1,098 results on '"calculus of variations"'
Search Results
2. Variational Methods for Evolution.
- Subjects
CALCULUS of variations ,VARIATIONAL principles ,PARTIAL differential equations ,NUMERICAL analysis ,HAMILTONIAN systems - Abstract
Variational principles for evolutionary systems arise in many settings, both in those describing the physical world and in man-made algorithms for data science and optimization tasks. Variational principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, as well as in time-incremental minimization techniques for more general evolutionary problems. Additional challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, and in multi-agent systems. Variational principles and associated evolutions are also at the core of the modern approaches to machine learning tasks, since many of them are posed as minimizing an objective functional that models the problem. The discrete and random nature of these problems and the need for accurate computation in high dimension present a set of challenges that require new mathematical insights. Variational methods for evolution allow for the usage of the rich toolbox provided by the calculus of variations, metric-space geometry, partial differential equations, and other branches of applied analysis. The variational methods for evolution have seen a rapid growth over the last two decades. This workshop continued the successful line of meetings (2011, 2014, 2017, and 2020), while evolving and branching into new directions. We have brought together a wide scope of mathematical researchers from calculus of variations, partial differential equations, numerical analysis, and stochastics, as well as researchers from data science and machine learning, to exchange ideas, foster interaction, develop new avenues, and generally bring these communities closer together. [ABSTRACT FROM AUTHOR]
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- 2023
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3. New approximations of space-time fractional Fokker-Planck equations.
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Singh, Brajesh Kumar, Kumar, Anil, and Gupta, Mukesh
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FOKKER-Planck equation ,APPROXIMATION theory ,CALCULUS of variations ,NUMERICAL analysis ,ACCURACY - Abstract
The present study focuses on the two new hybrid methods: variational iteration J-transform technique (J-VIT) and J-transform method with optimal homotopy analysis (OHAJTM) for analytical assessment of space-time fractional Fokker-Planck equations (STF-FPE), appearing in many realistic physical situations, e.g., in ultra-slow kinetics, Brownian motion of particles, anomalous diffusion, polymerases of Ribonucleic acid, deoxyribonucleic acid, continuous random movement, and formation of wave patterns. OHAJTM is developed via optimal homotopy analysis after implementing the properties of J-transform while (J-VIT) is produced by implementing properties of the J-transform and the theory of variational iteration. Banach approach is utilized to analyze the convergence of these methods. In addition, it is demonstrated that J-VIT is T-stable. Computed new approximations are reported as a closed form expression of the Mittag-Leffler function, and in addition, the effectiveness/validity of the proposed new approximations is demonstrated via three test problems of STF-FPE by computing the error norms: L
2 and absolute errors. The numerical assessment demonstrates that the developed techniques perform better for STF-FPE and for a given iteration, andO HAJTM produces new approximations with better accuracy as compared to J-VIT as well as the techniques developed recently. [ABSTRACT FROM AUTHOR]- Published
- 2023
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4. Periodic parabolic problem with discontinuous coefficients: Mathematical analysis and numerical simulation.
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Alaa, Nour Eddine, Charkaoui, Abderrahim, and Elaassri, Abdelwahab
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DISCONTINUOUS coefficients , *NUMERICAL analysis , *MATHEMATICAL analysis , *COST functions , *CALCULUS of variations , *GALERKIN methods - Abstract
This work presents a new approach for the mathematical analysis and numerical simulation of a class of periodic parabolic equations with discontinuous coefficients. Our technique is based on the minimization of a least squares cost function. By the means of variational calculus, we prove that the considered optimization problem admits an optimal solution. Using the Lagrangian method, we compute the gradient of the cost function associated with our problem. Finally, we give several numerical simulations that show the efficiency and robustness of our method. [ABSTRACT FROM AUTHOR]
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- 2022
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5. Optimization, Simulation and Control : ICOSC 2022, Ulaanbaatar, Mongolia, June 20–22
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Rentsen Enkhbat, Altannar Chinchuluun, Panos M. Pardalos, Rentsen Enkhbat, Altannar Chinchuluun, and Panos M. Pardalos
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- Mathematical optimization, Calculus of variations, Numerical analysis, Statistics, Computer science
- Abstract
This volume gathers selected, peer-reviewed works presented at the 7th International Conference on Optimization, Simulation and Control, ICOSC 2022, held at the National University of Mongolia, Ulaanbaatar, June 20–22, 2022. Topics covered include (but are not limited to) mathematical programming; network, global, linear, nonlinear, parametric, stochastic, and multi-objective optimization; control theory; biomathematics; and deep and machine learning, to name a few. Held every three years since 2002, the ICOSC conference has become a traditional gathering for experienced and young researchers in optimization and control to share recent findings in these fields and discuss novel applications in myriad sectors. Researchers and graduate students in the fields of mathematics, engineering, and computer science can greatly benefit from this book, which can also be enjoyed by advanced practitioners in research laboratories and the industry. The 2022 edition of the ICOSC conference was sponsored by the Mongolian Academy of Sciences, the National University of Mongolia and the German-Mongolian Institute for Resources and Technology.
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- 2023
6. Approximation and Computation in Science and Engineering
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Nicholas J. Daras, Themistocles M. Rassias, Nicholas J. Daras, and Themistocles M. Rassias
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- Mathematical optimization, Calculus of variations, Data mining, Number theory, Numerical analysis, Algebraic geometry, Mathematics
- Abstract
In recent years, extensive research has been conducted by eminent mathematicians and engineers whose results and proposed problems are presented in this new volume. It is addressed to graduate students, research mathematicians, physicists, and engineers. Individual contributions are devoted to topics of approximation theory, functional equations and inequalities, fixed point theory, numerical analysis, theory of wavelets, convex analysis, topology, operator theory, differential operators, fractional integral operators, integro-differential equations, ternary algebras, super and hyper relators, variational analysis, discrete mathematics, cryptography, and a variety of applications in interdisciplinary topics. Several of these domains have a strong connection with both theories and problems of linear and nonlinear optimization. The combination of results from various domains provides the reader with a solid, state-of-the-art interdisciplinary reference to theory and problems. Some of the works provide guidelines for further research and proposals for new directions and open problems with relevant discussions.
