, Oliver & Boyd, 1965. [AM11] E. Aranda and R.J. Meziat, The method of moments for some onedimensional, non-local, non-convex variational problems, Journal of Mathematical Analysis and Applications 382 (2011), no. 1, 314–323. [CFM06] Raul Curto, Lawrence Fialkow, and H. Moeller, The extremal truncated moment problem, Integral Equations and Operator Theory 60 (2006). [Fia11] Lawrence A. Fialkow, Solution of the truncated moment problem, Transactions of the American Mathematical Society 363 (2011), no. 6, 3133–3165. [HH13] J.R. Hook and H.E. Hall, Solid state physics, Manchester Physics Series, Wiley, 2013. [Ioh82] I.S. Iohvidov, Hankel and toeplitz matrices and forms: Algebraic theory, Birkhauser, 1982. [Lan56] Lev Davídovich Landau, Theory of fermi-liquids, 1956. [MBW08] Nele Moelans, Bart Blanpain, and Patrick Wollants, An introduction to phase-field modeling of microstructure evolution, Calphad 32 (2008), no. 2, 268–294. [MV06] Rene Meziat and Jorge Villalobos, Analysis of microstructures and phase transition phenomena in one-dimensional, non-linear elasticity by convex optimization, Structural and Multidisciplinary Optimization 32 (2006), 507–519. [Nes00] Yurii Nesterov, Squared functional systems and optimization problems. [VB01] Robert Vanderbei and Yurttan Benson, On formulating semidefinite programming problems as smooth convex nonlinear optimization problems. |
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45 páginas |
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application/pdf |
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Universidad de los Andes |
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Matemáticas |
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Facultad de Ciencias |
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Departamento de Matemáticas |
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Universidad de los Andes |
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Meziat Vélez, René Joaquínvirtual::20160-1Afanador Rodríguez, Daniel FelipeGiniatoulline, Andreivirtual::20161-12024-12-02T13:27:16Z2024-12-02T13:27:16Z2024-01-15https://hdl.handle.net/1992/75214instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/Within the theoretical framework of phase transformation physics, one of the most studied and applied strategies to comprehend the universal behavior of many different physical systems is given by the Landau theory, which provides a phenomenological theory for the system's second order phase transformations, where the second derivative of the free energy with respect to temperature is discontinuous at the transition temperature. Furthermore, in many study cases, it is of utter importance to find global minima of the free energy density for the identification of such transition temperatures. In accordance with this problematic, due to the possible high complexity of the system's free energy polynomial, this work proposes an approach to locate its global minima by means of its convex envelope, considering the optimization problem given by the Method of Moments. The development of this approach utilizes semidefinite programming techniques to construct a self-contained algorithm that is applicable under the scope of applied mathematics, physics and mechanical engineering.Pregrado45 páginasapplication/pdfengUniversidad de los AndesMatemáticasFacultad de CienciasDepartamento de MatemáticasAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Convex optimization on Landau Theory ModelsTrabajo de grado - Pregradoinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_7a1fTexthttp://purl.org/redcol/resource_type/TPConvex OptimizationLandau TheoryMatemáticas[Akh65] N.I. Akhiezer, The classical moment problem: And some related questions in analysis, University mathematical monographs, Oliver & Boyd, 1965.[AM11] E. Aranda and R.J. Meziat, The method of moments for some onedimensional, non-local, non-convex variational problems, Journal of Mathematical Analysis and Applications 382 (2011), no. 1, 314–323.[CFM06] Raul Curto, Lawrence Fialkow, and H. Moeller, The extremal truncated moment problem, Integral Equations and Operator Theory 60 (2006).[Fia11] Lawrence A. Fialkow, Solution of the truncated moment problem, Transactions of the American Mathematical Society 363 (2011), no. 6, 3133–3165.[HH13] J.R. Hook and H.E. Hall, Solid state physics, Manchester Physics Series, Wiley, 2013.[Ioh82] I.S. Iohvidov, Hankel and toeplitz matrices and forms: Algebraic theory, Birkhauser, 1982.[Lan56] Lev Davídovich Landau, Theory of fermi-liquids, 1956.[MBW08] Nele Moelans, Bart Blanpain, and Patrick Wollants, An introduction to phase-field modeling of microstructure evolution, Calphad 32 (2008), no. 2, 268–294.[MV06] Rene Meziat and Jorge Villalobos, Analysis of microstructures and phase transition phenomena in one-dimensional, non-linear elasticity by convex optimization, Structural and Multidisciplinary Optimization 32 (2006), 507–519.[Nes00] Yurii Nesterov, Squared functional systems and optimization problems.[VB01] Robert Vanderbei and Yurttan Benson, On formulating semidefinite programming problems as smooth convex nonlinear optimization 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