Linear Variational Problems Research Articles

A solution method for linear variational relation problems

In this paper, we consider a particular class of variational relation problem namely linear variational relation problem wherein the sets are defined by linear inequalities. The purpose is to study...

Hardy—Poincaré inequalities with boundary singularities

We are interested in variational problems involving weights that are singular at a point of the boundary of the domain. More precisely, we study a linear variational problem related to the Poincaré inequality and to the Hardy inequality for maps in H01(Ω), where Ω is a bounded domain in ℝN, N ≥ 2, with 0 ∈ ∂Ω. In particular, we give sufficient and necessary conditions so that the best constant is achieved.

A new result for projection neural networks to solve linear variational inequalities and related optimization problems

In recent years, a projection neural network was proposed for solving linear variational inequality (LVI) problems and related optimization problems, which required the monotonicity of LVI to guarantee its convergence to the optimal solution. In this paper, we present a new result on the global exponential convergence of the projection neural network. Unlike existing convergence results for the projection neural network, our main result does not assume the monotonicity of LVI problems. Therefore, the projection neural network can be further guaranteed to solve a class of non-monotone LVI and non-convex optimization problems. Numerical examples illustrate the effectiveness of the obtained result.

A Discrete-Time Neural Network for Optimization Problems With Hybrid Constraints

Recurrent neural networks have become a prominent tool for optimizations including linear or nonlinear variational inequalities and programming, due to its regular mathematical properties and well-defined parallel structure. This brief presents a general discrete-time recurrent network for linear variational inequalities and related optimization problems with hybrid constraints. In contrary to the existing discrete-time networks, this general model can operate not only on bound constraints, but also on hybrid constraints comprised of inequality, equality and bound constraints. The model has dynamical properties of global convergence, asymptotical and exponential convergences under some weaker conditions. Numerical examples demonstrate its efficacy and performance.

Functional a posteriori error estimates for incremental models in elasto-plasticity

Abstract We consider incremental problem arising in elasto-plastic models with isotropic hardening. Our goal is to derive computable and guaranteed bounds of the difference between the exact solution and any function in the admissible (energy) class of the problem considered. Such estimates are obtained by an advanced version of the variational approach earlier used for linear boundary-value problems and nonlinear variational problems with convex functionals [24, 30]. They do no contain mesh-dependent constants and are valid for any conforming approximations regardless of the method used for their derivation. It is shown that the structure of error majorant reflects properties of the exact solution so that the majorant vanishes only if an approximate solution coincides with the exact one. Moreover, it possesses necessary continuity properties, so that any sequence of approximations converging to the exact solution in the energy space generates a sequence of positive numbers (explicitly computable by the majorant functional) that tends to zero.

Uniqueness and existence for bounded boundary value problems

The existence and uniqueness of solutions for the boundary value problems with general linear point evaluation boundary conditions is established. We assume that f is bounded and that there is uniqueness on a homogeneous problem and on the linear variational problems.

An application of babuska-brezzi's theory to a class of variational problems

In this paper we apply the Babuska-Brezzi theory to obtain new results on the solvability and Galerkin approximations of some operator equations arising in linear and nonlinear variational problems with linear constraints. This constitutes an extension of the usual theory to a class of variational formulations that can be written in a dual-dual form

Decomposition of energy spaces and applications

In this Note we discuss the solution of linear variational problems in Hilbert spaces V, such that V = Σ i=1 m , V t with the possibility that V t ∩ V j ≠ {0} if i ≠ j. We take advantage of the above decomposition of V to solve the linear variational problems by conjugate gradient algorithms operating in Π t=1 m V t For some situations, the method discussed in this note leads to novel domain decomposition methods with good convergence and parallelization properties. We conclude this note with the results of numerical experiments concerning the solution of elliptic problems on an L-shaped domain considered as the union of two overlapping rectangles and on a two-dimensional domain which is the union of a rectangle and a disk.

