@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)},
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}
}
@article{GarloffAdmTiti,
author = {Garloff, J{\"u}rgen and Adm, Mohammad and Titi, Jihad},
title = {A Survey of Classes of Matrices Possessing the Interval Property and Related Properties},
series = {Konstanzer Schriften in Mathematik},
number = {344},
pages = {1 -- 14},
abstract = {This paper considers intervals of real matrices with respect to partial orders and the problem to infer from some exposed matrices lying on the boundary of such an interval that all real matrices taken from the interval possess a certain property. In many cases such a property requires that the chosen matrices have an identically signed inverse. We also briefly survey related problems, e.g., the invariance of matrix properties under entry-wise perturbations.},
language = {de}
}
@inproceedings{AdmGarloffTiti,
author = {Adm, Mohammad and Garloff, J{\"u}rgen and Titi, Jihad},
title = {Intervals of sign regular matrices},
series = {In: 8th Small Workshop in Interval Methods (SWIM 2015) June 9 - 11, 2015, Prague, ISBN: 978-80-7378-293-1 IUUK-CE-ITI series ; 2015-620},
pages = {1 -- 4},
language = {en}
}
@inproceedings{TitiGarloff,
author = {Titi, Jihad and Garloff, J{\"u}rgen},
title = {Matrix methods for the Bernstein form and their application in global optimization},
series = {8th Small Workshop in Interval Methods (SWIM 2015) June 9 - 11, 2015, Prague ; IUUK-CE-ITI series ; 2015-620},
pages = {113 -- 116},
language = {en}
}
@article{GarloffAdmTiti,
author = {Garloff, J{\"u}rgen and Adm, Mohammad and Titi, Jihad},
title = {A survey of classes of matrices possessing the interval property and related properties},
series = {Reliable computing},
volume = {2016},
number = {22},
publisher = {University of Louisiana at Lafayette},
address = {Lafayette, Louisiana},
pages = {1 -- 14},
abstract = {This paper considers intervals of real matrices with respect to partial orders and the problem to infer from some exposed matrices lying on the boundary of such an interval that all real matrices taken from the interval possess a certain property. In many cases such a property requires that the chosen matrices have an identically signed inverse. We also briefly survey related problems, e.g., the invariance of matrix properties under entry-wise perturbations.},
language = {en}
}
@inproceedings{GarloffTiti,
author = {Garloff, J{\"u}rgen and Titi, Jihad},
title = {Fast determination of the tensorial and simplicial Bernstein enclosure},
series = {SCAN 2016 ; 17th International Symposium on Scientific Computing, Computer Arithmetic and Verified Numerics - September, 26 - 29, 2016, Book of Abstracts},
pages = {51 -- 52},
language = {en}
}
@inproceedings{TitiGarloff,
author = {Titi, Jihad and Garloff, J{\"u}rgen},
title = {Enclosure of the Range of a Complex Polynomial Over a Complex Interval},
series = {SCAN 2018, The 18th International Symposium on Scientific Computing, Computer Arithmetic, and Verified Numerical Computations, 10 - 15 September 2018, Waseda University, Tokyo, Japan},
publisher = {Waseda University},
address = {Tokyo, Japan},
pages = {40 -- 41},
language = {en}
}
@inproceedings{TitiGarloff,
author = {Titi, Jihad and Garloff, J{\"u}rgen},
title = {Enclosure of the Range of a Complex Polynomial Over a Complex Interval},
series = {Sixth Palestinian Conference on Modern Trends in Mathematics and Physics (PCMTMP-VI), 5. - 8. August 2018, Palestine Technical University Kadoorei, Pal{\"a}stina},
pages = {1},
language = {en}
}
@article{TitiGarloff,
author = {Titi, Jihad and Garloff, J{\"u}rgen},
title = {Fast determination of the tensorial and simplicial Bernstein forms of multivariate polynomials and rational functions},
series = {Reliable Computing Journal},
number = {25},
pages = {24 -- 37},
abstract = {Tests for speeding up the determination of the Bernstein enclosure of the range of a multivariate polynomial and a rational function over a box and a simplex are presented. In the polynomial case, this enclosure is the interval spanned by the minimum and the maximum of the Bernstein coefficients which are the coefficients of the polynomial with respect to the tensorial or simplicial Bernstein basis. The methods exploit monotonicity properties of the Bernstein coefficients of monomials as well as a recently developed matrix method for the computation of the Bernstein coefficients of a polynomial over a box.},
language = {en}
}
@article{TitiGarloff,
author = {Titi, Jihad and Garloff, J{\"u}rgen},
