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Journal articles on the topic 'Variational methods'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Cipu, Elena Corina, and Cosmin Dănuţ Barbu. "Variational Estimation Methods for Sturm–Liouville Problems." Mathematics 10, no. 20 (2022): 3728. http://dx.doi.org/10.3390/math10203728.

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In this paper, we are concerned with approach solutions for Sturm–Liouville problems (SLP) using variational problem (VP) formulation of regular SLP. The minimization problem (MP) is also set forth, and the connection between the solution of each formulation is then proved. Variational estimations (the variational equation associated through the Euler–Lagrange variational principle and Nehari’s method, shooting method and bisection method) and iterative variational methods (He’s method and HPM) for regular RSL are unitary presented in final part of the paper, which ends with applications.

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2

Pedregal, Pablo. "Variational methods for non-variational problems." SeMA Journal 74, no. 3 (2017): 299–317. http://dx.doi.org/10.1007/s40324-017-0119-z.

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3

Sohaly, M. A., M. T. Yassen, and I. M. Elbaz. "The Variational Methods for Solving Random Models." International Journal of Innovative Research in Computer Science & Technology 5, no. 2 (2017): 214–25. http://dx.doi.org/10.21276/ijircst.2017.5.2.1.

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4

Benhadid, Ayache. "Iterative methods for extended general variational inequalities." General Mathematics 29, no. 1 (2021): 95–102. http://dx.doi.org/10.2478/gm-2021-0008.

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Abstract In this paper, we suggest and analyze a new approximation schemes (3) to solve the extended general variational inequalities (2), which were introduced by Muhammad Aslam Noor (see[7, 9]). Using the projection operator technique, we establish the equivalence between the extended general variational inequalities and the fixed-point problem. This equivalent formulation is used to discuss the existence of a solution of the extended general variational inequalities. Several special cases are also discussed.

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5

Ceng, L. C., A. Latif, C. F. Wen, and A. E. Al-Mazrooei. "Hybrid Steepest-Descent Methods for Triple Hierarchical Variational Inequalities." Journal of Function Spaces 2015 (2015): 1–22. http://dx.doi.org/10.1155/2015/980352.

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We introduce and analyze a relaxed iterative algorithm by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions, and the solution set of general system of variational inequalities (GSVI), which is just a unique solution of a triple hie

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6

Noor, Muhammad Aslam, and Khalida Inayat Noor. "Iterative resolvent methods for general mixed variational inequalities." Journal of Applied Mathematics and Stochastic Analysis 16, no. 3 (2003): 283–94. http://dx.doi.org/10.1155/s1048953303000236.

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In this paper, we use the technique of updating the solution to suggest and analyze a class of new self-adaptive splitting methods for solving general mixed variational inequalities. It is shown that these modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. Proof of convergence is very simple. Since general mixed variational include variational inequalities and complementarity problems as special cases, our results continue to hold for these problems.

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7

Gibson, Andrew A. P. "Circuit Analogy to Introduce Variational Methods." International Journal of Electrical Engineering & Education 31, no. 2 (1994): 144–47. http://dx.doi.org/10.1177/002072099403100206.

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Circuit analogy to introduce variational methods Variational procedures, energy expressions and the minimum energy principle are often difficult steps to introduce in advanced undergraduate and postgraduate courses in Electrical Engineering. A simple preamble, relying only on a basic circuit analogy, can be used to overcome these difficulties and is presented here.

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8

Noor, Muhammad Aslam, Muzaffar Akhter, and Khalida Inayat Noor. "Forward-backward resolvent splitting methods for general mixed variational inequalities." International Journal of Mathematics and Mathematical Sciences 2003, no. 43 (2003): 2759–70. http://dx.doi.org/10.1155/s0161171203210462.

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We use the technique of updating the solution to suggest and analyze a class of new splitting methods for solving general mixed variational inequalities. It is shown that these modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. Our methods differ from the known three-step forward-backward splitting of Glowinski, Le Tallec, and M. A. Noor for solving various classes of variational inequalities and complementarity problems. Since general mixed variational inequalities include variational inequalities and complementarity problems as special case

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9

He, Ji-Huan. "Asymptotic Methods for Solitary Solutions and Compactons." Abstract and Applied Analysis 2012 (2012): 1–130. http://dx.doi.org/10.1155/2012/916793.

