Error estimates for approximate dynamic systems

Two approaches are investigated for obtaining estimates on the error between approximate and exact solutions of dynamic systems. The first method is primarily useful if the system is non-linear and of low dimension. The second requires construction of a system of u-functions but is useful for higher dimensional systems, either linear or non-linear.     

Error estimates for conforming finite element solutions

A method for determining realistic error estimates for conforming finite element solutions is presented. The method requires solution of the problem by at least two, and preferably three mesh schemes that yield monotonic solution covergence. This in turn will automatically yield one solution bound, upper or lower. The paper describes a simple and practical scheme for obtaining the other bound by utilizing the solutions from the multiple mesh schemes. These bounds bracket the exact solution within relatively narrow limits and provide the basis of the error estimate. The solution quantities considered are the system energy quantities; and for eigenvalue problems these correspond to the eigenvalues themselves. As in convergence proofs, it is expected that the displacement and stress quantities will follow the behavior of the energy quantities. The proposed bounding method is applicable to eigenvalue and static problems devoid of stress singularities, and considers only the discretization error of conforming finite element models. The validity of the proposed bounding method has not been proved mathematically; however, extensive numerical applications of the method indicate its workability in every case tested. Results of some applications are included in this article.     

Numerical fixed grid methods for differential inclusions

Summary Numerical methods for initial value problems for differential inclusions usually require a discretization of time as well as of the set valued right hand side. In this paper, two numerical fixed grid methods for the approximation of the full solution set are proposed and analyzed. Convergence results are proved which show the combined influence of time and (phase) space discretization. Supported by CRC 701 “Spectral Analysis and Topological Methods in Mathematics”.     

Error estimates for least squares finite element methods

A least squares finite element scheme for a boundary value problem associated with a second-order partial differential equation is considered. Previous work on this subject is generalized and improved by considering a larger class of equations, by working in the natural context, without additional smoothness conditions, and by deriving error estimates, not only in the H¹-norm and the H_div-norm, but also in the L₂-norm. Some of these estimates are sharpened by using finite element spaces with the grid decomposition property. The error estimates are supported by numerical results which extend previous numerical work.     

Straightening out rectangular differential inclusions

In this paper, the classic straightening out theorem from differential geometry is used to derive necessary and sufficient conditions for locally converting rectangular differential inclusions to constant rectangular differential inclusions. Both scalar and coupled differential inclusions are considered. The results presented in this paper have use in the area of computer aided verification of hybrid systems where they represent the frontier of the known decidable models of infinite state systems.     

Fuzzy differential inclusions as substitutes for stochastic differential equations in population biology

The familiar procedure of adding white noise to deterministic systems of equations may not be appropriate or even possible in some modelling problems arising in the biological sciences. Although some mathematical handle on indeterminant factors (i.e. “noise”) may be necessary, sometimes the probabilistic requirements involved cannot be rigorously verified for the data set in hand. As an alternative, we discuss here the modelling utility of fuzzy differential inclusions associated with given systems of nonlinear ODE's. We give concrete examples and give account of the conservative stochastic mechanics of E. Nelson applied to growth of a dimorphic clone, and its fuzzy differential inclusion analogue.     

Error estimates for iterative algorithms for minimizing regularized quadratic subproblems

ABSTRACT

We derive bounds for the objective errors and gradient residuals when finding approximations to the solution of common regularized quadratic optimization problems within evolving Krylov spaces. These provide upper bounds on the number of iterations required to achieve a given stated accuracy. We illustrate the quality of our bounds on given test examples.     

Error estimates for Gauss-Chebyshev and Clenshaw Curtis quadrature formulas

For functionsf analytic in a circle we obtain bounds for the Chebyshev-Fourier coefficients off. These results are then used to obtain bounds for the errors of Gauss-Chebyshev quadratures, and the quadrature formula of Clenshaw and Curtis. Two examples are given to illustrate the error bounds.     

Time-dependent differential inclusions, cocycle attractors andfuzzy differential equations

Traditional formulations of fuzzy differential equations do not reproduce the rich and varied behavior of crisp differential equations (DEs). A recent interpretation in terms of differential inclusions, expressed level setwise, overcomes this deficiency and opens up for profitable investigation such properties as stability, attraction, periodicity, and the like. This is especially important for investigating continuous systems which are uncertain or incompletely specified. This paper studies attractors of fuzzy DEs in terms of cocycles and encompasses both the time-dependent and autonomous cases     

Approximation of the optimal-time problem for controlled differential inclusions

Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 172–176, May–June, 1994     

Existence results for impulsive differential inclusions with nonlocal conditions

In this paper, we shall establish sufficient conditions for the existence of mild solutions for nonlocal impulsive differential inclusions. On the basis of the fixed point theorems for multivalued maps and the technique of approximate solutions, new results are obtained. Examples are also provided to illustrate our results.     

Existence of monotone and slow solutions for differential inclusions

The existence of slow and monotone solutions with respect to a continuous pre-order is proved for differential inclusions defined on a closed convex subset of R⁴. First we examine the ‘projected system’, and prove that it is equivalent to a differential variational inequality. We then establish the existence of monotone trajectories. Subsequently, we prove the existence of slow monotone solutions for a class of differential inclusions, satisfying a Nagumo-type condition. Finally we prove a convergence result.     

Conjugate convex Lyapunov functions for dual linear differential inclusions

Tools from convex analysis are used to show how stability properties and Lyapunov inequalities translate when passing from a linear differential inclusion (LDI) to its dual. In particular, it is proved that a convex, positive definite function is a Lyapunov function for an LDI if and only if its convex conjugate is a Lyapunov function for the LDIs dual. Examples show how such duality effectively doubles the number of tools available for assessing stability of LDIs.     

Lagrange-type extremal trajectories in differential inclusions

Necessary conditions for optimal, or boundary solutions of differential inclusions usually state that such solutions are extremal in some sense. There are several possible concepts of extremality, which lead to different, often difficult to compare necessary conditions. In this paper, we give a complete comparison of three classes of extremal trajectories: two different Lagrange-type extremals, and Hamiltonian extremals. In the second part, we consider a nonconvex differential inclusion, and prove that every boundary trajectory is a Lagrangian extremal.