A variational model and its numerical solution for local,selective and automatic segmentation

Variational region-based segmentation models can serve as effective tools for identifying all features and their boundaries in an image. To adapt such models to identify a local feature defined by geometric constraints, re-initializing iterations towards the feature offers a solution in some simple cases but does not in general lead to a reliable solution. This paper presents a dual level set model that is capable of automatically capturing a local feature of some interested region in three dimensions. An additive operator spitting method is developed for accelerating the solution process. Numerical tests show that the proposed model is robust in locally segmenting complex image structures.     

The numerical solution of parabolic differential equations using variational methods

Summary The question of constructing stable numerical representations for the solutions of initial-boundary value problems for parabolic differential equations is examined.An earlier formulation and discussion of this work can be found in the Author's Ph.D. Thesis (University of Adelaide, South Australia, 1967).     

On the variational formulation of a transmission problem for the Biot equations

In this article, a variational formulation for the transmission problem of the fluid–bone interaction is formulated. The formulation is based on a modified Biot system of equations for the cancellous bone together with a boundary integral equation formulation of the pressure in the water. Existence and uniqueness for the weak solution of the interaction problem are established in appropriate Sobolev spaces.     

On the numerical analysis of non-convex variational problems

Summary. We discuss a numerical method for finding Young-measure-valued minimizers of non-convex variational problems. To have any hope of a convergence theorem, one must work in a setting where the minimizer is unique and minimizing sequences converge strongly. This paper has two main goals: (i) we specify a method for producing strongly-convergent minimizing sequences, despite the failure of strict convexity; and (ii) we show how uniqueness of the Young measure can be parlayed into a numerical convergence theorem. The treatment of (ii) is done in the setting of two model problems, one involving scalar valued functions and a multiwell energy, the other from micromagnetics. Received July 29, 1995     

On the solution stability of variational inequalities

In the present paper, we will study the solution stability of parametric variational conditions

On the existence of a solution to a class of variational inequalities

In this note we consider a class of semilinear elliptic variational inequalities on H ¹(Ω) space. With the aid of the mountain-pass principle and the Ekeland variational principle we prove the existence of solutions.     

On the solution existence of pseudomonotone variational inequalities

As shown by Thanh Hao [Acta Math. Vietnam 31, 283–289, 2006], the solution existence results established by Facchinei and Pang [Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I (Springer, Berlin, 2003) Prop. 2.2.3 and Theorem 2.3.4] for variational inequalities (VIs) in general and for pseudomonotone VIs in particular, are very useful for studying the range of applicability of the Tikhonov regularization method. This paper proposes some extensions of these results of Facchinei and Pang to the case of generalized variational inequalities (GVI) and of variational inequalities in infinite-dimensional reflexive Banach spaces. Various examples are given to analyze in detail the obtained results. B. T. Kien: On leave from Hanoi University of Civil Engineering. The online version of the original article can be found at .     

On the approximate solution of nonlinear variational inequalities

Summary Nonlinear locally coercive variational inequalities are considered and especially the minimal surface over an obstacle. Optimal or nearly optimal error estimates are proved for a direct discretization of the problem with linear finite elements on a regular triangulation of the not necessarily convex domain. It is shown that the solution may be computed by a globally convergent relaxation method. Some numerical results are presented.     

On the numerical solution of a quasilinear algebraic-differential system

We consider a quasilinear algebraic-differential system, suggest a spline collocation method for its solution, and prove a convergence theorem for this method. Results of numerical experiments are given.     

On the variational formulation of the dynamics of systems with friction

We discuss the basic problem of the dynamics of mechanical systems with constraints, namely, the problem of finding accelerations as a function of the phase variables. It is shown that in the case of Coulomb friction, this problem is equivalent to solving a variational inequality. The general conditions for the existence and uniqueness of solutions are obtained. A number of examples are considered. For systems with ideal constraints the problem under discussion was solved by Lagrange in his “Analytical Dynamics” (1788), which became a turning point in the mathematization of mechanics. In 1829, Gauss gave his principle, which allows one to obtain the solution as the minimum of a quadratic function of acceleration, called the constraint. In 1872 Jellett gave examples of non-uniqueness of solutions in systems with static friction, and in 1895 Painlevé showed that in the presence of friction, the absence of solutions is possible along with the nonuniqueness. Such situations were a serious obstacle to the development of theories, mathematical models and the practical use of systems with dry friction. An elegant, and unexpected, advance can be found in the work [1] by Pozharitskii, where the author extended the Gauss principle to the special case where the normal reaction can be determined from the dynamic equations regardless of the values of the coefficients of friction. However, for systems with Coulomb friction, where the normal reaction is a priori unknown, there are still only partial results on the existence and uniqueness of solutions [2–4]. The approach proposed here is based on a combination of the Gauss principle in the form of reactions with the representation of the nonlinear algebraic system of equations for the normal reactions in the form of a variational inequality. The theory of such inequalities [5] includes results on the existence and uniqueness, as well as the developed methods of solution.     

On the variational formulation and implementation of Allen-Cahn and Cahn-Hilliard-type phase field theories

Phase field theory is a promising framework for analyzing evolving microstructures in materials. Phenomena like those related to microstructures in Ni-based superalloys, twin structures in martensites or precipitation in Al-alloys can be predicted by phase field theory. While phase transformations such as those characterizing twinning are captured by an Allen-Cahn-type approach, a Cahn-Hilliard-type formulation is used, if the respective interface motion is driven by the concentration of the species. Although the Allen-Cahn and the Cahn-Hilliard formulation are indeed different, they do share some similarities. To be more precise, a Cahn-Hilliard model is obtained by enforcing balance of mass in the Allen-Cahn approach. Within an energy-based formulation this can be implemented by adding additional energy terms to the underlying Allen-Cahn energy. Such a universal energy-based framework is elaborated in this presentation. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the numerical solution of a class of Stackelberg problems

This study tries to develop two new approaches to the numerical solution of Stackelberg problems. In both of them the tools of nonsmooth analysis are extensively exploited; in particular we utilize some results concerning the differentiability of marginal functions and some stability results concerning the solutions of convex programs. The approaches are illustrated by simple examples and an optimum design problem with an elliptic variational inequality.Prepared while the author was visiting the Department of Mathematics, University of Bayreuth as a guest of the FSP Anwendungsbezogene Optimierung und Steuerung.     

On the numerical solution of Plateau's problem

The present paper is dedicated to the numerical computation of minimal surfaces by the boundary element method. Having a parametrization γ of the boundary curve over the unit circle at hand, the problem is reduced to seeking a reparametrization κ of the unit circle. The Dirichlet energy of the harmonic extension of γ○κ has to be minimized among all reparametrizations. The energy functional is calculated as boundary integral that involves the Dirichlet-to-Neumann map. First and second order necessary optimality conditions of the underlying minimization problem are formulated. Existence and convergence of approximate solutions is proven. An efficient algorithm is proposed for the computation of minimal surfaces and numerical results are presented.     

A mixed variational formulation of a contact problem with adhesion

A mathematical model describing the contact between a viscoplastic body and a deformable foundation is analyzed under small deformation hypotheses. The process is quasistatic and in normal direction the contact is with adhesion, normal compliance, memory effects and unilateral constraint. We derive a mixed-variational formulation of the problem using Lagrange multipliers. Finally, we prove the unique weak solvability of the contact problem.