On the L2 improperly posed problems

This paper considers a problem of approximation of functions proposed by Bellman [1]. The results include a representation lemma for the solutions and an algorithm for computing such solutions. A sufficient condition for the convergence of the algorithm to the optimal solutions is shown to be related to the uniqueness of solutions of a pair of functional equations. To the author's knowledge, not much is known about these functional equations and these may prove to be of interest in future research. An example is included to illustrate the algorithm.     

On the approximate solution of time-optimal control problems

In this paper we are concerned with the approximate solution of time-optimal control problems in a nonreflexive Banach SpaceE by sequences of similar problems in Banach spacesE _n which are assumed to approximateE in a fairly general sense. The problems under consideration are such that the solution operator of the associated evolution equation is a strongly continuous holomorphic contraction semigroup and the class of controls is taken from the dual of the Phillips adjoint space with respect to the infinitesimal generator of that semigroup. The main object is to establish convergence of optimal controls, transition times and corresponding trajectories of the approximating control problems which can be done by means of some results from the theory of approximation of semigroups of operators. Finally, these abstract convergence results will be applied to time-optimal control problems arising from heat transfer and diffusion processes.Research supported in part by the Deutsche Forschungsgemeinschaft (DFG)     

Ritz method for approximate solution of optimization problems

The article investigates convergence of the minimum points of special classes of functionals defined on reflexive Banach spaces.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 171–178, 2002.     

Iterative solution of elliptic problems by approximate factorization

An iterative method for the numerical solution of singularly perturbed second-order linear elliptic problems is presented. It is a defect correction iteration in which the approximate operator is the product of two first-order operators, which is readily inverted numerically. The approximate operator is generated by formal asymptotic factorization of the original operator. Hence this is a QUasi Analytic Defect correction iteration (QUAD). Both its continuous and discrete versions are analyzed in one dimension. The scheme is extended to a variety of two dimensional operators and it is analyzed for a model advection-diffusion equation. Numerical calculations show the effectiveness of the scheme over a wide range of values of the small parameter.     

On the optimization of projection-iterative methods for the approximate solution of ill-posed problems

We consider a new version of the projection-iterative method for the solution of operator equations of the first kind. We show that it is more economical in the sense of amount of used discrete information.     

On singular perturbation in non well posed problems

Summary Let L_εu and L ₀ v be the elliptic and “backward” heat operators defined by(1.1) and(1.2), respectively. The following question is considered for a pair of “non-well posed” initial-boundary value problems for L_ε and L ₀ : if u and v are the respective solutions, under what restrictions on the classes of admissible solutions and in what sense, if any, does u converge to v as ɛ →0? This research was supported in part by the National Science Foundation Grant No. GP 5882 with Cornell University.     

Continuity of approximate solution mappings for parametric equilibrium problems

In this paper, we obtain sufficient conditions for Hausdorff continuity and Berge continuity of an approximate solution mapping for a parametric scalar equilibrium problem. By using a scalarization method, we also discuss the Berge lower semicontinuity and Berge continuity of a approximate solution mapping for a parametric vector equilibrium problem.     

On the approximate solution of operator equations

Numerische Mathematik - Prolongations and restrictions are used to derive error estimates when linear operator equations of the second kind are solved by discretisation methods.     

On approximate solutions in set-valued optimization problems

In this paper we introduce several concepts of approximate solutions of set-valued optimization problems with vector and set optimization. We prove existence results and necessary and sufficient conditions by using limit sets.