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- 2022
7. Non-Smooth and Complementarity-Based Distributed Parameter Systems : Simulation and Hierarchical Optimization
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Michael Hintermüller, Roland Herzog, Christian Kanzow, Michael Ulbrich, Stefan Ulbrich, Michael Hintermüller, Roland Herzog, Christian Kanzow, Michael Ulbrich, and Stefan Ulbrich
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- Mathematical optimization, Calculus of variations, Numerical analysis, System theory, Control theory, Mathematics—Data processing
- Abstract
Many of the most challenging problems in the applied sciences involve non-differentiable structures as well as partial differential operators, thus leading to non-smooth distributed parameter systems. This edited volume aims to establish a theoretical and numerical foundation and develop new algorithmic paradigms for the treatment of non-smooth phenomena and associated parameter influences. Other goals include the realization and further advancement of these concepts in the context of robust and hierarchical optimization, partial differential games, and nonlinear partial differential complementarity problems, as well as their validation in the context of complex applications. Areas for which applications are considered include optimal control of multiphase fluids and of superconductors, image processing, thermoforming, and the formation of rivers and networks.Chapters are written by leading researchers and present results obtained in the first funding phase of the DFG Special Priority Program on Nonsmooth and Complementarity Based Distributed Parameter Systems: Simulation and Hierarchical Optimization that ran from 2016 to 2019.
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- 2022
8. Research in Mathematics of Materials Science
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Malena I. Español, Marta Lewicka, Lucia Scardia, Anja Schlömerkemper, Malena I. Español, Marta Lewicka, Lucia Scardia, and Anja Schlömerkemper
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- Differential equations, Mechanics, Applied, Dynamical systems, Mathematical optimization, Calculus of variations, Numerical analysis
- Abstract
This volume highlights contributions of women mathematicians in the study of complex materials and includes both original research papers and reviews. The featured topics and methods draw on the fields of Calculus of Variations, Partial Differential Equations, Functional Analysis, Differential Geometry and Topology, as well as Numerical Analysis and Mathematical Modelling. Areas of applications include foams, fluid-solid interactions, liquid crystals, shape-memory alloys, magnetic suspensions, failure in solids, plasticity, viscoelasticity, homogenization, crystallization, grain growth, and phase-field models.
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- 2022
9. Controller Tuning Optimization Methods for Multi-Constraints and Nonlinear Systems : A Metaheuristic Approach
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Maude Josée Blondin and Maude Josée Blondin
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- Mathematical optimization, Calculus of variations, Control engineering, Numerical analysis, Differential equations
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This book covers controller tuning techniques from conventional to new optimization methods for diverse control engineering applications. Classical controller tuning approaches are presented with real-world challenges faced in control engineering. Current developments in applying optimization techniques to controller tuning are explained. Case studies of optimization algorithms applied to controller tuning dealing with nonlinearities and limitations like the inverted pendulum and the automatic voltage regulator are presented with performance comparisons. Students and researchers in engineering and optimization interested in optimization methods for controller tuning will utilize this book to apply optimization algorithms to controller tuning, to choose the most suitable optimization algorithm for a specific application, and to develop new optimization techniques for controller tuning.
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- 2021
10. Optimization on Solution Sets of Common Fixed Point Problems
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Alexander J. Zaslavski and Alexander J. Zaslavski
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- Mathematical optimization, Calculus of variations, Operations research, Management science, Numerical analysis
- Abstract
This book is devoted to a detailed study of the subgradient projection method and its variants for convex optimization problems over the solution sets of common fixed point problems and convex feasibility problems. These optimization problems are investigated to determine good solutions obtained by different versions of the subgradient projection algorithm in the presence of sufficiently small computational errors. The use of selected algorithms is highlighted including the Cimmino type subgradient, the iterative subgradient, and the dynamic string-averaging subgradient. All results presented are new. Optimization problems where the underlying constraints are the solution sets of other problems, frequently occur in applied mathematics. The reader should not miss the section in Chapter 1 which considers some examples arising in the real world applications. The problems discussed have an important impact in optimization theory as well. The book will be useful for researches interested in the optimization theory and its applications.
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- 2021
11. The Projected Subgradient Algorithm in Convex Optimization
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Alexander J. Zaslavski and Alexander J. Zaslavski
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- Mathematical optimization, Calculus of variations, Numerical analysis
- Abstract
This focused monograph presents a study of subgradient algorithms for constrained minimization problems in a Hilbert space. The book is of interest for experts in applications of optimization to engineering and economics. The goal is to obtain a good approximate solution of the problem in the presence of computational errors. The discussion takes into consideration the fact that for every algorithm its iteration consists of several steps and that computational errors for different steps are different, in general. The book is especially useful for the reader because it contains solutions to a number of difficult and interesting problems in the numerical optimization. The subgradient projection algorithm is one of the most important tools in optimization theory and its applications. An optimization problem is described by an objective function and a set of feasible points. For this algorithm each iteration consists of two steps. The first step requires a calculation of a subgradient of the objective function; the second requires a calculation of a projection on the feasible set. The computational errors in each of these two steps are different. This book shows that the algorithm discussed, generates a good approximate solution, if all the computational errors are bounded from above by a small positive constant. Moreover, if computational errors for the two steps of the algorithm are known, one discovers an approximate solution and how many iterations one needs for this. In addition to their mathematical interest, the generalizations considered in this book have a significant practical meaning.
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- 2020
12. Nonlinear Analysis, Geometry and Applications : Proceedings of the First NLAGA-BIRS Symposium, Dakar, Senegal, June 24–28, 2019
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Diaraf Seck, Kinvi Kangni, Philibert Nang, Marie Salomon Sambou, Diaraf Seck, Kinvi Kangni, Philibert Nang, and Marie Salomon Sambou
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- Mathematical optimization, Calculus of variations, Numerical analysis, Global analysis (Mathematics), Manifolds (Mathematics), Differential equations, Functions of complex variables, Number theory
- Abstract
This book gathers nineteen papers presented at the first NLAGA-BIRS Symposium, which was held at the Cheikh Anta Diop University in Dakar, Senegal, on June 24–28, 2019.The four-day symposium brought together African experts on nonlinear analysis and geometry and their applications, as well as their international partners, to present and discuss mathematical results in various areas. The main goal of the NLAGA project is to advance and consolidate the development of these mathematical fields in West and Central Africa with a focus on solving real-world problems such as coastal erosion, pollution, and urban network and population dynamics problems.The book addresses a range of topics related to partial differential equations, geometrical analysis of optimal shapes, geometric structures, optimization and optimal transportation, control theory, and mathematical modeling.