On efficiency calculations for nonholonomic locomotion problems: An application to microswimming

We compare two notions of efficiency for locomotion processes modelled by linear variational problems with nonholonomic constraints. We revisit Lighthill's work on squirming motions of a sphere inside a viscous fluid, also studied by Blake and by Shapere and Wilczek. Our approach uses the spectral basis of the propulsion operator (sending velocity to surface force fields) to describe the surface deformations. We show that Shapere and Wilczek's extremal solutions, using high order surface deformations, are optimal only in a naive sense. In fact, using Lighthill's efficiency definition, high order deformations are not efficient. Finally, we present a new criterion which combines the advantages of the two notions.

Parallelization model for successive approximations to the Rayleigh-Ritz linear variational problem

Many of the differential equations arising in science and engineering can be recast in the form of a matrix eigenvalue problem. Solution of this equation within the context of the Rayleigh-Ritz variational method may be viewed as one of the fundamental tasks of numerical analysis. Successive approximation approaches to the Rayleigh-Ritz problem seek to improve eigenvectors and eigenfunctions by sequentially refining a trial function. Parallelization of successive approximation approaches has been demonstrated numerous times in the literature; these studies addressed either the successive approximations or the matrix diagonalization levels of the algorithm. It is shown in this paper that these two strategies may be applied independently of one another, and the advantages of applying both parallelization levels simultaneously to the problem are discussed. Performance estimates for a two-tiered parallelization strategy are obtained by extrapolating from existing published performance data for which the two levels of parallelization were applied separately.

Coupling of Mixed Finite Elements and Boundary Elements for A Hyperelastic Interface Problem

We apply the coupling of mixed finite elements and boundary integral methods, using two integral equations on the coupling boundary, to study the weak solvability of a nonlinear elliptic problem arising in elastostatics. Our procedure is based on both the usual Brezzi's theory for linear constrained variational problems and some fundamental tools from nonlinear functional analysis. We derive existence and uniqueness of solution for the continuous variational formulations and provide general approximation results for the corresponding Galerkin schemes. Although we consider bounded domains, the same analysis applies for the case in which the boundary element region is unbounded.

A posteriori error estimation for nonlinear variational problems

We introduce a new modus operandi for a posteriori error estimation for nonlinear (and linear) variational problems based on the duality theory of the calculus of variations. We derive what we call duality error estimates and show that they yield computable a posteriori error estimates without directly solving the dual problem.

Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems

In this paper we apply the coupling of mixed finite element and boundary integral methods for solving some class of linear and nonlinear elliptic boundary value problems. As a model case we consider the two-domensional Laplacian coupled with a second order elliptic equation in divergence frorm which may also be nonlinear. Such problems arise, e.g., in nonlinear magnetic field computations. Our analysis is based on Brezzi's theory for linear constrained variational problems, and on some nonlinear functional analysis. We show existence and uniqueness for the continuous variational formulations, and derive general approximation results for the corresponding Galerkin achemes.

Regularity for a non linear variational problem in dimension two

Regularity for a non linear variational problem in dimension two

Mathematical Bioeconomics: The Optimal Management of Renewable Resources.

