Equivalence of variational inequality problems to unconstrained minimization

In this paper we propose a class of merit functions for variational inequality problems (VI). Through these merit functions, the variational inequality problem is cast as unconstrained minimization problem. We estimate the growth rate of these merit functions and give conditions under which the stationary points of these functions are the solutions of VI. This work was supported by the state key project “Scientific and Engineering Computing”.     

On inertial proximal algorithm for split variational inclusion problems

ABSTRACT

In this paper, a hybrid proximal algorithm with inertial effect is introduced to solve a split variational inclusion problem in real Hilbert spaces. Under mild conditions on the parameters, we establish weak convergence results for the proposed algorithm. Unlike the earlier iterative methods, we do not impose any conditions on the sequence generated by the proposed algorithm. Also, we extend our results to find a common solution of a split variational inclusion problem and a fixed-point problem. Finally, some numerical examples are given to discuss the convergence and superiority of the proposed iterative methods.     

An optimal control problem for two-dimensional Schrödinger equation

In this paper we consider an optimal control problem controlled by three functions which are in the coefficients of a two-dimensional Schrödinger equation. After proving the existence and uniqueness of the optimal solution, we get the Frechet differentiability of the cost functional using Hamilton-Pontryagin function. Then we state a necessary condition to an optimal solution in the variational inequality form using the gradient.     

Identification of nonlinear heat transfer laws from boundary observations

We consider the problem of identifying a nonlinear heat transfer law at the boundary, or of the temperature-dependent heat transfer coefficient in a parabolic equation from boundary observations. As a practical example, this model applies to the heat transfer coefficient that describes the intensity of heat exchange between a hot wire and the cooling water in which it is placed. We reformulate the inverse problem as a variational one which aims to minimize a misfit functional and prove that it has a solution. We provide a gradient formula for the misfit functional and then use some iterative methods for solving the variational problem. Thorough investigations are made with respect to several initial guesses and amounts of noise in the input data. Numerical results show that the methods are robust, stable and accurate.     

The numerical solution of Newton’s problem of least resistance

In this paper we consider Newton’s problem of finding a convex body of least resistance. This problem could equivalently be written as a variational problem over concave functions in \({\mathbb {R}}^{2}\) . We propose two different methods for solving it numerically. First, we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions. The discretized problem could be solved by standard optimization software efficiently. Second, we conjecture that the optimal body has a certain structure. We exploit this structure and obtain a variational problem in \({\mathbb {R}}^{1}\) . Deriving its Euler–Lagrange equation yields a program with two unknowns, which can be solved quickly.     

Perturbation Theory for Density Matrices and the Thermodynamic Potential of a Quantum System

We use the generating functional method for the matrix elements of second quantization operators to obtain a high-temperature expansion of the thermodynamic potential of a quantum system. This method permits isolating irreducible parts of matrices, including the particle-density matrices. We derive an equation for the full unary density matrix, which is equivalent to the variational principle for the thermodynamic potential. The thermodynamic functions and the density matrix can thus be found in the framework of the same variational problem.     

Fractional variational problems with the Riesz-Caputo derivative

In this paper we investigate optimality conditions for fractional variational problems, with a Lagrangian depending on the Riesz-Caputo derivative. First we prove a generalized Euler-Lagrange equation for the case when the interval of integration of the functional is different from the interval of the fractional derivative. Next we consider integral dynamic constraints on the problem, for several different cases. Finally, we determine optimality conditions for functionals depending not only on the admissible functions, but on time also, and we present a necessary condition for a pair function-time to be an optimal solution to the problem.     

Lagrange multipliers for regular and singular problems in the calculus of variations

A Lagrange multiplier rule is presented for a variational problem of Bolza type under hypotheses that allow certain components of the coefficient matrices involved in the functional being minimized to fail to be integrable near an endpoint of the interval on which the relevant functions are defined. The problem is also addressed when all coefficients are of classL ², but not necessarily bounded. Applications are made to ascertain properties of functions providing equality to certain singular and regular integral inequalities appearing in the literature.     

A uniqueness theorem for a free boundary problem

In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the limit of an equation that serves as a basic model in population biology. Apart from the interest in the problem itself, the techniques used in this paper, which are based on the regularity theory of variational inequalities and of harmonic functions, are of independent interest, and may have other applications.

     

Investigation of the behavior of weak waves in the solution of a vector variational wave equation

We consider nonsmooth solutions of the system of Euler-Lagrange equations corresponding to a variational problem with several unknown functions of several variables and with a quadratic functional. The propagation of weak discontinuities is described by the equations of the method of singular characteristics developed by Melikyan. The onset and interaction of weak discontinuities of the solution caused by nonsmooth initial conditions are studied by numerical-analytic methods. We develop two computer programs for shock-fitting and shock-capturing computations. The approach was earlier applied by the authors to the analysis of a variational wave equation, namely, to the solution of the Euler-Lagrange equation for a variational problem with a single unknown function.     