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This paper is an elementary introduction to some new asymptotic methods for the search for the solitary solutions of nonlinear differential equations, nonlinear differential-difference equations, and nonlinear fractional differential equations. Particular attention is paid throughout the paper to giving an intuitive grasp for the variational approach, the Hamiltonian approach, the variational iteration method, the homotopy perturbation method, the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform, and ancient Chinese mathematics. Hamilton principle and variatio

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10

Mielke, Alexander, Felix Otto, Giuseppe Savaré, and Ulisse Stefanelli. "Variational Methods for Evolution." Oberwolfach Reports 8, no. 4 (2011): 3145–216. http://dx.doi.org/10.4171/owr/2011/55.

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11

Ambrosio, Luigi, Alexander Mielke, Mark Peletier, and Giuseppe Savaré. "Variational Methods for Evolution." Oberwolfach Reports 11, no. 4 (2014): 3177–254. http://dx.doi.org/10.4171/owr/2014/57.

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12

Mielke, Alexander, Mark Peletier, and Dejan Slepčev. "Variational Methods for Evolution." Oberwolfach Reports 14, no. 4 (2018): 3185–261. http://dx.doi.org/10.4171/owr/2017/54.

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13

Mielke, Alexander, Mark A. Peletier, and Dejan Slepčev. "Variational Methods for Evolution." Oberwolfach Reports 17, no. 2 (2021): 1391–467. http://dx.doi.org/10.4171/owr/2020/29.

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14

Struwe, Michael. "Variational Methods in Geometry." Jahresbericht der Deutschen Mathematiker-Vereinigung 119, no. 2 (2016): 71–91. http://dx.doi.org/10.1365/s13291-016-0155-0.

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15

Hoffmann, Franca, Alexander Mielke, Mark A. Peletier, and Dejan Slepčev. "Variational Methods for Evolution." Oberwolfach Reports 20, no. 4 (2024): 3173–247. http://dx.doi.org/10.4171/owr/2023/57.

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16

Laura, P. A. A. "Optimization of variational methods." Ocean Engineering 22, no. 3 (1995): 235–50. http://dx.doi.org/10.1016/0029-8018(94)p2695-y.

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17

G., W., and C. A. Brebbia. "Variational Methods in Engineering." Mathematics of Computation 47, no. 175 (1986): 383. http://dx.doi.org/10.2307/2008113.

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18

W., L. B., and P. M. Prenter. "Splines and Variational Methods." Mathematics of Computation 56, no. 193 (1991): 384. http://dx.doi.org/10.2307/2008555.

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19

Keller, S. H., F. Lauze, and M. Nielsen. "Deinterlacing Using Variational Methods." IEEE Transactions on Image Processing 17, no. 11 (2008): 2015–28. http://dx.doi.org/10.1109/tip.2008.2003394.

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20

Mura, Toshio, Tatsuhito Koya, and S. M. Heinrich. "Variational Methods in Mechanics." Journal of Pressure Vessel Technology 115, no. 1 (1993): 102–3. http://dx.doi.org/10.1115/1.2929487.

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21

Long, Yiming. "Progress in variational methods." Frontiers of Mathematics in China 3, no. 2 (2008): 149–50. http://dx.doi.org/10.1007/s11464-008-0020-2.

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22

Chacon, Monique Revoredo, and Michael C. Zerner. "Perturbation-variational methods revisited." International Journal of Quantum Chemistry 47, no. 2 (1993): 103–18. http://dx.doi.org/10.1002/qua.560470202.

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23

SHARUN, I. V., and M. E. OVCHINNIKOV. "SOLVING VARIATIONAL INEQUALITIES BY ITERATIVE METHODS." Applied Mathematics and Fundamental Informatics 11, no. 4 (2024): 10–16. https://doi.org/10.25206/2311-4908-2024-11-4-10-16.

