Moduli of smoothness of vector valued functions of a real variable and applications

Farkhi

2007

Set-Valued Anal

A family of probability measures on the unit ball in R n generates a family of generalized Steiner (GS-)points for every convex compact set in R n . Such a "rich" family of probability measures determines a representation of a convex compact set by GS-points. In this way, a representation of a set-valued map with convex compact images is constructed by GS-selections (which are defined by the GS-points of its images). The properties of the GS-points allow to represent Minkowski sum, Demyanov difference and Demyanov distance between sets in terms of their GSpoints, as well as the Aumann integral of a set-valued map is represented by the integrals of its GS-selections. Regularity properties of set-valued maps (measurability, Lipschitz continuity, bounded variation) are reduced to the corresponding uniform properties of its GS-selections. This theory is applied to formulate regularity conditions for the first-order of convergence of iterated set-valued quadrature formulae approximating the Aumann integral.

Section: Approximate Set-valued Integrationmentioning

confidence: 51%

“…Thanks to these properties, it is successfully applied to set-valued numerical approximation and integration (see e.g. [5,8,20,21]) .…”

Section: Introductionmentioning

confidence: 99%

Regularity and Integration of Set-Valued Maps Represented by Generalized Steiner Points

Farkhi

2007

Set-Valued Anal

“…[10, Chapter IX]) involved in the definition of M(·|Θ) to establish the statement. In [14,17] a different idea for the proofs has been pursued instead. Basically, the scalarisation, through means of functionals, of the functions taking their values in Banach spaces allows to apply well-known results for real-valued functions; finally, the separation of points by functionals is exploited to finish the proofs.…”

Section: Proposition 5 Letmentioning

confidence: 99%

“…Some of the results achieved in [36,14,34,17,39] for polynomial interpolation in Banach spaces can be applied, since − → D n is itself a Banach space. These pioneering works were aimed at more theoretical results, whereas here we focus on the numerical analysis; in fact, error estimates are not provided in [36] or demand in [34,17,39] too much regularity.…”

Section: Introductionmentioning

confidence: 97%

“…These pioneering works were aimed at more theoretical results, whereas here we focus on the numerical analysis; in fact, error estimates are not provided in [36] or demand in [34,17,39] too much regularity. We shall point out that, although the regularity assumptions in [14] are rather weak, the conditions (i), (ii), H1 in [14] for deriving error estimates (in − → D n ) still demand research. Furthermore, no numerical results of set-valued interpolation are visualised in these works unlike in [26,33].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Set-valued Hermite interpolation

Journal of Approximation Theory

Perria

2011

The problem of interpolating a set-valued function with convex images is addressed by means of directed sets. A directed set will be visualised as a usually non-convex set in R n consisting of three parts together with its normal directions: the convex, the concave and the mixed-type part. In the Banach space of the directed sets, a mapping resembling the Kergin map is established. The interpolating property and error estimates similar to the point-wise case are then shown; the representation of the interpolant through means of divided differences is given. A comparison to other set-valued approaches is presented. The method developed within the article is extended to the scope of the Hermite interpolation by using the derivative notion in the Banach space of directed sets. Finally, a numerical analysis of the explained technique corroborates the theoretical results.

Selection Strategies for Set-Valued Runge-Kutta Methods

Lecture Notes in Computer Science

2005

A general framework for proving an order of convergence for set-valued Runge Kutta methods is given in the case of linear differential inclusions, if the attainable set at a given time should be approximated. The set-valued method is interpreted as a (set-valued) quadrature method with disturbed values for the fundamental solution at the nodes of the quadrature method. If the precision of the quadrature method and the order of the disturbances fit together, then an overall order of convergence could be guaranteed. The results are applied to modified Euler method to emphasize the dependence on a suitable selection strategy (one strategy leads to an order breakdown).