Dual Variational Problem Research Articles

Extragradient type projection algorithm for solving quasimonotone variational inequalities

In this paper, we propose an extragradient type projection algorithm for solving quasimonotone variational inequalities in R n . Under the assumption that f is a quasimonotone and continuous mapping and the solution set of the dual variational inequality problem is nonempty, the globally convergence of the algorithm is established. Numerical experiments are also reported.

Biharmonic Obstacle Problem: Guaranteed and Computable Error Bounds for Approximate Solutions

The paper is concerned with a free boundary problem generated by the biharmonic operator and an obstacle. The main goal is to deduce a fully guaranteed upper bound of the difference between the exact minimizer u and any function (approximation) from the corresponding energy class (which consists of the functions in $H^2$ satisfying the prescribed boundary conditions and the restrictions stipulated by the obstacle). For this purpose we use the duality method of the calculus of variations and general type error identities earlier derived for a wide class of convex variational problems. By this method, we define a combined primal--dual measure of error. It contains four terms of different nature. Two of them are the norms of the difference between the exact solutions (of the direct and dual variational problems) and corresponding approximations. Two others are nonlinear measures, related to approximation of the coincidence set (they vanish if the coincidence set defined by means of the approximate solution coincides with the exact one). The measure satisfies the error identity, which right hand side depends on approximate solutions only and, therefore, is fully computable. Thus, the identity provides direct estimation of the primal--dual errors. However, it contains a certain restriction on the form of the dual approximation. In the second part of the paper, we present a way to skip the restriction. As a result, we obtain a fully guaranteed and directly computable error majorant valid for a wide class of approximations regardless of the method used for their construction. The estimates are verified in a series of tests with different approximate solutions. Some of them are quite close to the exact solution and others are rather coarse and have coincidence sets that differ much from the exact one. The results show that the estimates are robust and effective in all the cases.

Computing the yield limit in three-dimensional flows of a yield stress fluid about a settling particle

Calculating the yield limit Yc (the critical ratio of the yield stress to the driving stress), of a viscoplastic fluid flow is a challenging problem, often needing iteration in the rheological parameters to approach this limit, as well as accurate computations that account properly for the yield stress and potentially adaptive meshing. For particle settling flows, in recent years calculating Yc has been accomplished analytically for many antiplane shear flow configurations and also computationally for many geometries, under either two dimensional (2D) or axisymmetric flow restrictions. Here we approach the problem of 3D particle settling and how to compute the yield limit directly, i.e. without iteratively changing the rheology to approach the yield limit. The presented approach develops tools from optimization theory, taking advantage of the fact that Yc is defined via a minimization problem. We recast this minimization in terms of primal and dual variational problems, develop the necessary theory and finally implement a basic but workable algorithm. We benchmark results against accurate axisymmetric flow computations for cylinders and ellipsoids, computed using adaptive meshing. We also make comparisons of accuracy in calculating Yc on comparable fixed meshes. This demonstrates the feasibility and benefits of directly computing Yc in multiple dimensions. Lastly, we present some sample computations for complex 3D particle shapes.

Primal-Dual Reduced Basis Methods for Convex Minimization Variational Problems: Robust True Solution a Posteriori Error Certification and Adaptive Greedy Algorithms

The a posteriori error estimate and greedy algorithms play central roles in the reduced basis method (RBM). In [M. Yano, Comput. Methods Appl. Mech. Engrg., 287 (2015), pp. 290--309; ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 163--185; SIAM J. Sci. Comput., 40 (2018), pp. A388--A420], several versions of RBMs based on exact error certifications and greedy algorithms with spatio-parameter adaptivities are developed. In this paper, with the parametric symmetric coercive elliptic boundary value problem as an example of the primal-dual variational problems satisfying the strong duality, we develop primal-dual RBMs (PD-RBM) with robust true error certifications and discuss three versions of greedy algorithms to balance the finite element error, the exact RB error, and the adaptive mesh refinements. For a class of convex minimization variational problems which has corresponding dual problems satisfying the strong duality, the primal-dual gap between the primal and dual functionals can be used as an a posteriori error estimator. This primal-dual gap error estimator is robust with respect to the parameters of the problem, and it can be used for both mesh refinements of finite element methods and the true RB error certification. With the help of an integration by parts formula, the primal-dual variational theory is developed for the symmetric coercive elliptic boundary value problems with nonhomogeneous boundary conditions by both the conjugate function and Lagrangian theories. A generalized Prager--Synge identity, which is the primal-dual gap error representation for this specific problem, is developed. RBMs for both the primal and dual problems with robust error estimates are developed. The dual variational problem often can be viewed as a constraint optimization problem. In the paper, different from the standard saddle-point finite element approximation, the dual RBM is treated as a Galerkin projection by constructing RB spaces satisfying the homogeneous constraint. Inspired by the greedy algorithm with spatio-parameter adaptivity of [M. Yano, SIAM J. Sci. Comput., 40 (2018), pp. A388--A420], adaptive balanced greedy algorithms with primal-dual finite element and RB error estimators are discussed. Numerical tests are presented to test the PD-RBM with adaptive balanced greedy algorithms.