On the existence of positive solutions and their continuous dependence on functional parameters for some class of elliptic problems

We discuss the existence and the dependence on functional parameters of solutions of the Dirichlet problem for a kind of the generalization of the balance of a membrane equation. Since we shall propose an approach based on variational methods, we treat our equation as the Euler-Lagrange equation for a certain integral functional J. We will not impose either convexity or coercivity of the functional. We develop a duality theory which relates the infimum on a special set X of the energy functional associated with the problem, to the infimum of the dual functional on a corresponding set X^d. The links between minimizers of both functionals give a variational principle and, in consequence, their relation to our boundary value problem. We also present the numerical version of the variational principle. It enables the numerical characterization of approximate solutions and gives a measure of a duality gap between primal and dual functional for approximate solutions of our problem.     

Relaxation and nonoccurrence of the Lavrentiev phenomenon for nonconvex problems

The paper studies a relaxation of the basic multidimensional variational problem, when the class of admissible functions is endowed with the Lipschitz convergence introduced by Morrey. It is shown that in this setup, the integral of a variational problem must satisfy a classical growth condition, unlike the case of uniform convergence. The relaxations constructed here imply the existence of a Lipschitz convergent minimizing sequence. Based on this observation, the paper also shows that the Lavrentiev phenomenon does not occur for a class of nonconvex problems.     

Surfaces of prescribed mean curvature in a cone

We show existence of surfaces of prescribed mean curvature in central projection for such values of the mean curvature for which estimates for the corresponding Euler–Lagrange equations are generally not known. This is achieved by extending the variational problem to the space \({BV(\Omega)}\), where graphs in a cone must satisfy a side condition, and using variational methods. Moreover, we give an example of a solution in \({BV(\Omega)}\) which does not solve the Dirichlet problem for the Euler-Lagrange equation.     

Dynamical Systems in the Variational Formulation of the Fokker—Planck Equation by the Wasserstein Metric

R. Jordan, D. Kinderlehrer, and F. Otto proposed the discrete-time approximation of the Fokker—Planck equation by the variational formulation. It is determined by the Wasserstein metric, an energy functional, and the Gibbs—Boltzmann entropy functional. In this paper we study the asymptotic behavior of the dynamical systems which describe their approximation of the Fokker—Planck equation and characterize the limit as a solution to a class of variational problems. Accepted 2 June 2000. Online publication 6 October 2000.     

A constrained variational problem arising in attractive Bose–Einstein condensate with ellipse-shaped potential

We consider a minimization problem for the variational functional associated with a Gross–Pitaevskii equation arising in the study of an attractive Bose–Einstein condensate. Under an ellipse-shaped trapping potential, that is, the bottom of the trapping potential is an ellipse, we prove that any minimizer of the minimization problem blows up at one of the endpoints of the major axis of the ellipse if the parameter associated to the attractive interaction strength approaches a critical value.     

A modified regularized algorithm for a semilinear space‐fractional backward diffusion problem

In this paper, we investigate a backward problem for a space‐fractional partial differential equation. The main purpose is to propose a modified regularization method for the inverse problem. The existence and the uniqueness for the modified regularized solution are proved. To derive the gradient of the optimization functional, the variational adjoint method is introduced, and hence, the unknown initial value is reconstructed. Finally, numerical examples are provided to show the effectiveness of the proposed algorithm. Copyright © 2017 John Wiley & Sons, Ltd.     

An inverse coefficient problem for a nonlinear parabolic variational inequality

This study is related to inverse coefficient problems for a nonlinear parabolic variational inequality with an unknown leading coefficient in the equation for the gradient of the solution. An inverse method, involving minimization of a least-squares cost functional, is developed to identify the unknown coefficient. It is proved that the solution of the corresponding direct problem depends continuously on the coefficient. On the basis of this, the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.     

On the Dirichlet problem for a nonlinear elastic beam equation

Using a direct variational approach with no global growth conditions on the nonlinear term, we consider the existence of solutions and their dependence on a functional parameter for the fourth order Dirichlet problem connected with the elastic beam equation. We investigate also the existence of an optimal process for such an optimal control problem in which the dynamics is described by the beam equation.     

Keplerian Action,Convexity Optimization,and the 4-Body Problem

In this paper we introduce a method to construct periodic solutions for the n-body problem with only boundary and topological constraints. Our approach is based on some novel features of the Keplerian action functional, constraint convex optimization techniques, and variational methods. We demonstrate the strength of this method by constructing relative periodic solutions for the planar four-body problem within a special topological class, and our results hold for an open set of masses.