Evolutionary variational inequality with long-term memory and applications to economic networks

The aim of this paper is to present the relation between an evolutionary variational inequality with long-term memory and Lagrange multipliers. More precisely, we study the oligopolistic market equilibrium problem in which the profit function depends also on previous events of the market by means of a long-term memory which takes into account the previous states of the equilibrium. Moreover, thanks to the variational formulation, we are able to show existence and regularity results for equilibrium solutions. Then, we apply the infinite dimensional duality theory through which we obtain the existence of Lagrange multipliers which are great utility in order to understand the behaviour of the market. Finally, an example is provided, which allows to analyse the influence of the long-term memory on the equilibrium solution.     

Image Segmentation with Depth Information via Simplified Variational Level Set Formulation

Image segmentation with depth information can be modeled as a minimization problem with Nitzberg–Mumford–Shiota functional, which can be transformed into a tractable variational level set formulation. However, such formulation leads to a series of complicated high-order nonlinear partial differential equations which are difficult to solve efficiently. In this paper, we first propose an equivalently reduced variational level set formulation without using curvatures by taking level set functions as signed distance functions. Then, an alternating direction method of multipliers (ADMM) based on this simplified variational level set formulation is designed by introducing some auxiliary variables, Lagrange multipliers via using alternating optimization strategy. With the proposed ADMM method, the minimization problem for this simplified variational level set formulation is transformed into a series of sub-problems, which can be solved easily via using the Gauss–Seidel iterations, fast Fourier transform and soft thresholding formulas. The level set functions are treated as signed distance functions during computation process via implementing a simple algebraic projection method, which avoids the traditional re-initialization process for conventional variational level set methods. Extensive experiments have been conducted on both synthetic and real images, which validate the proposed approach, and show advantages of the proposed ADMM projection over algorithms based on traditional gradient descent method in terms of computational efficiency.     

Asynchronous multi-domain variational integrators for nonlinear hyperelastic solids

We present the asynchronous multi-domain variational time integrators with a dual domain decomposition method for the initial hyperbolic boundary-value problem in hyperelasticity. Variational time integration schemes, based on the principle of minimal action within the Lagrangian framework, are constructed for the equation of motion and implemented into a variational finite element framework, which is systematically derived from the three-field de Veubeke-Hu-Washizu variational principle to accommodate the incompressibility constraint present in an analysis of nearly-incompressible materials. For efficient parallel computing, we use the dual domain decomposition method with local Lagrange multipliers to ensure the continuity of the displacement field at the interface between subdomains. The α-method for time discretization and the multi-domain spatial decomposition enable us to use different types of integrators (explicit vs. implicit) and different time steps on different parts of a computational domain, and thus efficiently capture the underlying physics with less computational effort. The energy conservation of our nonlinear, midpoint, asynchronous integration scheme is investigated using the Energy method, and both local and the global energy error estimates are derived. We illustrate the performance of proposed variational multi-domain time integrators by means of three examples. First, the method of manufactured solutions is used to examine the consistency of the formulation. In the second example, we investigate energy conservation and stability. Finally, we apply the method to the motion of a heterogeneous plane domain, where different integrators and time discretization steps are used accordingly with disparate material data of individual parts.     

Total FETI based algorithm for contact problems with additional non-linearities

The paper is concerned with application of a new variant of the FETI domain decomposition method called Total FETI to the solution to contact problems. Its basic idea is that both the compatibility between adjacent sub-domains and Dirichlet boundary conditions are enforced by Lagrange multipliers. We introduce the Total FETI technique for solution to the variational inequalities governing the equilibrium of system of bodies in contact. Moreover, we show implementation of the method into a code which treats the material and geometric non-linear effects. Numerical experiments were carried out with our in-house general purpose finite element package PMD.     