Exponential type duality for η-approximated variational problems

In this article, we use the so-called ?-approximation method for solving a new class of nonconvex variational problems with exponential (p, r)-invex functionals. In this approach, we construct ?-approximated variational problem and ?-approximated Mond- Weir dual variational problem for the considered variational problem and its Mond-Weir dual variational problem. Then several duality results for considered variational problem and its Mond-Weir dual variational problem are proved by the help of duality results established between ?-approximated variational problems mentioned above.

Second-order symmetric duality in multiobjective variational problems

In this work, we introduce a pair of multiobjective second-order symmetric dual variational problems. Weak, strong, and converse duality theorems for this pair are established under the assumption of ?-bonvexity/?-pseudobonvexity. At the end, the static case of our problems has also been discussed.

Existence of Generalized Minimizers and Dual Solutions for a Class of Variational Problems with Linear Growth Related to Image Recovery

We continue the analysis of modifications of the total variation image inpainting method formulated on the space BV (Ω) M and treat the case of vector-valued images where we do not impose any structure condition on the density F and the dimension of the domain Ω is arbitrary. We discuss the existence of generalized solutions of the corresponding variational problem and show the unique solvability of the associated dual variational problem. We establish the uniqueness of the absolutely continuous part ∇ a u of the gradient of BV -solutions u on the domain Ω and get the uniqueness of BV -solutions outside the damaged region D. We also prove new density results for functions of bounded variation and for Sobolev functions. Bibliography: 36 titles.

Stresses in equilibrium configurations of inextensible nets with slack

The paper deals with nets formed by two families of fibers (cords) which can grow shorter but not longer, in a deformation. The nets are treated as two-dimensional continua in the three-dimensional space. The inextensibility condition places unilateral constraint on the partial derivatives y,1 and y,2 of the deformation of the form | y , 1 ( x ) | ≤ 1 , | y , 2 ( x ) | ≤ 1 , There is no deformation energy, the total energy reduces to the potential energy of the net under external forces. Equilibrium configurations are those of minimum energy. The stresses in equilibrium configurations thus reduce to the reactions to the constraints. Nonzero stresses occur only in tense regions where one or two constraints are satisfied with the equality sign. The paper follows the work of Paroni in treating the stress problem via the dual variational problem in the sense of convex analysis. Unlike in the work of Paroni, where stresses are modeled as finitely additive set functions, here a (perhaps more economic) choice of spaces is made that leads to more accessible stresses represented by (countably additive) measures. The present development is made possible by an observation, of independent value, that the space of measures with divergence measure is the dual of another Banach space, in the present context naturally interpreted as the space of strains. Our measures generalize stress fields represented by ordinary functions to account for stress concentrations along folded lines in tension, frequently occurring in equilibrium configurations of the net.

On Some Perturbations of the total variation image inpainting method. Part II: Relaxation and Dual Variational Formulation

We continue the analysis of some strongly elliptic modifications of the total variation image inpainting model formulated in the space BV(Ω) and investigate the corresponding dual variational problems. Remarkable features are the uniqueness of the dual solution and the uniqueness of the absolutely continuous part ∇ a u of the gradient of BV-solutions u on the whole domain. Additionally, any BV-minimizer u automatically satisfies the inequality 0 ≤ u ≤ 1, which means that u measures the intensity of the grey level. Outside of the damaged region we even have the uniqueness of BV-solutions, whereas on the damaged domain the L 2-deviation $$ {\left\Vert u-\upsilon \right\Vert}_{L^2} $$ of different solutions is governed by the total variation of the singular part ∇ s (u − υ) of the vector measure ∇(u − υ). Moreover, the dual solution is related to the BV-solutions through an equation of stress-strain type. Bibliography: 23 titles.

One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: remarks on duality and flows

This paper is devoted to one-dimensional interpolation Gagliardo–Nirenberg–Sobolev inequalities. We study how various notions of duality, transport and monotonicity of functionals along flows defined by some non-linear diffusion equations apply. We start by reducing the inequality to a much simpler dual variational problem using mass transportation theory. Our second main result is devoted to the construction of a Lyapunov functional associated with a non-linear diffusion equation, that provides an alternative proof of the inequality. The key observation is that the inequality on the line is equivalent to Sobolev's inequality on the sphere, at least when the dimension is an integer, or to the critical interpolation inequality for the ultraspherical operator in the general case. The time derivative of the functional along the flow is itself very interesting. It explains the machinery of some rigidity estimates for non-linear elliptic equations and shows how eigenvalues of a linearized problem enter into the computations. Notions of gradient flows are then discussed for various notions of distances. Throughout this paper, we shall deal with two classes of inequalities corresponding either to p > 2 or to 1 < p < 2 . The algebraic part in the computations is very similar in both cases, although the case 1 < p < 2 is definitely less standard.