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- 2020
13. Approximation and Optimization : Algorithms, Complexity and Applications
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Ioannis C. Demetriou, Panos M. Pardalos, Ioannis C. Demetriou, and Panos M. Pardalos
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- Approximation theory, Mathematical optimization, Calculus of variations, Algorithms, Numerical analysis, Probabilities
- Abstract
This book focuses on the development of approximation-related algorithms and their relevant applications. Individual contributions are written by leading experts and reflect emerging directions and connections in data approximation and optimization. Chapters discuss state of the art topics with highly relevant applications throughout science, engineering, technology and social sciences. Academics, researchers, data science practitioners, business analysts, social sciences investigators and graduate students will find the number of illustrations, applications, and examples provided useful. This volume is based on the conference Approximation and Optimization: Algorithms, Complexity, and Applications, which was held in the National and Kapodistrian University of Athens, Greece, June 29–30, 2017. The mix of survey and research content includes topics in approximations to discrete noisy data; binary sequences; design of networks and energy systems; fuzzy control; large scale optimization; noisy data; data-dependent approximation; networked control systems; machine learning ; optimal design; no free lunch theorem; non-linearly constrained optimization; spectroscopy.
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- 2019
14. Splitting Algorithms, Modern Operator Theory, and Applications
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Heinz H. Bauschke, Regina S. Burachik, D. Russell Luke, Heinz H. Bauschke, Regina S. Burachik, and D. Russell Luke
- Subjects
- Operator theory, Mathematical optimization, Calculus of variations, Differential equations, Functional analysis, Numerical analysis
- Abstract
This book brings together research articles and state-of-the-art surveys in broad areas of optimization and numerical analysis with particular emphasis on algorithms. The discussion also focuses on advances in monotone operator theory and other topics from variational analysis and nonsmooth optimization, especially as they pertain to algorithms and concrete, implementable methods. The theory of monotone operators is a central framework for understanding and analyzing splitting algorithms. Topics discussed in the volume were presented at the interdisciplinary workshop titled Splitting Algorithms, Modern Operator Theory, and Applications held in Oaxaca, Mexico in September, 2017. Dedicated to Jonathan M. Borwein, one of the most versatile mathematicians in contemporary history, this compilation brings theory together with applications in novel and insightful ways.
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- 2019
15. A NEWTON ALGORITHM FOR SEMIDISCRETE OPTIMAL TRANSPORT WITH STORAGE FEES.
- Author
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BANSIL, MOHIT and JUN KITAGAWA
- Subjects
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ADMINISTRATIVE fees , *STORAGE , *CALCULUS of variations , *NUMERICAL analysis - Abstract
We introduce and prove convergence of a damped Newton algorithm to approximate solutions of the semidiscrete optimal transport problem with storage fees, corresponding to a problem with hard capacity constraints. This is a variant of the optimal transport problem arising in queue penalization problems and has applications to data clustering. Our result is novel, as it is the first numerical method with proven convergence for this variant problem; additionally, the algorithm applies to the classical semidiscrete optimal transport problem but does not require any connectedness assumptions on the support of the source measure, in contrast with existing results. Furthermore we find some stability results of the associated Laguerre cells. All of our results come with quantitative rates. We also present some numerical examples. [ABSTRACT FROM AUTHOR]
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- 2021
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16. Control Systems and Mathematical Methods in Economics : Essays in Honor of Vladimir M. Veliov
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Gustav Feichtinger, Raimund M. Kovacevic, Gernot Tragler, Gustav Feichtinger, Raimund M. Kovacevic, and Gernot Tragler
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- Economics, Environmental management, Population, Operations research, Numerical analysis, Calculus of variations, Control theory
- Abstract
Since the days of Lev Pontryagin and his associates, the discipline of Optimal Control has enjoyed a tremendous upswing – not only in terms of its mathematical foundations, but also with regard to numerous fields of application, which have given rise to highly active research areas. Few scholars, however, have been able to make contributions to both the mathematical developments and the (socio-)economic applications; Vladimir Veliov is one of them. In the course of his scientific career, he has contributed highly influential research on mathematical aspects of Optimal Control Theory, as well as applications in Economics and Operations Research. One of the hallmarks of his research is its impressive breadth. This volume, published on the occasion of his 65th birthday, accurately reflects that diversity. The mathematical aspects covered include stability theory for difference inclusions, metric regularity, generalized duality theory, the Bolza problem from a functional analytic perspective, and fractional calculus. In turn, the book explores various applications of control theory, such as population dynamics, population economics, epidemiology, optimal growth theory, resource and energy economics, environmental management, and climate change. Further topics include optimal liquidity, dynamics of the firm, and wealth inequality.
- Published
- 2018
17. Algorithms for Solving Common Fixed Point Problems
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Alexander J. Zaslavski and Alexander J. Zaslavski
- Subjects
- Calculus of variations, Mathematics, Operator theory, Numerical analysis
- Abstract
This book details approximate solutions to common fixed point problems and convex feasibility problems in the presence of perturbations. Convex feasibility problems search for a common point of a finite collection of subsets in a Hilbert space; common fixed point problems pursue a common fixed point of a finite collection of self-mappings in a Hilbert space. A variety of algorithms are considered in this book for solving both types of problems, the study of which has fueled a rapidly growing area of research. This monograph is timely and highlights the numerous applications to engineering, computed tomography, and radiation therapy planning. Totaling eight chapters, this book begins with an introduction to foundational material and moves on to examine iterative methods in metric spaces. The dynamic string-averaging methods for common fixed point problems in normed space are analyzed in Chapter 3. Dynamic string methods, for common fixed point problemsin a metric space are introduced and discussed in Chapter 4. Chapter 5 is devoted to the convergence of an abstract version of the algorithm which has been called component-averaged row projections (CARP). Chapter 6 studies a proximal algorithm for finding a common zero of a family of maximal monotone operators. Chapter 7 extends the results of Chapter 6 for a dynamic string-averaging version of the proximal algorithm. In Chapters 8 subgradient projections algorithms for convex feasibility problems are examined for infinite dimensional Hilbert spaces.
