Fundamental solution for linear two-point boundary value problem

The quadratic functional minimization with differential restrictions represented by the command linear systems is considered. The optimal solution determination implies the solving of a linear problem with two points boundary values. The proposed method implies the construction of a fundamental solution S(t)—a n×n matrix—and of a vector h(t) defining an adjoint variable λ(t) depending of the state variable x(t). From the extremum necessary conditions it is obtained the Riccati matrix differential equation having the S(t) as unknown fundamental solution is obtained. The paper analyzes the existence of the Riccati equation solution S(t) and establishes as the optimal solution of the proposed optimum problem. Also a superior limit of the minimum for the considered quadratic functionals class are evaluated.     

Curvature variation minimizing cubic Hermite interpolants

In this paper, planar parametric Hermite cubic interpolants with small curvature variation are studied. By minimization of an appropriate approximate functional, it is shown that a unique solution of the interpolation problem exists, and has a nice geometric interpretation. The best solution of such a problem is a quadratic geometric interpolant. The optimal approximation order 4 of the solution is confirmed. The approach is combined with strain energy minimization in order to obtain G¹ cubic interpolatory spline.     

Identification of distributed systems and the theory of the regularization

The identification problems, i.e., the problems of finding unknown parameters in distributed systems from the observations are very important in modern control theory. The solutions of these identification problems can be obtained by solving the equations of the first kind. However, the solutions are often unstable. In other words, they are not continuously dependent on the data. The regularization or Tihonov's regularization is known as one of the stabilizing algorithms to solve these non well-posed problems. In this paper is studied the regularization method for identification of distributed systems. Several approximation theorems are proved to solve the equations of the first kind. Then, identification problems are reduced to the minimization of quadratic cost functionals by virtue of these theorems. On the other hand, it is known that the statistical methods for identification such as the maximum likelihood lead to the minimization problems of certain quadratic functionals. Comparing these quadratic cost functionals, the relations between the regularization and the statistical methods are discussed. Further, numerical examples are given to show the effectiveness of this method.     

Numerical experiments on dual matrix algorithms for function minimization

In Ref. 2, four algorithms of dual matrices for function minimization were introduced. These algorithms are characterized by the simultaneous use of two matrices and by the property that the one-dimensional search for the optimal stepsize is not needed for convergence. For a quadratic function, these algorithms lead to the solution in at mostn+1 iterations, wheren is the number of variables in the function. Since the one-dimensional search is not needed, the total number of gradient evaluations for convergence is at mostn+2. In this paper, the above-mentioned algorithms are tested numerically by using five nonquadratic functions. In order to investigate the effects of the stepsize on the performances of these algorithms, four schemes for the stepsize factor are employed, two corresponding to small-step processes and two corresponding to large-step processes. The numerical results show that, in spite of the wide range employed in the choice of the stepsize factor, all algorithms exhibit satisfactory convergence properties and compare favorably with the corresponding quadratically convergent algorithms using one-dimensional searches for optimal stepsizes.     

A parametrization method for the numerical solution of singular differential equations

The paper explains the numerical parametrization method (PM), originally created for optimal control problems, for classical calculus of variation problems that arise in connection with singular implicit (IDEs) and differential-algebraic equations (DAEs) in frame of their regularization. The PM for IDEs is based on representation of the required solution as a spline with moving knots and on minimization of the discrepancy functional with respect to the spline parameters. Such splines are named variational splines. For DAEs only finite entering functions can be represented by splines, and the functional under minimization is the discrepancy of the algebraic subsystem. The first and the second derivatives of the functionals are calculated in two ways – for DAEs with the help of adjoint variables, and for IDE directly. The PM does not use the notion of differentiation index, and it is applicable to any singular equation having a solution.     

Projection-iterative version of the pointwise relaxation method

We propose a projection-iterative modification of the pointwise relaxation method for the solution of variational problems of mechanics of deformable solids. For the general case of the problem of conventional minimization of functionals in Hilbert spaces, convergence is proved. For the problem of minimization of quadratic functional, a theoretical justification is performed by the finite-element method.     

On a representation of additive functionals of zero quadratic variation

For a Markov process X associated to a Dirichlet form, we use continuous additive functionals obtained by Fukushima decompositions in order to represent the class of additive functionals of zero quadratic variation. We do not assume that X is symmetric.     

Parameter identification problem for the equation of motion of membrane with strong viscosity

The parameter identification problem of constant parameters in the equation of membrane with strong viscosity is studied. The problem is formulated by a minimization of quadratic cost functionals by distributive measurements. The existence of optimal parameters and necessary optimality conditions for the parameters are proved.     

Numerical solutions of the m-membranes problem

In this paper numerical approximation for the m-membrane problem is considered. We make a change of variables that leads to a different expression of the quadratic functional that allows after discretizing the problem to reformulate it as finite dimensional bound constrained quadratic problem. To our knowledge this is the first paper on numerical approximation of the m-membrane problem. We reformulate the m-membrane problem as a bound constraint quadratic minimization problem. The bound constraint quadratic form is solved with the gradient projection method.     