Reduced-order forward dynamics of multiclosed-loop systems

In this work, a reduced-order forward dynamics of multiclosed-loop systems is proposed by exploiting the associated inherent kinematic constraints at acceleration level. First, a closed-loop system is divided into an equivalent open architecture consisting of several serial and tree-type subsystems by introducing cuts at appropriate joints. The resulting cut joints are replaced by appropriate constraint forces also referred to as Lagrange multipliers. Next, for each subsystem, the governing equations of motion are derived in terms of the Lagrange multipliers, which are based on the Newton–Euler formulation coupled with the concept of Decoupled Natural Orthogonal Complement (DeNOC) matrices, introduced elsewhere. In the proposed forward dynamics formulation, Lagrange multipliers are calculated sequentially at the subsystem level, and later treated as external forces to the resulting serial or tree-type systems of the original closed-loop system, for the recursive computation of joint accelerations. Note that such subsystem-level treatment allows one to use already existing algorithms for serial and tree-type systems. Hence, one can perform the dynamic analyses relatively quickly without rewriting the complete model of the closed-loop system at hand. The proposed methodology is in contrast to the conventional approaches, where the Lagrange multipliers are calculated together at the system level or simultaneously along with the joint accelerations, both of which incur higher order computational complexities, and thereby a greater number of arithmetic operations. Due to the smaller size of matrices involved in evaluating Lagrange multipliers in the proposed methodology, and the recursive computation of the joint accelerations, the overall numerical performances like computational efficiency, etc., are likely to improve. The proposed reduced-order forward dynamics formulation is illustrated with two multiclosed-loop systems, namely, a 7-bar carpet scrapping mechanism and a 3-RRR parallel manipulator.     

A mixed hemivariational–variational problem and applications

We consider an abstract system with Lagrange multipliers which consists of a hemivariational inequality and a variational inequality. The hemivariational inequality is governed by a hemicontinuous, generalized monotone, possibly nonlinear operator as well as by a bilinear form. This bilinear form also governs the variational inequality. We are looking for a pair solution in a subset of a product of two real reflexive Banach spaces. In order to illustrate the applicability of the abstract results, two examples in terms of PDEs are delivered. Each example is related to the weak solvability of a boundary value problem arising in contact mechanics.     

The lagrange multipliers in the equilibrium finite element method: Error estimates and postprocessing

We study the use of the Lagrange multipliers to impose interelement continuity in the resolution of the biharmonic problem Δ² p=f by an equilibrium finite element method. We prove the convergence of the Lagrange multipliers to the displacement and to the normal derivative of the displacement. By means of a suitable postprocessing we show that the multipliers can be used to provide a better (with respect to the discrete displacement) approximation of the continuous displacement.     

A symmetric Galerkin BEM variational framework for multi-domain interface problems

In this paper we discuss an energy-based variational framework for the solution of interior problems in multiply-connected domains comprising multiple piecewise homogeneous subdomains, using exclusively boundary integral equations. The primary goal is to provide a unified variational setting that lends itself naturally to symmetric Galerkin boundary element formulations in terms of Dirichlet-type variables only.The approach hinges on the explicit imposition of the normal derivative of the classical integral representation of the interior solution on each subdomain via Lagrange multipliers in the augmented Lagrangian of the system. We use Maue-type identities to resolve the hypersingular kernels, leading to a scheme that requires only standard single- and double-layer evaluations. In addition, the usual difficulty with multi-valued normals at subdomain corners is treated here within the same variational framework, by incorporating into the variational formulation the constraint equation between the limiting normal derivatives at either side of the corner. The resulting scheme remains fully symmetric.The numerical implementation avoids the explicit presence of Neumann-type unknowns on the boundaries, through condensation at the subdomain level. In all integral evaluations, three- or four-point Gauss quadrature rules are sufficient for accurate results. We describe the theory and present illustrative examples for thermal and acoustic problems governed by Laplace and Helmholtz equations, respectively. This technique, however, can be applied without essential modification to more general problems.     

Numerical approximation of incompressible flows with net flux defective boundary conditions by means of penalty techniques

We consider incompressible flow problems with defective boundary conditions prescribing only the net flux on some inflow and outflow sections of the boundary. As a paradigm for such problems, we simply refer to Stokes flow. After a brief review of the problem and of its well posedness, we discretize the corresponding variational formulation by means of finite elements and looking at the boundary conditions as constraints, we exploit a penalty method to account for them. We perform the analysis of the method in terms of consistency, boundedness and stability of the discrete bilinear form and we show that the application of the penalty method does not affect the optimal convergence properties of the finite element discretization. Since the additional terms introduced to account for the defective boundary conditions are non-local, we also analyze the spectral properties of the equivalent algebraic formulation and we exploit the analysis to set up an efficient solution strategy. In contrast to alternative discretization methods based on Lagrange multipliers accounting for the constraints on the boundary, the present scheme is particularly effective because it only mildly affects the structure and the computational cost of the numerical approximation. Indeed, it does not require neither multipliers nor sub-iterations or additional adjoint problems with respect to the reference problem at hand.     