Introduction. 1. Elementary Dynamics of Exploited Populations. 1.1 The Logistic Growth Model. 1.2 Generalized Logistic Models: Depensation. 1.3 Summary and Critique. 2. Economic Models of Renewable-Resource Harvesting. 2.1 The Open-Access Fishery. 2.2 Economic Overfishing. 2.3 Biological Overfishing. 2.4 Optimal Fishery Management. 2.5 The Optimal Harvest Policy. 2.6 Examples Based on the Schaefer Model. 2.7 Linear Variational Problems. 2.8 The Possibility of Extinction. 2.9 Summary and Critique. 3. Capital-Theoretic Aspects of Resource Management. 3.1 Interest and Discount Rates. 3.2 Capital Theory and Renewable Resources. 3.3 Nonautonomous Models. 3.4 Applications to Policy Problems: Labor Mobility in the Fishery. 4. Optimal Control Theory. 4.1 One-Dimensional Control Problems. 4.2 A Nonlinear Fishery Model. 4.3 Economic Interpretation of the Maximum Principle. 4.4 Multidimensional Optimal Control Problem. 4.5 Optimal Investment in Renewable-Resource Harvesting. 5. Supply and Demand: Nonlinear Models. 5.1 The Elementary Theory of Supply and Demand. 5.2 Supply and Demand in Fisheries. 5.3 Nonlinear Cost Effects: Pulse Fishing. 5.4 Game-Theoretic Models. 5.5 Transboundary Fishery Resources: A Further Application of the Theory. 5.6 Summary and Critique. 6. Dynamical Systems. 6.1 Basic Theory. 6.2 Dynamical Systems in the Plane: Linear Theory. 6.3 Isoclines. 6.4 Nonlinear Plane-Autonomous Systems. 6.5 Limit Cycles. 6.6 Gause's Model of Interspecific Competition. 7. Discrete-Time and Metered Models. 7.1 A General Metered Stock-Recruitment Model. 7.2 The Beverton-Holt Stock-Recruitment Model. 7.3 Depensation Models. 7.4 Overcompensation. 7.5 A Simple Cohort Model. 7.6 The Production Function of a Fishery. 7.7 Optimal Harvest Policies. 7.8 The Discrete Maximum Principle. 7.9 Dynamic Programming. 8. The Theory of Resource Regulation. 8.1 A Behavioral Model. 8.2 Optimization Analysis. 8.3 Limited Entry. 8.4 Taxes and Allocated Transferable Quotas. 8.5 Total Catch Quotas. 8.6 Summary and Critique. 9. Growth and Aging. 9.1 Forestry Management: The Faustmann Model. 9.2 The Beverton-Holt Fisheries Model. 9.3 Dynamic Optimization in the Beverton-Holt Model. 9.4 The Case of Bounded F. 9.5 Multiple, Cohorts: Nonselective Gear. 9.6 Pulse Fishing. 9.7 Multiple Cohorts: Selective Gear. 9.8 Regulation. 9.9 Summary and Critique. 10. Multispecies Models. 10.1 Differential Productivity. 10.2 Harvesting Competing Populations. 10.3 Selective Harvesting. 10.4 A Diffusion Model: The Inshore-Offshore Fishery. 10.5 Summary and Critique. 11. Stochastic Resource Models. 11.1 Stochastic Dynamic Programming. 11.2 A Stochastic Forest Rotation Model. 11.3 Uncertainty and Learning. 11.4 Searching for Fish. 11.5 Summary and Critique. Supplementary Reading. References. Index.

The Asymptotics of the Solutions of Linear Elliptic Variational Problems in Domains with Edges

The Asymptotics of the Solutions of Linear Elliptic Variational Problems in Domains with Edges

Geometric nonlinearity: potential energy, complementary energy, and the gap function

Dual minimum principles for displacements and stresses are well established for linear variational problems and also for nonlinear (and monotone) constitutive laws. This paper studies the problem of geometric nonlinearity. By introducing a gap function, we recover complementary variational principles in the equilibrium problems of mathematical physics. When the gap function is nonnegative those become minimum principles. The theory is based on convex analysis, and the applications made here are to nonlinear mechanics.

Periodic solutions and homogenization of non linear variational problems

We prove an homogenization formula for some non linear variational problems that extends the analogous one known in the linear case. Namely the solution ue of the problem $$\int\limits_\Omega {\left\{ {f\left( {\frac{x}{\varepsilon },Du_\varepsilon } \right) - \varphi \left( x \right)u_\varepsilon } \right\}dx} = minimum,$$ where the boundary data are independent of ɛ and f=f(x, ξ) is periodic in x, converges in some Lp space as ɛ goes to zero to the solution of an analogous variational problem whose integrand can be evaluated from f.

Duality Methods for Linear Variational Problems in $L^\infty $

Minimization problems of the form $\{ \| Lu \|_{L^\infty } :u \in U\} $ are considered where L is a linear operator and U is a convex subset of some Hilbert space determined by a finite number of linear functional constraints. The usual variational techniques which provide an Euler equation for such problems in U for $p < \infty$ are not applicable in $L^\infty $; the solution is obtained and necessary conditions that it must satisfy are found by dualizing the problem to an appropriate extremal problem in a subspace of $L^1 $. Several applications are given where L is a linear differential operator acting on a Sobolev space.