Piecewise-linearized methods for oscillators with limit cycles

A piecewise linearization method based on the linearization of nonlinear ordinary differential equations in small intervals, that provides piecewise analytical solutions in each interval and smooth solutions everywhere, is developed for the study of the limit cycles of smooth and non-smooth, conservative and non-conservative, nonlinear oscillators. It is shown that this method provides nonlinear maps for the displacement and velocity which depend on the previous values through the nonlinearity and its partial derivatives with respect to time, displacement and velocity, and yields non-standard finite difference formulae. It is also shown by means of five examples that the piecewise linearization method presented here is more robust and yields more accurate (in terms of displacement, energy and frequency) solutions than the harmonic balance procedure, the method of slowly varying amplitude and phase, and other non-standard finite difference equations.     

Optimal sliding mode control of single degree-of-freedom hysteretic structural system

This paper addresses the controller design problem of a nonlinear single degree-of-freedom structural system excited by the earthquake. Bouc–Wen model, as an efficient hysteresis modeling method, is used to model the system nonlinearity. Sliding mode control (SMC), due to its robustness in dealing with uncertainty, is utilized as the main control strategy. An optimal sliding surface is presented which minimizes the displacement and control force in terms of a quadratic cost function. Two numerical examples are given to illustrate the effectiveness of the proposed strategy subject to three earthquakes of El-Centro, Rinaldi and Kobe. Simulation results show a significant and considerable reduction in structural response and indicate that the performance of suggested optimal SMC strategy is remarkable.     

Interval methods for fixed-point problems

Interval analysis is applied to the fixed-point problem x=?(x) for continuous ?:S→S, where the space S is constructed from Cartesian products of the set R of real numbers, with componentwise definitions of arithmetic operations, ordering, and the product topology. With the aid of an interval inclusion φ:IS → IS in the interval space IS corresponding to S, interval iteration is used to establish the existence or nonexistence of a fixed point x^? of ? in the initial interval X₀. Each step of the interval iteration provides lower and upper bounds for fixed points of ? in the initial interval, from which approximate values and guaranteed error bounds can be obtained directly. In addition to interval iteration, operator equation and dissection methods are considered briefly.

The theory of interval iteration applies directly when only finite subsets of S, IS are used, so this method is adaptable immediately to actual computation. A numerical example is given of the use of interval iteration for the computational solution of a nonlinear integral equation of radiative transfer. It is shown that numerical results with acceptable, guaranteed accuracy can be obtained with a modest amount of computation for an extended range of the parameter involved.     

High-order methods for parabolic problems

Error estimates valid for all t ? 0 for the semi-discrete Galerkin approximation of a parabolic mixed boundary-initial value problem are presented. The solution of the resulting system of ordinary differential equations by implicit Runge-Kutta formulae for arbitrarily high order of accuracy, are discussed. Strongly A-stable methods are found to be advantageous. Theoretical and experimental results for the solution of the resulting system of algebraic equations using a preconditioned outer iteration scheme are discussed. Even the inner linear algebraic equations are preferably solved by iteration.     

Stochastic methods for Dirichlet problems

Using an equivalent expression for solutions of second order Dirichlet problems in terms of Ito type stochastic differential equations, we develop a numerical solution method for Dirichlet boundary value problems. It is possible with this idea to solve for solution values of a partial differential equation at isolated points without having to construct any kind of mesh and without knowing approximations for the solution at any other points. Our method is similar to a recently published approach, but differs primarily in the handling of the boundary. Some numerical examples are presented, applying these techniques to model Laplace and Poisson equations on the unit disk. Visiting Professor, Universidad de Salamanca.     

Difference methods for problems of magnetostatics

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostiki, pp. 79–87, Izd. Moskovskogo Universiteta, Moscow, 1990.     

Boundary element methods for potential problems

The boundary element formulation of potential problems is presented using weighted residual techniques. The paper emphasizes the simplicity of the boundary methods and the way in which they can be applied in engineering. The advantage of using this method in preference to finite elements is discussed in the applications.     

Spectral element methods for parabolic problems

A spectral element method for solving parabolic initial boundary value problems on smooth domains using parallel computers is presented in this paper. The space domain is divided into a number of shape regular quadrilaterals of size h and the time step k   is proportional to h²$h^2$. At each time step we minimize a functional which is the sum of the squares of the residuals in the partial differential equation, initial condition and boundary condition in different Sobolev norms and a term which measures the jump in the function and its derivatives across inter-element boundaries in certain Sobolev norms. The Sobolev spaces used are of different orders in space and time. We can define a preconditioner for the minimization problem which allows the problem to decouple. Error estimates are obtained for both the h and p versions of this method.     

Primal-dual subgradient methods for convex problems

In this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure. Our methods are primal-dual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem. Besides other advantages, this useful feature provides the methods with a reliable stopping criterion. The proposed schemes differ from the classical approaches (divergent series methods, mirror descent methods) by presence of two control sequences. The first sequence is responsible for aggregating the support functions in the dual space, and the second one establishes a dynamically updated scale between the primal and dual spaces. This additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set (however, we always assume the uniform boundedness of subgradients). We present the variants of subgradient schemes for nonsmooth convex minimization, minimax problems, saddle point problems, variational inequalities, and stochastic optimization. In all situations our methods are proved to be optimal from the view point of worst-case black-box lower complexity bounds.     