- Published
- 2018
18. UNSTABILIZED HYBRID HIGH-ORDER METHOD FOR A CLASS OF DEGENERATE CONVEX MINIMIZATION PROBLEMS.
- Author
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CARSTENSEN, CARSTEN and TRAN, TIEN
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CALCULUS of variations , *NUMERICAL analysis , *ENERGY density , *BINDING energy , *TOPOLOGY - Abstract
The relaxation in the calculus of variation motivates the numerical analysis of a class of degenerate convex minimization problems with nonstrictly convex energy densities with some convexity control and two-sided p-growth. The minimizers may be nonunique in the primal variable but lead to a unique stress σ ∈ H (div, Ω; 필). Examples include the p-Laplacian, an optimal design problem in topology optimization, and the convexified double-well problem. The approximation by hybrid high-order methods (HHO) utilizes a reconstruction of the gradients with piecewise Raviart-Thomas or Brezzi-Douglas-Marini finite elements without stabilization on a regular triangulation into simplices. The application of this HHO method to the class of degenerate convex minimization problems allows for a unique H(div) conforming stress approximation σh. The main results are a priori and a posteriori error estimates for the stress error σ-σh in Lebesgue norms and a computable lower energy bound. Numerical benchmarks display higher convergence rates for higher polynomial degrees and include adaptive mesh-refining with the first superlinear convergence rates of guaranteed lower energy bounds. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
19. Micromechanical Analysis in Applications of Active Mono-Slip and Continuum Dislocations in the MDCM.
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Kasa, Temesgen Takele and Amaro, Ana Paula Betencourt Martins
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ENERGY dissipation ,CALCULUS of variations ,MATHEMATICAL continuum ,NUMERICAL analysis ,COMPOSITE structures ,KINEMATICS - Abstract
The key purpose of this paper is to propose a mono-slip-dependent continuum dislocation method for matrix-dominated composite structure (MDCS) analysis. The methodology focuses on dissipation energy theories utilizing a continuum dislocation method (CDM) integrated with small-strain kinematics. The mathematical modeling of the CDM comprises active mono-slip system formulations, thermodynamic dislocation analysis (TDA), free energy dissipation analysis, and the progression of dislocations. Furthermore, zero and non-zero energy dissipation due to dislocation progression is formulated by using an energy minimization technique with variational calculus. The numerical analysis, performed with Wolfram Mathematica©, is presented using zero and non-zero energy dissipation energy formulations. The outcomes indicate that the formulated approach can be effective for obtaining optimal analysis results for matrix-dominated composite (MDC) materials with a mono-slip system. In sum, this study confirms the feasibility of using the proposed approach to investigate MDCS with inclusions. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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20. A convex approach to the Gilbert-Steiner problem.
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BONAFINI, MAURO and OUDET, ÉDOUARD
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- *
CALCULUS of variations , *ALGORITHMS , *FUNCTIONAL analysis , *NUMERICAL analysis , *MACHINE theory - Abstract
We describe a convex relaxation for the Gilbert-Steiner problem both in Rd and on manifolds, extending the framework proposed in [10], and we discuss its sharpness by means of calibration type arguments. The minimization of the resulting problem is then tackled numerically and we present results for an extensive set of examples. In particular we are able to address the Steiner tree problem on surfaces. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
21. Analysis and numerical approximation of tempered fractional calculus of variations problems.
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Almeida, Ricardo and Morgado, M. Luísa
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FRACTIONAL calculus , *NUMERICAL analysis , *CAPUTO fractional derivatives , *HOLONOMIC constraints , *ISOPERIMETRICAL problems , *CALCULUS of variations - Abstract
In this paper, we study variational problems where the cost functional involves the tempered Caputo fractional derivative. Several important optimization conditions are derived to find the optimal solution. Sufficient and necessary conditions are presented for different variational problems. For example, the cases of integral (isoperimetric problem) and holonomic constraints are considered, as well as problems with high order derivatives. A numerical scheme is proposed to determine approximations of the solution and it is illustrated through some examples. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
22. Time-dependent density-functional theory calculations of triplet-triplet absorption.
- Author
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Cronstrand, Peter, Rinkevicius, Zilvinas, Luo, Yi, and Ågren, Hans
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DENSITY functionals , *FUNCTIONAL analysis , *CALCULUS of variations , *HARTREE-Fock approximation , *MATHEMATICAL analysis , *NUMERICAL analysis - Abstract
We present density-functional theory calculations of triplet-triplet absorption by three different approaches based on time-dependent density-functional theory (DFT): unrestricted DFT linear response, open-shell restricted DFT linear response applied to the triplet state, and quadratic response with triplet excitations applied to the ground state. Comparison is also made with corresponding results obtained by Hartree–Fock and multiconfiguration self-consistent-field response theory. Two main conclusions concerning triplet-triplet transitions are drawn in this study: First, the very good agreement between unrestricted and restricted DFT results indicates that spin contamination of the triplet state is not a serious problem when computing triplet-triplet spectra of common organic molecules. Second, DFT response calculations of triplet-triplet transitions can be affected by triplet instability problems, especially for the combination of DFT quadratic response with functionals containing fractional exact Hartree–Fock exchange. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
23. Finite Element and Boundary Element Techniques From Mathematical and Engineering Point of View
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E. Stein, W. Wendland, E. Stein, and W. Wendland
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- Numerical analysis, Mathematical analysis, System theory, Control theory, Mathematical optimization, Calculus of variations, Computer simulation, Mechanics
- Abstract
Traditional FEM and the more recent BEM underlie many engineering computational methods and corresponding software. Both methods have their merits and also their limitations. The combination of both methods will provide an improved numerical tool in the future. The aim of this book is to present significant basic formulations of FEM and BEM and to show their common practical and mathematical foundations, their differences and possibilities for their combination. These include variational foundations, FEM and BEM for linear and non-linear elasticity and potential problems, the combination of FEM-BEM asymptotic error analysis, modifications due to corner and crack singularities and corresponding improvement of convergence, plastic analysis, numerical algorithms and engineering applications.