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This article considers a class of iterative algorithms, during the study of which the results are presented in the form of implemented some iterative algorithms from the considered algorithms for solving variational inequalities, applied to solve some types of problems, such as numerical approximation of solving a system of linear algebraic equations, linear complementarity problem, nonlinear variational inequalities. A comparative analysis of the effectiveness of the methods based on graphs of the dependence of the values of the loss function on the number of iterations is carried out.

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24

Simo, J. C., and T. J. R. Hughes. "On the Variational Foundations of Assumed Strain Methods." Journal of Applied Mechanics 53, no. 1 (1986): 51–54. http://dx.doi.org/10.1115/1.3171737.

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So-called assumed strain methods are based on the a-priori assumption of an interpolation for the discrete gradient operator, not necessarily derivable from the displacement interpolation. It is shown that this class of methods falls within the class of variational methods based on the Hu-Washizu principle. The essential point of this equivalence lies in the statement of the appropriate stress recovery procedure compatible with this variational structure. It is noted that most currently existing assumed strain methods fail to perform the stress recovery in a manner consistent with the variatio

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25

Noor, Muhammad Aslam, Khalida Inayat Noor, and Saira Zainab. "Some Iterative Methods for Solving Nonconvex Bifunction Equilibrium Variational Inequalities." Journal of Applied Mathematics 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/280451.

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We introduce and consider a new class of equilibrium problems and variational inequalities involving bifunction, which is called the nonconvex bifunction equilibrium variational inequality. We suggest and analyze some iterative methods for solving the nonconvex bifunction equilibrium variational inequalities using the auxiliary principle technique. We prove that the convergence of implicit method requires only monotonicity. Some special cases are also considered. Our proof of convergence is very simple. Results proved in this paper may stimulate further research in this dynamic field.

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26

Tsang, B., S. W. Taylor, and G. C. Wake. "Variational methods for boundary value problems." Journal of Applied Mathematics and Decision Sciences 4, no. 2 (2000): 193–204. http://dx.doi.org/10.1155/s1173912600000158.

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The variational formulation of boundary value problems is valuable in providing remarkably easy computational algorithms as well as an alternative framework with which to prove existence results. Boundary conditions impose constraints which can be annoying from a computational point of view. The question is then posed: what is the most general boundary value problem which can be posed in variational form with the boundary conditions appearing naturally? Special cases of two-point problems in one-dimension and some higher dimensional problems are addressed. There is a deep connection with self-

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27

Feng, Xiaobing, and Miun Yoon. "Numerical Methods for Genetic Regulatory Network Identification Based on a Variational Approach." Computational Methods in Applied Mathematics 16, no. 1 (2016): 77–103. http://dx.doi.org/10.1515/cmam-2015-0019.

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AbstractThis paper studies differential equation-based mathematical models and their numerical solutions for genetic regulatory network identification. The primary objectives are to design, analyze, and test a general variational framework and numerical methods for seeking its approximate solutions for reverse engineering genetic regulatory networks from microarray datasets. In the proposed variational framework, no structure assumption on the genetic network is presumed, instead, the network is solely determined by the microarray profile of the network components and is identified through a w

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28

Blatter, Joachim, and Markus Haverland. "Methodologies-theories-praxis." Qualitative & Multi-Method Research 11, no. 1 (2013): 23–24. https://doi.org/10.5281/zenodo.910219.

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We are grateful for the thoughtful discussion of our book by Goetz and Mahoney and the valuable remarks of other contributors. In our final statement we want to address three points. First, we would like to revisit our distinction between three approaches to explanatory case studies. This is triggered by Goertz and Mahoney’s statement that our congruence approach usually requires elements of one of our two other approaches, the co-variational approach and causal-process tracing. Secondly, in a response to comments by Rohlfing we want to briefly clarify what we mean by a co-variational approach

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29

Ablaev, S. S., F. S. Stonyakin, M. S. Alkousa, and D. A. Pasechnyk. "Adaptive Methods or Variational Inequalities with Relatively Smooth and Reletively Strongly Monotone Operators." Программирование, no. 6 (November 1, 2023): 5–13. http://dx.doi.org/10.31857/s013234742306002x.