title = {Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials},
series = {Applied Mathematics and Computation},
number = {315},
pages = {246 -- 256},
abstract = {In this paper, multivariate polynomials in the Bernstein basis over a simplex (simplicial Bernstein representation) are considered. Two matrix methods for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented. Also matrix methods for the calculation of the Bernstein coefficients over subsimplices generated by subdivision of the standard simplex are proposed and compared with the use of the de Casteljau algorithm. The evaluation of a multivariate polynomial in the power and in the Bernstein basis is considered as well. All the methods solely use matrix operations such as multiplication, transposition, and reshaping; some of them rely also on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. The latter one enables the use of the Fast Fourier Transform hereby reducing the amount of arithmetic operations.},
language = {en}
}
@inproceedings{TitiGarloff,
author = {Titi, Jihad and Garloff, J{\"u}rgen},
title = {Efficient methods for computation of the simplicial Bernstein coefficients},
series = {Pacific Institute for the Mathematical Sciences Young Researchers Conference (PIMS YRC 2017), University of Saskatchewan, Saskatoon, Canada, 2017, June 5-8},
pages = {12 -- 12},
language = {en}
}
@inproceedings{TitiGarloff,
author = {Titi, Jihad and Garloff, J{\"u}rgen},
title = {Efficient methods for computation of tensorial Bernstein coefficients},
series = {The Prairie Discrete Math Workshop (PDMW 2017), 2-5 June, 2017, Living Skies Conference Centre Lumsden, Saskatchewan, Canada},
pages = {9 -- 9},
language = {en}
}
@misc{TitiGarloff,
author = {Titi, Jihad and Garloff, J{\"u}rgen},
title = {Fast determination of the tensorial and simplicial Bernstein enclosure},
series = {Meeting of the International Linear Algebra Society (ILAS 2017: Connections), Ames, Iowa State University, July 24-28 2017},
pages = {92 -- 93},
abstract = {Contributed Talks},
language = {en}
}
@article{TitiGarloff,
author = {Titi, Jihad and Garloff, J{\"u}rgen},
title = {Matrix methods for the tensorial Bernstein form},
series = {Applied Mathematics and Computation},
volume = {346},
pages = {254 -- 271},
abstract = {In this paper, multivariate polynomials in the Bernstein basis over a box (tensorial Bernstein representation) are considered. A new matrix method for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, is presented and compared with existing methods. Also matrix methods for the calculation of the Bernstein coefficients over subboxes generated by subdivision of the original box are proposed. All the methods solely use matrix operations such as multiplication, transposition and reshaping; some of them rely on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. In the case that the coefficients of the polynomial are due to uncertainties and can be represented in the form of intervals it is shown that the developed methods can be extended to compute the set of the Bernstein coefficients of all members of the polynomial family.},
language = {en}
}
@phdthesis{Titi,
author = {Titi, Jihad},
title = {Matrix methods for the tensorial and simplicial Bernstein forms with application to global optimization},
language = {en}
}
@inproceedings{GarloffTiti,
author = {Garloff, J{\"u}rgen and Titi, Jihad},
title = {Bounds for the range of a complex polynomial over a rectangular region},
series = {Book of Abstracts of the 3rd International Conference and Summer School 'Numerical Computations: Theory and Algorithms', June 15-21, 2019, Crotone, Italy},
publisher = {University of Calabria},
address = {Rende, Italy},
pages = {185 -- 185},
abstract = {Abstract},
language = {en}
}
@inproceedings{TitiGarloff,
author = {Titi, Jihad and Garloff, J{\"u}rgen},
title = {Symbolic-numeric computation of the Bernstein coefficients of a polynomial from those of one of its partial derivatives and of the product of two polynomials},
series = {Computer Algebra in Scientific Computing, 22nd International Workshop, CASC 2020, September 14-18, 2020, Linz, Austria, Proceedings},
publisher = {Springer},
address = {Cham},
pages = {583 -- 599},
abstract = {The expansion of a given multivariate polynomial into Bernstein polynomials is considered. Matrix methods for the calculation of the Bernstein expansion of the product of two polynomials and of the Bernstein expansion of a polynomial from the expansion of one of its partial derivatives are provided which allow also a symbolic computation.},
language = {en}
}