- Published
- 2014
24. Convex Analysis and Global Optimization
- Author
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Hoang Tuy and Hoang Tuy
- Subjects
- Mathematical optimization, Calculus of variations, Numerical analysis, Mathematical models, Computer science, Business, Management science
- Abstract
Due to the general complementary convex structure underlying most nonconvex optimization problems encountered in applications, convex analysis plays an essential role in the development of global optimization methods. This book develops a coherent and rigorous theory of deterministic global optimization from this point of view. Part I constitutes an introduction to convex analysis, with an emphasis on concepts, properties and results particularly needed for global optimization, including those pertaining to the complementary convex structure. Part II presents the foundation and application of global search principles such as partitioning and cutting, outer and inner approximation, and decomposition to general global optimization problems and to problems with a low-rank nonconvex structure as well as quadratic problems. Much new material is offered, aside from a rigorous mathematical development. Audience: The book is written as a text for graduate students in engineering, mathematics, operations research, computer science and other disciplines dealing with optimization theory. It is also addressed to all scientists in various fields who are interested in mathematical optimization.
- Published
- 2013
25. Numerical Optimization : Theoretical and Practical Aspects
- Author
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Joseph-Frédéric Bonnans, Jean Charles Gilbert, Claude Lemarechal, Claudia A. Sagastizábal, Joseph-Frédéric Bonnans, Jean Charles Gilbert, Claude Lemarechal, and Claudia A. Sagastizábal
- Subjects
- Operations research, Management science, Mathematical optimization, Calculus of variations, Numerical analysis, Algorithms, Computer science—Mathematics
- Abstract
Just as in its 1st edition, this book starts with illustrations of the ubiquitous character of optimization, and describes numerical algorithms in a tutorial way. It covers fundamental algorithms as well as more specialized and advanced topics for unconstrained and constrained problems. Most of the algorithms are explained in a detailed manner, allowing straightforward implementation. Theoretical aspects of the approaches chosen are also addressed with care, often using minimal assumptions. This new edition contains computational exercises in the form of case studies which help understanding optimization methods beyond their theoretical, description, when coming to actual implementation. Besides, the nonsmooth optimization part has been substantially reorganized and expanded.
- Published
- 2013
26. The Theory of Anisotropic Elastic Plates
- Author
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T.S. Vashakmadze and T.S. Vashakmadze
- Subjects
- Mechanics, Mathematical analysis, Numerical analysis, Mathematical models, Mathematical optimization, Calculus of variations
- Abstract
The main purpose of this work is construction of the mathematical theory of elastic plates and shells, by means of which the investigation of basic boundary value problems of the spatial theory of elasticity in the case of cylindrical do mains reduces to the study of two-dimensional boundary value problems (BVP) of comparatively simple structure. In this respect in sections 2-5 after the introductory material, methods of re duction, known in the literature as usually being based on simplifying hypotheses, are studied. Here, in contradiction to classical methods, the problems, connected with construction of refined theories of anisotropic nonhomogeneous plates with variable thickness without the assumption of any physical and geometrical re strictions, are investigated. The comparative analysis of such reduction methods was carried out, and, in particular, in section 5, the following fact was established: the error transition, occuring with substitution of a two-dimensional model for the initial problem on the class of assumed solutions is restricted from below. Further, in section 6, Vekua's method of reduction, containing regular pro cess of study of three-dimensional problem, is investigated. In this direction, the problems, connected with solvability, convergence of processes, and construction of effective algorithms of approximate solutions are studied.
- Published
- 2013
27. Applied Mathematics: Body and Soul : Volume 1: Derivatives and Geometry in IR3
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Kenneth Eriksson, Donald Estep, Claes Johnson, Kenneth Eriksson, Donald Estep, and Claes Johnson
- Subjects
- Mathematics, Algebras, Linear, Differential equations, Differential equations, Partial, Calculus of variations, Numerical analysis, Mathematical optimization, Fluid mechanics
- Abstract
Applied Mathematics: Body & Soul is a mathematics education reform project developed at Chalmers University of Technology and includes a series of volumes and software. The program is motivated by the computer revolution opening new possibilities of computational mathematical modeling in mathematics, science and engineering. It consists of a synthesis of Mathematical Analysis (Soul), Numerical Computation (Body) and Application. Volumes I-III present a modern version of Calculus and Linear Algebra, including constructive/numerical techniques and applications intended for undergraduate programs in engineering and science. Further volumes present topics such as Dynamical Systems, Fluid Dynamics, Solid Mechanics and Electro-Magnetics on an advanced undergraduate/graduate level. The authors are leading researchers in Computational Mathematics who have written various successful books.
- Published
- 2013
28. Geometric Methods and Optimization Problems
- Author
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Vladimir Boltyanski, Horst Martini, V. Soltan, Vladimir Boltyanski, Horst Martini, and V. Soltan
- Subjects
- Mathematical optimization, Calculus of variations, Convex geometry, Discrete geometry, Numerical analysis, Discrete mathematics
- Abstract
VII Preface In many fields of mathematics, geometry has established itself as a fruitful method and common language for describing basic phenomena and problems as well as suggesting ways of solutions. Especially in pure mathematics this is ob vious and well-known (examples are the much discussed interplay between lin ear algebra and analytical geometry and several problems in multidimensional analysis). On the other hand, many specialists from applied mathematics seem to prefer more formal analytical and numerical methods and representations. Nevertheless, very often the internal development of disciplines from applied mathematics led to geometric models, and occasionally breakthroughs were b~ed on geometric insights. An excellent example is the Klee-Minty cube, solving a problem of linear programming by transforming it into a geomet ric problem. Also the development of convex programming in recent decades demonstrated the power of methods that evolved within the field of convex geometry. The present book focuses on three applied disciplines: control theory, location science and computational geometry. It is our aim to demonstrate how methods and topics from convex geometry in a wider sense (separation theory of convex cones, Minkowski geometry, convex partitionings, etc.) can help to solve various problems from these disciplines.
- Published
- 2013
29. Interactive Decision Maps : Approximation and Visualization of Pareto Frontier
- Author
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Alexander V. Lotov, Vladimir A. Bushenkov, Georgy K. Kamenev, Alexander V. Lotov, Vladimir A. Bushenkov, and Georgy K. Kamenev
- Subjects
- Mathematical optimization, Numerical analysis, Convex geometry, Discrete geometry, Environmental management, Calculus of variations
- Abstract
Since the volume may be of interest to a broad variety of people, it is arranged in parts that require different levels of mathematical background. Part I is written in a simple form and can be assessed by any computer-literate person interested in the application of visualization methods in decision making. This part will be of interest to specialists and students in various fields related to decision making including environmental studies, management, business, engineering, etc. In Part II computational methods are introduced in a relatively simple form. This part will be of interest to specialists and students in the field of applied optimization, operations research and computer science. Part III is written for specialists and students in applied mathematics interested in the theoretical basis of modern optimization. Due to this structure, the parts can be read independently. For example, students interested in environmental applications could restrict themselves to Part I and the Epilogue. In contrast, those who are interested in computational methods can skip Part I and read Part II only. Finally, specialists, who are interested in the theory of approximation of multi-dimensional convex sets or in estimation of disturbances of polyhedral sets, can read the corresponding chapters of Part III.