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The article is devoted to some adaptive methods for variational inequalities with relatively smooth and relatively strongly monotone operators. Based on the recently proposed proximal version of the extragradient method for this class of problems, we study in detail the method with adaptively selected parameter values. An estimate for the rate of convergence of this method is proved. The result is generalized to a class of variational inequalities with relatively strongly monotone δ-generalized smooth variational inequality operators. For the problem of ridge regression and variational inequal

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30

Robinson, Harold C. "An overview of variational methods." Journal of the Acoustical Society of America 98, no. 5 (1995): 2902. http://dx.doi.org/10.1121/1.414287.

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31

Nesbet, R. K. "Variational methods for cellular models." Physical Review A 38, no. 10 (1988): 4955–60. http://dx.doi.org/10.1103/physreva.38.4955.

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32

Lu, Yan, and Ankit Srivastava. "Variational methods for phononic calculations." Wave Motion 60 (January 2016): 46–61. http://dx.doi.org/10.1016/j.wavemoti.2015.08.004.

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33

Fonseca, Irene. "Variational methods for elastic crystals." Archive for Rational Mechanics and Analysis 97, no. 3 (1987): 189–220. http://dx.doi.org/10.1007/bf00250808.

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34

Drugan, Gregory, and Xuan Hien Nguyen. "Shrinking Doughnuts via Variational Methods." Journal of Geometric Analysis 28, no. 4 (2018): 3725–46. http://dx.doi.org/10.1007/s12220-017-9976-z.

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35

Haymaker, R. W. "Variational methods for composite operators." La Rivista del Nuovo Cimento 14, no. 8 (1991): 1–89. http://dx.doi.org/10.1007/bf02811226.

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36

NIRENBERG, LOUIS. "Variational Methods in Nonlinear Problems." Annals of the New York Academy of Sciences 661, no. 1 Frontiers of (1992): 365. http://dx.doi.org/10.1111/j.1749-6632.1992.tb26059.x.

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37

Quéau, Yvain, Jean-Denis Durou, and Jean-François Aujol. "Variational Methods for Normal Integration." Journal of Mathematical Imaging and Vision 60, no. 4 (2017): 609–32. http://dx.doi.org/10.1007/s10851-017-0777-6.

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38

Mohan, S. R. "Combined Relaxation Methods for Variational." OPSEARCH 38, no. 4 (2001): 441–42. http://dx.doi.org/10.1007/bf03398649.

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39

Borwein, Jonathan M., and Qiji J. Zhu. "Variational Methods in Convex Analysis." Journal of Global Optimization 35, no. 2 (2006): 197–213. http://dx.doi.org/10.1007/s10898-005-3835-3.

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40

Bobylev, N. A. "Homotopic methods in variational problems." Journal of Soviet Mathematics 67, no. 1 (1993): 2659–712. http://dx.doi.org/10.1007/bf01455150.

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41

Gozzi, E., M. Reuter, and W. D. Thacker. "Variational methods via supersymmetric techniques." Physics Letters A 183, no. 1 (1993): 29–32. http://dx.doi.org/10.1016/0375-9601(93)90883-2.

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42

Gibson, Andrew A. P., and Bernice M. Dillon. "Introductory approach for variational methods." Computer Applications in Engineering Education 3, no. 4 (1995): 245–49. http://dx.doi.org/10.1002/cae.6180030406.

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43

Ceng, Lu-Chuan, and Xiaoye Yang. "Some Mann-Type Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems Variational Inequalities and Fixed Point Problems." Mathematics 7, no. 3 (2019): 218. http://dx.doi.org/10.3390/math7030218.

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This paper discusses a monotone variational inequality problem with a variational inequality constraint over the common solution set of a general system of variational inequalities (GSVI) and a common fixed point (CFP) of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping in Hilbert spaces, which is called the triple hierarchical constrained variational inequality (THCVI), and introduces some Mann-type implicit iteration methods for solving it. Norm convergence of the proposed methods of the iteration methods is guaranteed under some suitable assumptions.