Optimality Conditions and Second-Order Duality for Nondifferentiable Multiobjective Continuous Programming Problems

Fritz John and Karush-Kuhn-Tucker type optimality conditions for a nondifferentiable multiobjective variational problem are derived. As an application of Karush-Kuhn-Tucker type optimality conditions, Mond-weir type second-order nondifferentiable multiobjective dual variational problems is constructed. Various duality results for the pair of Mond-Weir type second-order dual variational problems are proved under second-order pseudoinvexity and second-order quasi-invexity. A pair of Mond-Weir type dual variational problems with natural boundary values is formulated to derive various duality results. Finally, it is pointed out that our results can be considered as dynamic generalizations of their static counterparts existing in the literature.

An existence theorem for inextensible nets with slack

Nets formed by two families of fibers that can grow shorter but not longer are studied variationally. It is shown that a minimizing deformation exists and, by means of the dual variational problem, that there exist also tension fibers which equilibrate the applied loads.

Canonical dual finite element method for solving post-buckling problems of a large deformation elastic beam

This paper presents a canonical dual mixed finite element method for the post-buckling analysis of planar beams with large elastic deformations. The mathematical beam model employed in the present work was introduced by Gao in 1996, and is governed by a fourth-order non-linear differential equation. The total potential energy associated with this model is a non-convex functional and can be used to study both the pre- and the post-buckling responses of the beams. Using the so-called canonical duality theory, this non-convex primal variational problem is transformed into a dual problem. In a proper feasible space, the dual variational problem corresponds to a globally concave maximization problem. A mixed finite element method involving both the transverse displacement field and the stress field as approximate element functions is derived from the dual variational problem and used to compute global optimal solutions. Numerical applications are illustrated by several problems with different boundary conditions.

A Posteriori Error Estimates for the Approximations of the Stresses in the Hencky Plasticity Problem

In this article, we derive a posteriori error estimates for the Hencky plasticity problem. These estimates are formulated in terms of the stresses and present guaranteed and computable bounds of the difference between the exact stress field and any approximation of it from the energy space of the dual variational problem. They consist of quantities that can be considered as penalties for the violations of the equilibrium equations, the yield condition and the constitutive relations that must hold for the exact stresses and strains. It is proved that the upper bound tends to zero for any sequence of stresses that tends to the exact solution of the Haar–Karman variational problem. An important ingredient of our analysis is a collection of Poincaré type inequalities involving the L 1 norms of the tensors of small deformation. Estimates of this form are not new, however we will present computable upper bounds for the constants being involved even for rather complicated domains.

Composition duality methods for quasistatic evolution elastoviscoplastic variational problems

Composition duality methods for dual quasistatic evolution elastoviscoplastic variational problems are studied. Dual evolution mixed analysis is performed, as well as corresponding primal static mixed analysis. For multi-constitutive modeling and parallel computing, macro-hybrid variational formulations are further considered at the continuous level.

Estimates of deviations from minimizers for variational problems with power growth functionals

We derive directly computable estimates of differences between approximate solutions and minimizers of the variational problem $$J_\alpha \left[ w \right]: = \int\limits_\Omega {\left[ {\frac{1}{\alpha }\left| {\nabla w} \right|^\alpha - fw} \right]dx} \to \min .$$ . If the functional has a superquadratic growth, then the estimate is given in terms of the natural energy norm. For problems with subquadratic growth, it is more convenient to derive such estimates in terms of the dual variational problem. The estimates are obtained for Dirichlet, Neumann, and mixed boundary conditions. Bibliography: 22 titles.

Mixed type duality for multiobjective variational problems with generalized [formula omitted]-convexity

A mixed type dual for multiobjective variational problems is formulated. Several duality theorems are established relating properly efficient solutions of the primal and dual variational problems under generalized ( F , ρ ) -convexity. Static mixed type dual multiobjective problems are particular cases of these problems.

New-invexity type conditions with applications to constrained dynamic games

Weaker invexity type properties are presented in the continuous case, and are then applied to develop duality results between a pair of symmetric dual variational problems. A constrained dynamic game is shown to be equivalent to this symmetric dual pair.

Regularity for Dual Solutions and for Weak Cluster Points of Minimizing Sequences of Variational Problems with Linear Growth

The minimum problem \(\int_\Omega {f(\nabla u)dx \to \min } \) among the mappings \(u:\mathbb{R}^n \supset \Omega \to \mathbb{R}^N \) with prescribed Dirichlet boundary data and for integrands f of linear growth in general fails to have solutions in the Sobolev space W 1 1 . We therefore concentrate on the dual variational problem, which admits a unique maximizer σ, and prove the partial Holder continuity of σ. Moreover, we study smoothness properties of the L1-limits of minimizing sequences of the original problem. Bibliography: 20 titles.

Estimates for Errors in Two-Dimensional Models of Elasticity Theory

Explicit (computable) estimates for errors in some simplified two-dimensional models of the theory of elastic body are obtained. Such models are usually used if one of the characteristic sizes of a domain is significantly less than two others. In particular, the model of plane stresses is considered. To obtain estimates for the deviation of reduced solutions from exact solutions, the direct and dual variational problems are considered and the duality methods are used. Bibliography: 9 titles.