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- 2013
30. Convex Analysis and Minimization Algorithms II : Advanced Theory and Bundle Methods
- Author
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Jean-Baptiste Hiriart-Urruty, Claude Lemarechal, Jean-Baptiste Hiriart-Urruty, and Claude Lemarechal
- Subjects
- Operations research, Management science, Mathematical optimization, Calculus of variations, Numerical analysis
- Abstract
From the reviews:'The account is quite detailed and is written in a manner that will appeal to analysts and numerical practitioners alike...they contain everything from rigorous proofs to tables of numerical calculations.... one of the strong features of these books...that they are designed not for the expert, but for those who whish to learn the subject matter starting from little or no background...there are numerous examples, and counter-examples, to back up the theory...To my knowledge, no other authors have given such a clear geometric account of convex analysis.''This innovative text is well written, copiously illustrated, and accessible to a wide audience'
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- 2013
31. Variational Theory of Splines
- Author
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Anatoly Yu. Bezhaev, Vladimir A. Vasilenko, Anatoly Yu. Bezhaev, and Vladimir A. Vasilenko
- Subjects
- Mathematical analysis, Functional analysis, Numerical analysis, Approximation theory, Mathematical optimization, Calculus of variations
- Published
- 2013
32. Surveys on Solution Methods for Inverse Problems
- Author
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David Colton, Heinz W. Engl, Alfred K. Louis, Joyce McLaughlin, William Rundell, David Colton, Heinz W. Engl, Alfred K. Louis, Joyce McLaughlin, and William Rundell
- Subjects
- Numerical analysis, System theory, Control theory, Mathematical optimization, Calculus of variations, Potential theory (Mathematics)
- Abstract
Inverse problems are concerned with determining causes for observed or desired effects. Problems of this type appear in many application fields both in science and in engineering. The mathematical modelling of inverse problems usually leads to ill-posed problems, i.e., problems where solutions need not exist, need not be unique or may depend discontinuously on the data. For this reason, numerical methods for solving inverse problems are especially difficult, special methods have to be developed which are known under the term'regularization methods'. This volume contains twelve survey papers about solution methods for inverse and ill-posed problems and about their application to specific types of inverse problems, e.g., in scattering theory, in tomography and medical applications, in geophysics and in image processing. The papers have been written by leading experts in the field and provide an up-to-date account of solution methods for inverse problems.
- Published
- 2012
33. Perspectives on Enclosure Methods
- Author
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Ulrich Kulisch, Rudolf Lohner, Axel Facius, Ulrich Kulisch, Rudolf Lohner, and Axel Facius
- Subjects
- Numerical analysis, Computer simulation, Mathematical analysis, Compilers (Computer programs), Mathematical optimization, Calculus of variations, Algorithms
- Abstract
Enclosure methods and their applications have been developed to a high standard during the last decades. These methods guarantee the validity of the computed results. This means they are of the same standard as the rest of mathematics. The book deals with a wide variety of aspects of enclosure methods. All contributions follow the common goal to push the limits of enclosure methods forward. Topics that are treated include basic questions of arithmetic, proving conjectures, bounds for Krylow type linear system solvers, bounds for eigenvalues, the wrapping effect, algorithmic differencing, differential equations, finite element methods, application in robotics, and nonsmooth global optimization.
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- 2012
34. Validation Numerics : Theory and Applications
- Author
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R. Albrecht, G. Alefeld, H.J. Stetter, R. Albrecht, G. Alefeld, and H.J. Stetter
- Subjects
- Numerical analysis, Algebras, Linear, Mathematical analysis, System theory, Control theory, Mathematical optimization, Calculus of variations
- Abstract
The articles in this book give a comprehensive overview on the whole field of validated numerics. The problems covered include simultaneous systems of linear and nonlinear equations, differential and integral equations and certain applications from technical sciences. Furthermore some papers which improve the tools are included. The book is a must for scientists working in numerical analysis, computer science and in technical fields.
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- 2012
35. Iterative Methods for Fixed Point Problems in Hilbert Spaces
- Author
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Andrzej Cegielski and Andrzej Cegielski
- Subjects
- Functional analysis, Mathematical optimization, Calculus of variations, Numerical analysis, Operator theory
- Abstract
Iterative methods for finding fixed points of non-expansive operators in Hilbert spaces have been described in many publications. In this monograph we try to present the methods in a consolidated way. We introduce several classes of operators, examine their properties, define iterative methods generated by operators from these classes and present general convergence theorems. On this basis we discuss the conditions under which particular methods converge. A large part of the results presented in this monograph can be found in various forms in the literature (although several results presented here are new). We have tried, however, to show that the convergence of a large class of iteration methods follows from general properties of some classes of operators and from some general convergence theorems.
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- 2012
36. Large-Scale Optimization with Applications : Part III: Molecular Structure and Optimization
- Author
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Lorenz T. Biegler, Thomas Coleman, Andrew r. Conn, Fadil N. Santosa, Lorenz T. Biegler, Thomas Coleman, Andrew r. Conn, and Fadil N. Santosa
- Subjects
- Mathematical optimization, Calculus of variations, System theory, Control theory, Numerical analysis, Operations research
- Abstract
Many important molecular conformation problems, such as protein folding, are expressed as global minimization problems. It is the fact that local minimization is insufficient, that markedly differentiates this volume from the previous two. Unfortunately, global minimization problems that result from models of molecular conformation are usually intractable. For example, simple 1-dimensional versions of distance conformation problems are NP-hard. Nevertheless, there has been significant recent progress in the design of promising heuristic strategies (often involving the use of high- performance parallel computers) for computing approximate global minimizers. The purpose of the sessions represented in this volume was to discuss the new algorithmic advances for global minimization in the context of protein folding and related molecular minimization problems. Emphasis was on practical shortcomings of current approaches, outstanding problems and questions, and the use of high-performance parallel computers.
