@article{TitiHamadnehGarloff,
author = {Titi, Jihad and Hamadneh, Tareq and Garloff, J{\"u}rgen},
title = {Convergence of the Simplicial Rational Bernstein Form},
series = {Modelling, Computation and Optimization in Information Systems and Management Sciences Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 359)},
doi = {10.1007/978-3-319-18161-5},
pages = {433 -- 441},
abstract = {Bernstein polynomials on a simplex V are considered. The expansion of a given polynomial p into these polynomials provides bounds for range of p over V. Bounds for the range of a rational function over V can easily be obtained from the Bernstein expansions of the numerator and denominator polynomials of this function. In this paper it is shown that these bounds converge monotonically and linearly to the range of the rational function if the degree of the Bernstein expansion is elevated. If V is subdivided then the convergence is quadratic with respect to the maximum of the diameters of the subsimplices.},
language = {en}
}
@inproceedings{GarloffHamadneh,
author = {Garloff, J{\"u}rgen and Hamadneh, Tareq},
title = {Convergence and inclusion isotonicity of the tensorial rational Bernstein form},
series = {Scientific Computing, Computer Arithmetic, and Validated Numerics, 16th International Symposium, SCAN 2014, W{\"u}rzburg, Germany, September 21-26, 2014, (Lecture Notes in Computer Science ; 9553)},
publisher = {Springer},
address = {Cham},
isbn = {978-3-319-31768-7},
doi = {10.1007/978-3-319-31769-4},
pages = {171 -- 179},
abstract = {A method is investigated by which tight bounds on the range of a multivariate rational function over a box can be computed. The approach relies on the expansion of the numerator and denominator polynomials in Bernstein polynomials. Convergence of the bounds to the range with respect to degree elevation of the Bernstein expansion, to the width of the box and to subdivision are proven and the inclusion isotonicity of the related enclosure function is shown.},
language = {en}
}
@phdthesis{Hamadneh,
author = {Hamadneh, Tareq},
title = {Bounding Polynomials and Rational Functions in the Tensorial and Simplicial Bernstein Forms},
pages = {153},
abstract = {This thesis considers bounding functions for multivariate polynomials and rational functions over boxes and simplices. It also considers the synthesis of polynomial Lyapunov functions for obtaining the stability of control systems. Bounding the range of functions is an important issue in many areas of mathematics and its applications like global optimization, computer aided geometric design, robust control etc.},
language = {en}
}