Analysis of Nonlinear Quadratic Control in Infinite Time

This study refers to minimization of quadratic functionals in infinite time. The coefficients of the quadratic form are quadratic matrices, function of the state variable. Dynamic constraints are represented by a bilinear differential systems of the form. The necessary extremum conditions determine the adjoint variables λ and the control variables u as functions of state variable, respectively the adjoint system corresponding to those functions. Thus it will be obtained a matrix differential equation where the solution representing the positive defined symmetric matrix P ( x ), verifies the Riccati algebraic equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergent Outer Approximation Algorithms for Solving Unary Programs

Interesting cutting plane approaches for solving certain difficult multiextremal global optimization problems can fail to converge. Examples include the concavity cut method for concave minimization and Ramana's recent outer approximation method for unary programs which are linear programming problems with an additional constraint requiring that an affine mapping becomes unary. For the latter problem class, new convergent outer approximation algorithms are proposed which are based on sufficiently deep l-norm or quadratic cuts. Implementable versions construct optimal simplicial inner approximations of Euclidean balls and of intersections of Euclidean balls with halfspaces, which are of general interest in computational convexity. Computational behavior of the algorithms depends crucially on the matrices involved in the unary condition. Potential applications to the global minimization of indefinite quadratic functions subject to indefinite quadratic constraints are shown to be practical only for very small problem sizes.     

Γ-convergence of multiscale periodic functionals depending on the time-derivative and the curl operator

We study the Γ-convergence of a family of multiscale periodic quadratic integral functionals defined in a product space, whose densities depend on the time-derivative and on the curl of solenoidal fields, through the multiscale convergence in time–space and the multiscale Young measures in time–space associated with relevant sequences of pairs. An explicit representation of the Γ-limit density is given by means of an infinite dimensional minimization problem.     

Variational calculus with sums of elementary tensors of fixed rank

In this article we introduce a calculus of variations for sums of elementary tensors and apply it to functionals of practical interest. The survey provides all necessary ingredients for applying minimization methods in a general setting. The important cases of target functionals which are linear and quadratic with respect to the tensor product are discussed, and combinations of these functionals are presented in detail. As an example, we consider the solution of a linear system in structured tensor format. Moreover, we discuss the solution of an eigenvalue problem with sums of elementary tensors. This example can be viewed as a prototype of a constrained minimization problem. For the numerical treatment, we suggest a method which has the same order of complexity as the popular alternating least square algorithm and demonstrate the rate of convergence in numerical tests.     

Variational splines for the numerical solution of singular differential equations

The numerical parametrization method (PM), originally created for optimal control problems, is specificated for classical calculus of variation problems that arise in connection with singular implicit (IDEs) and differential-algebraic equations (DAEs). The PM for IDEs is based on representation of the required solution as a spline with moving knots and on minimization of the discrepancy functional with respect to the spline parameters. Such splines are named variational splines. For DAEs only finite entering functions can be represented by splines, and the functional under minimization is the discrepancy of the algebraic subsystem. The first and the second derivatives of the functionals are calculated in two ways – for DAEs with the help of adjoint variables, and for IDE directly. The PM does not use the notion of differentiation index, and it is applicable to any singular equation having a solution. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A mathematical investigation of the Car-Parrinello method

The Car-Parrinello method for ab-initio molecular dynamics avoids the explicit minimization of energy functionals given by functional density theory in the context of the quantum adiabatic approximation (time-dependent Born-Oppenheimer approximation). Instead, it introduces a fictitious classical dynamics for the electronic orbitals. For many realistic systems this concept allowed first-principle computer simulations for the first time. In this paper we study the quantitative influence of the involved parameter , the fictitious electronic mass of the method. In particular, we prove by use of a carefully chosen two-time-scale asymptotics that the deviation of the Car-Parrinello method from the adiabatic model is of order – provided one starts in the ground state of the electronic system and the electronic excitation spectrum satisfies a certain non-degeneracy condition. Analyzing a two-level model problem we prove that our result cannot be improved in general. Received June 14, 1996     

Rate of convergence bound for approximating sequences in minimization problems for functionals

A general approximation scheme for minimization of functionals in a Banach space is considered. Inequalities are proved which supply bounds on the rate of convergence of the approximate solutions to the exact solution. These bounds are applied to an optimal control problem for an abstract operator equation in a Hilbert space with control in the right-hand side.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 117–121, 1986.     

A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems

Summary In this paper, we shall be concerned with the solution of constrained convex minimization problems. The constrained convex minimization problems are proposed to be transformable into a convex-additively decomposed and almost separable form, e.g. by decomposition of the objective functional and the restrictions. Unconstrained dual problems are generated by using Fenchel-Rockafellar duality. This decomposition-dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable. Moreover, the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return-formulas. In quadratic programming the decomposition-dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated byH-matrices. Numerical tests for solving obstacle problems in ¹ discretized by using piecewise quadratic finite elements and in ² by using the five-point difference approximation are presented.     

An efficient gradient method with approximate optimal stepsize for the strictly convex quadratic minimization problem

In this paper, a new type of stepsize, approximate optimal stepsize, for gradient method is introduced to interpret the Barzilai–Borwein (BB) method, and an efficient gradient method with an approximate optimal stepsize for the strictly convex quadratic minimization problem is presented. Based on a multi-step quasi-Newton condition, we construct a new quadratic approximation model to generate an approximate optimal stepsize. We then use the two well-known BB stepsizes to truncate it for improving numerical effects and treat the resulted approximate optimal stepsize as the new stepsize for gradient method. We establish the global convergence and R-linear convergence of the proposed method. Numerical results show that the proposed method outperforms some well-known gradient methods.