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- 2012
37. Ill-posed Variational Problems and Regularization Techniques : Proceedings of the “Workshop on Ill-Posed Variational Problems and Regulation Techniques” Held at the University of Trier, September 3–5, 1998
- Author
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Michel Thera, Rainer Tichatschke, Michel Thera, and Rainer Tichatschke
- Subjects
- Operations research, Mathematical analysis, System theory, Control theory, Mathematical optimization, Calculus of variations, Numerical analysis
- Published
- 2012
38. Variational Methods for Discontinuous Structures : International Workshop at Villa Erba (Cernobbio), Italy, July 2001
- Author
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Gianni Dal Maso, Franco Tomarelli, Gianni Dal Maso, and Franco Tomarelli
- Subjects
- Differential equations, Functional analysis, Numerical analysis, Mathematical optimization, Calculus of variations
- Abstract
This volume contains the Proceedings of the International Workshop Variational Methods For Discontinuous Structures, which was jointly organized by the Dipar timento di Matematica Francesco Brioschi of Milano Politecnico and the Interna tional School for Advanced Studies (SISSA) of Trieste. The Conference took place at Villa Erba Antica (Cernobbio) on the Lago di Como on July 4- 6, 2001. In past years the calculus of variations faced mainly the study of continuous structures, say particularly problems with smooth solutions. One of the deepest and more delicate problems was the regularity of weak solutions. More recently, new sophisticated tools have been introduced in order to study discontinuities: in many variational problems solutions develop singularities, and sometimes the most interesting part of a solution is the singularity itself. The conference intended to focus on recent developments in this direction. Some of the talks were devoted to differential or variational modelling of image segmentation, occlusion and textures synthesizing in image analysis, varia tional description of micro-magnetic materials, dimension reduction and structured deformations in elasticity and plasticity, phase transitions, irrigation and drainage, evolution of crystalline shapes; in most cases theoretical and numerical analysis of these models were provided. viii Preface Other talks were dedicated to specific problems of the calculus of variations: variational theory of weak or lower-dimensional structures, optimal transport prob lems with free Dirichlet regions, higher order variational problems, symmetrization in the BV framework.
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- 2012
39. Large-Scale Optimization with Applications : Part II: Optimal Design and Control
- Author
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Lorenz T. Biegler, Thomas F. Coleman, Andrew R. Conn, Fadil N. Santosa, Lorenz T. Biegler, Thomas F. Coleman, Andrew R. Conn, and Fadil N. Santosa
- Subjects
- Mathematical optimization, Calculus of variations, System theory, Control theory, Numerical analysis, Operations research
- Abstract
This IMA Volume in Mathematics and its Applications LARGE-SCALE OPTIMIZATION WITH APPLICATIONS, PART II: OPTIMAL DESIGN AND CONTROL is one of the three volumes based on the proceedings of the 1995 IMA three week Summer Program on'Large-Scale Optimization with Applications to Inverse Problems, Optimal Control and Design, and Molecular and Struc tural Optimization.'The other two related proceedings appeared as Vol ume 92: Large-Scale Optirpization with Applications, Part I: Optimization in Inverse Problems and Design and Volume 94: Large-Scale Optimization with Applications, Part III: Molecular Structure and Optimization. We would like to thank Lorenz T. Biegler, Thomas F. Coleman, An drew R. Conn, and Fadil N. Santosa for their excellent work as organizers of the meetings and for editing the proceedings. We also take this opportunity to thank the National Science Founda tion (NSF), the Department of Energy (DOE), and the Alfred P. Sloan support made the workshops possible.
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- 2012
40. Large-Scale Optimization with Applications : Part I: Optimization in Inverse Problems and Design
- Author
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Lorenz T. Biegler, Thomas F. Coleman, Andrew R. Conn, Fadil N. Santosa, Lorenz T. Biegler, Thomas F. Coleman, Andrew R. Conn, and Fadil N. Santosa
- Subjects
- Mathematical optimization, Calculus of variations, System theory, Control theory, Numerical analysis, Operations research
- Abstract
Inverse problems and optimal design have come of age as a consequence of the availability of better, more accurate, and more efficient simulation packages. Many of these simulators, which can run on small workstations, can capture the complicated behavior of the physical systems they are modeling, and have become commonplace tools in engineering and science. There is a great desire to use them as part of a process by which measured field data are analyzed or by which design of a product is automated. A major obstacle in doing precisely this is that one is ultimately confronted with a large-scale optimization problem. This volume contains expository articles on both inverse problems and design problems formulated as optimization. Each paper describes the physical problem in some detail and is meant to be accessible to researchers in optimization as well as those who work in applied areas where optimization is a key tool. What emerges in the presentations is that there are features about the problem that must be taken into account in posing the objective function, and in choosing an optimization strategy. In particular there are certain structures peculiar to the problems that deserve special treatment, and there is ample opportunity for parallel computation. THIS IS BACK COVER TEXT!!! Inverse problems and optimal design have come of age as a consequence of the availability of better, more accurate, and more efficient, simulation packages. The problem of determining the parameters of a physical system from
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- 2012
41. Solving Problems in Scientific Computing Using Maple and MATLAB®
- Author
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Walter Gander, Jiri Hrebicek, Walter Gander, and Jiri Hrebicek
- Subjects
- Numerical analysis, Computer software, Engineering mathematics, Engineering—Data processing, Mathematical physics, Mathematical optimization, Calculus of variations, Compilers (Computer programs)
- Abstract
From the reviews:'.. An excellent reference on undergraduate mathematical computing.'(American Mathematical Monthly)'... manuals for such systems (Maple and MATLAB) tend to use trivial examples, making it difficult for new users of such systems to quickly apply their power to real problems. The authors have written a good book to address this need.... the book is worth buying if you want guidance in applying Maple and MATLAB to problems in the workplace...'(Computing Reviews)'.. The presentation is unique, and extremely interesting. I was thrilled to read this text, and to learn the powerful problem-solving skills presented by these authors. I recommend the text highly, as a learning experience, not only to engineering students, but also to anyone interested in computation.'(Mathematics of Computation)