Mortar Finite Elements for Interface Problems

Mortar techniques provide a flexible tool for the coupling of different discretization schemes or triangulations. Here, we consider interface problems within the framework of mortar finite element methods. We start with a saddle point formulation and show that the interface conditions enter into the right-hand side. Using dual Lagrange multipliers, we can work with scaled sparse matrices, and static condensation gives rise to a symmetric and positive definite system on the unconstrained product space. The iterative solver is based on a modified multigrid approach. Numerical results illustrate the performance of our approach.This work was supported in part by the Deutsche Forschungsgemeinschaft, SFB 404, C12.     

Modeling the benefit of e-recruiting process integration

While e-recruiting has been widely adopted as one of the most successful e-business applications, it constitutes an under-researched area in e-business research. This study reviews the integration issues in e-recruiting and presents an e-recruiting integration decision model. The benefits of the investment in e-recruiting process integration are discussed in comparison to separate e-recruiting investments. We show that the optimal investment in the e-recruiting process integration results in a lower total cost than the separate e-recruiting investments. In addition, in light of the widely practiced resource constrained investments, we present the method of Lagrange multipliers which is used to find the optimal investment under a budget constraint.     

The Variational Garrote

We analyze the variational method for sparse regression using ? ₀ regularization. The variational approximation results in a model that is similar to Breiman’s Garrote model. We refer to this method as the Variational Garrote (VG). The VG has the effect of making the problem effectively of maximal rank even when the number of samples is small compared to the number of variables. We propose a naive mean field approximation combined with a maximum a posteriori (MAP) approach to estimate the model parameters and use an annealing and reheating schedule of the sparsity hyper-parameter to avoid local minima. The hyper-parameter is set by cross-validation. We compare the VG with the lasso, ridge regression and the recently introduced Bayesian paired mean field method (PMF) (Titsias and Lázaro-Gredilla in Advances in neural information processing systems, vol. 24, pp. 2339–2347, 2011). For fair comparison, we implemented a similar annealing-reheating schedule for the PMF sparsity parameter. Numerical results show that the VG and PMF yield more accurate predictions and more accurately reconstruct the true model than the other methods. The VG finds correct solutions when the lasso solution is inconsistent due to large input correlations. In the experiments that we consider we find that the VG, although based on a simpler approximation than the PMF, yields qualitatively similar or better results and is computationally more efficient. The naive implementation of the VG scales cubic with the number of features. By introducing Lagrange multipliers we obtain a dual formulation of the problem that scales cubic in the number of samples, but close to linear in the number of features.     

A Dynamic Network Oligopoly Model with Transportation Costs,Product Differentiation,and Quality Competition

This paper develops a new dynamic model of Cournot–Nash oligopolistic competition that includes production and transportation costs, product differentiation, and quality levels in a network framework. The production costs capture the total quality cost, which, in turn, can also represent the R&D cost. We first present the equilibrium version and derive alternative variational inequality formulations. We then construct the projected dynamical systems model, which provides a continuous-time evolution of the firms’ product shipments and product quality levels, and whose set of stationary points coincides with the set of solutions to the variational inequality problem. We establish stability analysis results using a monotonicity approach and construct a discrete-time version of the continuous-time adjustment process, which yields an algorithm, with closed form expressions at each iteration. The algorithm is then utilized to compute solutions to several numerical examples. The framework can serve as the foundation for the modeling and analysis of competition among firms in industries ranging from food to pharmaceuticals to durable goods and high tech products, as well as Internet services, where quality and product differentiation are seminal.     

Variational and extremum properties of homogeneous chemical kinetics. II. Minimum dissipation approaches

We continue our work on variational and extremum approaches to homogeneous chemical kinetics of complex reaction systems far from equilibrium and consider minimum dissipation approaches in both the energy and entropy representations. We generalize the linear results of Onsager and Onsager-Machlup to nonlinear (transient and nonisothermal) situations and show that the generalization obey their general thermodynamic schemes.With the help of an error criterion it is shown that the Onsager-Machlup linear variational scheme can be generalized to arbitrary nonlinear systems described by a set of coordinates not all of which need be independent and with dissipation quadratic with respect to rates; an outcome of this generalization is an integral principle of least entropy growth along the natural path governed by the dissipative Lagrange equations of motion and the balance constraints (in the entropy representation). The Lagrangian multipliers associated with the constraints are interpreted as the uniquenonequilibrium temperature and (negative) Planck potentials. They replace their well known equilibrium counterparts in extended expressions describing entropy flow far from equilibrium. The absolute nature of the minimum of the related power expressions is shown; this again corresponds to the dissipative Lagrange equations of motion.On leave from: Institute of Chemical Engineering, Warsaw Technical University, Waryskiego 1, 00-645 Warsaw, Poland.     