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44

Tyranowski, Tomasz M., and Mathieu Desbrun. "Variational Partitioned Runge–Kutta Methods for Lagrangians Linear in Velocities." Mathematics 7, no. 9 (2019): 861. http://dx.doi.org/10.3390/math7090861.

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In this paper, we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the “Hamiltonian” equations of motion can be formulated as an index-1 differential-algebraic system. We also construct variational Runge–Kutta methods and analyze their properties. The general properties of Runge–Kutta methods depend on the “veloci

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45

Bonnet, Marc, Giulio Maier, and Castrenze Polizzotto. "Symmetric Galerkin Boundary Element Methods." Applied Mechanics Reviews 51, no. 11 (1998): 669–704. http://dx.doi.org/10.1115/1.3098983.

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This review article concerns a methodology for solving numerically, for engineering purposes, boundary and initial-boundary value problems by a peculiar approach characterized by the following features: the continuous formulation is centered on integral equations based on the combined use of single-layer and double-layer sources, so that the integral operator turns out to be symmetric with respect to a suitable bilinear form. The discretization is performed either on a variational basis or by a Galerkin weighted residual procedure, the interpolation and weight functions being chosen so that th

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46

Noor, Muhammad Aslam. "Some Aspects of Extended General Variational Inequalities." Abstract and Applied Analysis 2012 (2012): 1–16. http://dx.doi.org/10.1155/2012/303569.

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Noor (“Extended general variational inequalities,” 2009, “Auxiliary principle technique for extended general variational inequalities,” 2008, “Sensitivity analysis of extended general variational inequalities,” 2009, “Projection iterative methods for extended general variational inequalities,” 2010) introduced and studied a new class of variational inequalities, which is called the extended general variational inequality involving three different operators. This class of variational inequalities includes several classes of variational inequalities and optimization problems. The main motivation

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47

Noor, Muhammad Aslam, and Zhenyu Huang. "Some Proximal Methods for Solving Mixed Variational Inequalities." Abstract and Applied Analysis 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/610852.

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It is well known that the mixed variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest some new proximal point methods for solving the mixed variational inequalities. These new methods include the explicit, the implicit, and the extragradient method as special cases. The convergence analysis of these new methods is considered under some suitable conditions. Our method of constructing these iterative methods is very simple. Results proved in this paper may stimulate further research in this direction.

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48

Atkins, Ethan, Matthias Morzfeld, and Alexandre J. Chorin. "Implicit Particle Methods and Their Connection with Variational Data Assimilation." Monthly Weather Review 141, no. 6 (2013): 1786–803. http://dx.doi.org/10.1175/mwr-d-12-00145.1.

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Abstract The implicit particle filter is a sequential Monte Carlo method for data assimilation that guides the particles to the high-probability regions via a sequence of steps that includes minimizations. A new and more general derivation of this approach is presented and the method is extended to particle smoothing as well as to data assimilation for perfect models. Minimizations required by implicit particle methods are shown to be similar to those that one encounters in variational data assimilation, and the connection of implicit particle methods with variational data assimilation is expl

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49

Yao, Yonghong, Yeong-Cheng Liou, Cun-Lin Li, and Hui-To Lin. "Extended Extragradient Methods for Generalized Variational Inequalities." Journal of Applied Mathematics 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/237083.

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We suggest a modified extragradient method for solving the generalized variational inequalities in a Banach space. We prove some strong convergence results under some mild conditions on parameters. Some special cases are also discussed.

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50

Ceng, Lu-Chuan, and Qing Yuan. "Hybrid Mann Viscosity Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities and Fixed Point Problems." Mathematics 7, no. 2 (2019): 142. http://dx.doi.org/10.3390/math7020142.

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In the present work, we introduce a hybrid Mann viscosity-like implicit iteration to find solutions of a monotone classical variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities and a problem of common fixed points of an asymptotically nonexpansive mapping and a countable of uniformly Lipschitzian pseudocontractive mappings in Hilbert spaces, which is called the triple hierarchical constrained variational inequality. Strong convergence of the proposed method to the unique solution of the problem is guarantee

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