- Published
- 2012
42. Numerical Resolution of Fluid Dynamic Problems using a Saddle Point Variational Formulation.
- Author
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Guirardello, Reginaldo
- Subjects
FLUID dynamics ,NUMERICAL analysis ,NUMERICAL solutions to differential equations ,CALCULUS of variations ,MATHEMATICAL optimization software ,TURBULENT flow ,GAS-liquid interfaces - Abstract
Variational methods are useful for finding numerical solutions of differential equations, which are the corresponding Euler-Lagrange equations to the stationary condition of the functional. Usually the functional is a maximum or a minimum with respect to some function, but in some cases the functional is a saddle point. In this work a saddle point variational formulation is proposed to solve fluid dynamic problems, and the saddle point is found through an iterative method using the optimization software GAMS. Two case studies are solved to show the applicability of the proposed method, one for a single fluid in a two dimensional laminar flow in a pipe and another for a one dimensional turbulent flow for gas-liquid column. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
43. On the asymptotic study of transmission problem in a thin domain.
- Author
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Benseghir, Aissa, Benseridi, Hamid, and Dilmi, Mourad
- Subjects
- *
BOUNDARY value problems , *INVISCID flow , *NUMERICAL analysis , *ELASTICITY , *CALCULUS of variations - Abstract
In this paper, we study the theoretical analysis of a frictionless contact between two general elastic bodies in a stationary regime in a three-dimensional thin domain Ω ε {\Omega^{\varepsilon}} with Tresca friction law. Firstly, the problem statement and variational formulation are presented. We then obtain the estimates on displacement independently of the parameter ε. Finally, we obtain the main results concerning the limit of a weak problem and its uniqueness. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
44. Inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach.
- Author
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Yan, Xiong Bin and Wei, Ting
- Subjects
- *
HEAT equation , *FRACTIONAL calculus , *BOUNDARY value problems , *NUMERICAL analysis , *CALCULUS of variations - Abstract
In this paper, we consider an inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach; that is, to determine the space-dependent source term from a noisy final data. Based on the series expression of the solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions, and we provide the existence and uniqueness for the adjoint problem. Further, we use the Tikhonov regularization method to solve the inverse source problem and provide a conjugate gradient algorithm to find an approximation to the minimizer of the Tikhonov regularization functional. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
45. The classical theory of calculus of variations for generalized functions.
- Author
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Lecke, Alexander, Luperi Baglini, Lorenzo, and Giordano, Paolo
- Subjects
- *
CALCULUS , *MATHEMATICAL functions , *FUNCTIONAL equations , *NUMERICAL analysis , *MATHEMATICAL analysis - Abstract
We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler–Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi's theorem on conjugate points and Noether's theorem. We close with an application to low regularity Riemannian geometry. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
46. A weak minima approach to the study of the existence of saddle points of integral functionals.
- Author
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Boccardo, Lucio and Orsina, Luigi
- Subjects
- *
INTEGRAL calculus , *FUNCTIONALS , *MATHEMATICAL singularities , *MATHEMATICAL regularization , *NUMERICAL analysis - Abstract
Abstract We study of the existence of saddle points of the functional J defined in (1.1) both in the regular case, i.e., if E belongs to (L N (Ω)) N , and in the singular one, i.e., if E belongs to (L 2 (Ω)) N. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
47. First-order, stationary mean-field games with congestion.
- Author
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Evangelista, David, Ferreira, Rita, Gomes, Diogo A., Nurbekyan, Levon, and Voskanyan, Vardan
- Subjects
- *
MEAN field theory , *HAMILTON'S principle function , *NUMERICAL analysis , *STRUCTURAL dynamics , *ALGEBRAIC field theory - Abstract
Mean-field games (MFGs) are models for large populations of competing rational agents that seek to optimize a suitable functional. In the case of congestion, this functional takes into account the difficulty of moving in high-density areas. Here, we study stationary MFGs with congestion with quadratic or power-like Hamiltonians. First, using explicit examples, we illustrate two main difficulties: the lack of classical solutions and the existence of areas with vanishing densities. Our main contribution is a new variational formulation for MFGs with congestion. With this formulation, we prove the existence and uniqueness of solutions. Finally, we consider applications to numerical methods. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
48. Direct numerical method for isoperimetric fractional variational problems based on operational matrix.
- Author
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Ezz–Eldien, Samer S., Bhrawy, Ali H., and El–Kalaawy, Ahmed A.
- Subjects
- *
ISOPERIMETRICAL problems , *CALCULUS of variations , *CAPUTO fractional derivatives , *LEGENDRE'S polynomials , *NUMERICAL analysis - Abstract
In this paper, we applied a direct method for a solution of isoperimetric fractional variational problems. We use shifted Legendre orthonormal polynomials as basis function of operational matrices of fractional differentiation and fractional integration in combination with the Lagrange multipliers technique for converting such isoperimetric fractional variational problems into solving a system of algebraic equations. Also, we show the convergence analysis of the presented technique and introduce some test problems with comparisons between our numerical results with those introduced using different methods. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
49. Stationarity of the crack-front for the Mumford–Shah problem in 3D.
- Author
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Lemenant, Antoine and Mikayelyan, Hayk
- Subjects
- *
FRACTURE mechanics , *FINITE element method , *GEOMETRY , *GROUP theory , *NUMERICAL analysis - Abstract
In this paper we exhibit a family of stationary solutions of the Mumford–Shah functional in R 3 , arbitrary close to a crack-front. Unlike other examples, known in the literature, those are topologically non-minimizing in the sense of Bonnet [4] . We also give a local version in a finite cylinder and prove an energy estimate for minimizers. Numerical illustrations indicate the stationary solutions are unlikely minimizers and show how the dependence on axial variable impacts the geometry of the discontinuity set. A self-contained proof of the stationarity of the cracktip function for the Mumford–Shah problem in 2D is presented. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
50. LARGE VOLUME MINIMIZERS OF A NONLOCAL ISOPERIMETRIC PROBLEM: THEORETICAL AND NUMERICAL APPROACHES.
- Author
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GÉNÉRAU, FRANÇOIS and OUDET, EDOUARD
- Subjects
- *
ISOPERIMETRIC inequalities , *NUMERICAL analysis , *PERIMETERS (Geometry) , *RIESZ spaces , *BOUNDARY value problems - Abstract
We consider the volume-constrained minimization of the sum of the perimeter and the Riesz potential. We add an external potential of the form ∥x∥β that provides the existence of a minimizer for any volume constraint, and we study the geometry of large volume minimizers. Then we provide a numerical method to address this variational problem. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
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