Sensitivity analysis for time dependent spatial price equilibrium problem

We present some sensitivity results for the spatial price equilibrium problem in the case of quantity formulation model and in presence of excess supply and excess demand. The equilibrium conditions that describe the above model are expressed in terms of a time dependent variational inequality. The variational inequality formulation plays a fundamental role in order to achieve the sensitivity results.     

Mixed coordination method for long-horizon optimal control problems

We present a new approach to solving long-horizon, discrete-time optimal control problems using the mixed coordination method. The idea is to decompose a long-horizon problem into subproblems along the time axis. The requirement that the initial state of a subproblem equal the terminal state of the preceding subproblem is relaxed by using Lagrange multipliers. The Lagrange multipliers and initial state of each subproblem are then selected as high-level variables. The equivalence of the two-level formulation and the original problem is proved for both convex and non-convex cases. The low-level subproblems are solved in parallel using extended differential dynamic programming (DDP). An efficient way to find the gradient and hessian of a low-level objective function with respect to high-level variables is developed. The high-level problem is solved using the modified Newton method. An effective procedure is developed to select initial values of multipliers based on the initial trajectory. The method can convexify the high-level problem while maintaining the separability of an originally non-convex problem. The method performs better and faster than one-level DDP for both convex and non-convex test problems.     

Augmented formulations for solving Maxwell equations

We consider augmented variational formulations for solving the static or time-harmonic Maxwell equations. For that, a term is added to the usual H (curl) conforming formulations. It consists of a (weighted) L² scalar product between the divergence of the EM and the divergence of test fields. In this respect, the methods we present are H (curl, div) conforming. We also build mixed, augmented variational formulations, with either one or two Lagrange multipliers, to dualize the equation on the divergence and, when applicable, the relation on the tangential or normal trace of the field. It is proven that one can derive formulations, which are equivalent to the original static or time-harmonic Maxwell equations. In the latter case, spurious modes are automatically excluded. Numerical analysis and experiments will be presented in the forthcoming paper [Augmented formulations for solving Maxwell equations: numerical analysis and experiments, in preparation].     

On the constraints violation in forward dynamics of multibody systems

It is known that the dynamic equations of motion for constrained mechanical multibody systems are frequently formulated using the Newton–Euler’s approach, which is augmented with the acceleration constraint equations. This formulation results in the establishment of a mixed set of partial differential and algebraic equations, which are solved in order to predict the dynamic behavior of general multibody systems. The classical solution of the equations of motion is highly prone to constraints violation because the position and velocity constraint equations are not fulfilled. In this work, a general and comprehensive methodology to eliminate the constraints violation at the position and velocity levels is offered. The basic idea of the described approach is to add corrective terms to the position and velocity vectors with the intent to satisfy the corresponding kinematic constraint equations. These corrective terms are evaluated as a function of the Moore–Penrose generalized inverse of the Jacobian matrix and of the kinematic constraint equations. The described methodology is embedded in the standard method to solve the equations of motion based on the technique of Lagrange multipliers. Finally, the effectiveness of the described methodology is demonstrated through the dynamic modeling and simulation of different planar and spatial multibody systems. The outcomes in terms of constraints violation at the position and velocity levels, conservation of the total energy and computational efficiency are analyzed and compared with those obtained with the standard Lagrange multipliers method, the Baumgarte stabilization method, the augmented Lagrangian formulation, the index-1 augmented Lagrangian, and the coordinate partitioning method.     

Uncertainty in Thermal Basin Modeling: An Interval Finite Element Approach

Uncertainty assessment in basin modeling and reservoir characterization is traditionally treated by geostatistical methods which are normally based on stochastic probabilistic approaches. In this paper, we present an alternative approach which is based on interval arithmetic. Here, we discuss a fnite element formulation which uses interval numbers rather than real numbers to solve the transient heat conduction in sedimentary basins. For this purpose, a novel formulation was developed to deal with both the special interval arithmetic properties and the transient term in the differential Equation governing heat transfer. In this formulation, the “stiffness” matrix resulting from the discretization of the heat conduction equation is assembled with an element-by-element technique in which the elements are globally independent and the continuity is enforced by Lagrange multipliers. This formulation is an alternative to traditional Monte Carlo method, where it is necessary to run a simulation several times to estimate the uncertainty in the results.We have applied the newly developed techniques to a one-dimensional thermal basin simulation to assess their potential and limitations.We also compared the quality of our formulation with other solution methods for interval linear systems of equations.