Hybrid proximal methods for equilibrium problems

This paper concerns developing two hybrid proximal point methods (PPMs) for finding a common solution of some optimization-related problems. First we construct an algorithm to solve simultaneously an equilibrium problem and a variational inequality problem, combing the extragradient method for variational inequalities with an approximate PPM for equilibrium problems. Next we develop another algorithm based on an alternate approximate PPM for finding a common solution of two different equilibrium problems. We prove the global convergence of both algorithms under pseudomonotonicity assumptions.     

Optimization methods for Dirichlet control problems

We discuss several optimization procedures to solve finite element approximations of linear-quadratic Dirichlet optimal control problems governed by an elliptic partial differential equation posed on a 2D or 3D Lipschitz domain. The control is discretized explicitly using continuous piecewise linear approximations. Unconstrained, control-constrained, state-constrained and control-and-state constrained problems are analysed. A preconditioned conjugate method for a reduced problem in the control variable is proposed to solve the unconstrained problem, whereas semismooth Newton methods are discussed for the solution of constrained problems. State constraints are treated via a Moreau–Yosida penalization. Convergence is studied for both the continuous problems and the finite dimensional approximations. In the finite dimensional case, we are able to show convergence of the optimization procedures even in the absence of Tikhonov regularization parameter. Computational aspects are also treated and several numerical examples are included to illustrate the theoretical results.     

Cascadic multigrid methods for parabolic problems

In this paper,we consider the cascadic multigrid method for a parabolic type equation.Backward Euler approximation in time and linear finite element approximation in space are employed.A stability result is established under some conditions on the smoother.Using new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter,these conditions are verified for a number of popular smoothers.Optimal error bound sare derived for both smooth and non-smooth data.Iteration strategies guaranteeing both the optimal accuracy and the optimal complexity are presented.     

Models and methods for standardization problems

When designing an information system, the so-called “Standardization Problem” (SP) arises. The (basic) problem can be described by means of a graph with n nodes and e edges. The nodes represent system elements which have to share information with other nodes. For each system element i, it is possible to introduce a (new) standard, causing fixed costs. In turn, information exchange becomes more efficient if sender i and receiver j introduce the (same) standard, which results in lower exchange costs. The task is to decide for any combination of a system element i and a standard k if k should be introduced in i, so that the sum of setup and exchange costs is minimized. Models and (exact) solution methods are presented for the basic SP as well as for a generalization. Complexity issues are also discussed.     

Interior point methods for equilibrium problems

In the present paper we discuss three methods for solving equilibrium-type fixed point problems. Concentrating on problems whose solutions possess some stability property, we establish convergence of these three proximal-like algorithms that promise a very high numerical tractability and efficiency. For example, due to the implemented application of zone coercive Bregman functions, all these methods allow to treat the generated subproblems as unconstrained and, partly, explicitly solvable ones.     

Defect-correction multigrid methods for nonlinear problems

Auzinger and Stetter [1] combine a multigrid method with defect-correction iteration and derive a composite iterative procedure which they call the DCMG (defect-correction multigrid) cycle. Using a high-order discrete operator in the coarsegrid correction and a lower-order operator in relaxation, the DCMG cycle achieves the higher-order approximation [4]. In an analogous way, DCMG can be used to solve nonlinear PDEs by using the nonlinear operator in correction and a related linear operator in relaxation. We prove convergence of such a DCMG scheme and give an error estimation.     

Quasi-Newton methods for multiobjective optimization problems

This work is an attempt to develop multiobjective versions of some well-known single objective quasi-Newton methods, including BFGS, self-scaling BFGS (SS-BFGS), and the Huang BFGS (H-BFGS). A comprehensive and comparative study of these methods is presented in this paper. The Armijo line search is used for the implementation of these methods. The numerical results show that the Armijo rule does not work the same way for the multiobjective case as for the single objective case, because, in this case, it imposes a large computational effort and significantly decreases the speed of convergence in contrast to the single objective case. Hence, we consider two cases of all multi-objective versions of quasi-Newton methods: in the presence of the Armijo line search and in the absence of any line search. Moreover, the convergence of these methods without using any line search under some mild conditions is shown. Also, by introducing a multiobjective subproblem for finding the quasi-Newton multiobjective search direction, a simple representation of the Karush–Kuhn–Tucker conditions is derived. The H-BFGS quasi-Newton multiobjective optimization method provides a higher-order accuracy in approximating the second order curvature of the problem functions than the BFGS and SS-BFGS methods. Thus, this method has some benefits compared to the other methods as shown in the numerical results. All mentioned methods proposed in this paper are evaluated and compared with each other in different aspects. To do so, some well-known test problems and performance assessment criteria are employed. Moreover, these methods are compared with each other with regard to the expended CPU time, the number of iterations, and the number of function evaluations.     

Tensor product methods for stochastic problems

In the solution of stochastic partial differential equations (SPDEs) the generally already large dimension N of the algebraic system resulting from the spatial part of the problem is blown up by the huge number of degrees of freedom P coming from the stochastic part. The number of degrees of freedom of the full system will be NP, which poses severe demands on memory and processor time. We present a method how to approximate the system by a data-sparse tensor product (based on the Karhunen-Loève decomposition with M terms), which uses only memory in the order of M (N + P), and how to keep this representation also inside the iterative solvers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)