TY - JOUR
U1 - Zeitschriftenartikel, wissenschaftlich - begutachtet (reviewed)
A1 - Titi, Jihad
A1 - Hamadneh, Tareq
A1 - Garloff, Jürgen
T1 - Convergence of the Simplicial Rational Bernstein Form
JF - Modelling, Computation and Optimization in Information Systems and Management Sciences
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 359)
N2 - 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.
KW - Bernstein polynomial
KW - simplex
KW - range bounds
KW - rational function
KW - degree elevation
Y1 - 2015
UR - http://www-home.htwg-konstanz.de/%7Egarloff/Paper95_MCO2015_Titi_Hamadneh_Garloff(1).pdf
U6 - https://dx.doi.org/10.1007/978-3-319-18161-5
DO - https://dx.doi.org/10.1007/978-3-319-18161-5
SP - 433
EP - 441
ER -
TY - JOUR
U1 - Zeitschriftenartikel, wissenschaftlich - begutachtet (reviewed)
A1 - Garloff, Jürgen
A1 - Adm, Mohammad
A1 - Titi, Jihad
T1 - A Survey of Classes of Matrices Possessing the Interval Property and Related Properties
JF - Konstanzer Schriften in Mathematik
N2 - 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.
KW - matrix interval
KW - vertex matrix
KW - entry-wise perturbation
Y1 - 2015
U6 - http://nbn-resolving.de/urn/resolver.pl?http://nbn-resolving.de/urn:nbn:de:bsz:352-0-310468
UN - http://nbn-resolving.de/urn/resolver.pl?http://nbn-resolving.de/urn:nbn:de:bsz:352-0-310468
SN - 1430-3558
SS - 1430-3558
IS - 344
SP - 1
EP - 14
ER -
TY - CHAP
U1 - Konferenzveröffentlichung
A1 - Adm, Mohammad
A1 - Garloff, Jürgen
A1 - Titi, Jihad
T1 - Intervals of sign regular matrices
T2 - 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
KW - Interval matrix
KW - Checkerboard ordering
KW - Totally nonnegative matrix
KW - Sign regular matrix
Y1 - 2015
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bsz:kon4-opus4-6979
UN - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bsz:kon4-opus4-6979
UR - https://iti.mff.cuni.cz/series/2015.html
SP - 1
EP - 4
ER -
TY - CHAP
U1 - Konferenzveröffentlichung
A1 - Titi, Jihad
A1 - Garloff, Jürgen
T1 - Matrix methods for the Bernstein form and their application in global optimization
T2 - 8th Small Workshop in Interval Methods (SWIM 2015) June 9 - 11, 2015, Prague ; IUUK-CE-ITI series ; 2015-620
KW - Bernstein polynomials
KW - Bernstein coefficients
KW - Range enclosure
KW - Subdivision
KW - Convex optimization
Y1 - 2015
UR - iti.mff.cuni.cz/series/2015/620.pdf
SN - 978-80-7378-293-1
SB - 978-80-7378-293-1
SP - 113
EP - 116
ER -
TY - JOUR
U1 - Zeitschriftenartikel, wissenschaftlich - begutachtet (reviewed)
A1 - Garloff, Jürgen
A1 - Adm, Mohammad
A1 - Titi, Jihad
T1 - A survey of classes of matrices possessing the interval property and related properties
JF - Reliable computing
N2 - 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.
KW - Matrix interval
KW - Vertex matrix
KW - Entry-wise perturbation
Y1 - 2016
U6 - http://nbn-resolving.de/urn/resolver.pl?http://nbn-resolving.de/urn:nbn:de:bsz:352-0-310051
UN - http://nbn-resolving.de/urn/resolver.pl?http://nbn-resolving.de/urn:nbn:de:bsz:352-0-310051
SN - 1385-3139
SS - 1385-3139
VL - 2016
IS - 22
SP - 1
EP - 14
PB - University of Louisiana at Lafayette
CY - Lafayette, Louisiana
ER -
TY - CHAP
U1 - Konferenzveröffentlichung
A1 - Garloff, Jürgen
A1 - Titi, Jihad
T1 - Fast determination of the tensorial and simplicial Bernstein enclosure
T2 - SCAN 2016 ; 17th International Symposium on Scientific Computing, Computer Arithmetic and Verified Numerics - September, 26 - 29, 2016, Book of Abstracts
Y1 - 2016
UR - www.math.uu.se/digitalAssets/635/635446_scan2016-book-of-abstracts3.pdf
SP - 51
EP - 52
ER -
TY - CHAP
U1 - Konferenzveröffentlichung
A1 - Titi, Jihad
A1 - Garloff, Jürgen
T1 - Enclosure of the Range of a Complex Polynomial Over a Complex Interval
T2 - SCAN 2018, The 18th International Symposium on Scientific Computing, Computer Arithmetic, and Verified Numerical Computations, 10 - 15 September 2018, Waseda University, Tokyo, Japan
KW - Complex interval
KW - Complex polynomial
KW - Enclosure of the range
KW - Bernstein polynomial
KW - Multivariate complex polynomial
Y1 - 2018
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bsz:kon4-opus4-14627
UN - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bsz:kon4-opus4-14627
SP - 40
EP - 41
S1 - 2
PB - Waseda University
CY - Tokyo, Japan
ER -
TY - CHAP
U1 - Konferenzveröffentlichung
A1 - Titi, Jihad
A1 - Garloff, Jürgen
T1 - Enclosure of the Range of a Complex Polynomial Over a Complex Interval
T2 - Sixth Palestinian Conference on Modern Trends in Mathematics and Physics (PCMTMP-VI), 5. - 8. August 2018, Palestine Technical University Kadoorei, Palästina
KW - Range enclosure
KW - Complex polynomial
KW - Complex interval
KW - Bernstein polynomial
Y1 - 2018
SP - 1
S1 - 1
ER -
TY - JOUR
U1 - Zeitschriftenartikel, wissenschaftlich - begutachtet (reviewed)
A1 - Titi, Jihad
A1 - Garloff, Jürgen
T1 - Fast determination of the tensorial and simplicial Bernstein forms of multivariate polynomials and rational functions
JF - Reliable Computing Journal
N2 - 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.
KW - Multivariate polynomial
KW - Multivariate rational function
KW - Bernstein coefficient
KW - Tensorial Bernstein form
KW - Simplicial Bernstein form
Y1 - 2017
UR - http://nbn-resolving.de/urn:nbn:de:bsz:352-0-410198
SN - 1573-1340
SS - 1573-1340
IS - 25
SP - 24
EP - 37
ER -
TY - JOUR
U1 - Zeitschriftenartikel, wissenschaftlich - begutachtet (reviewed)
A1 - Titi, Jihad
A1 - Garloff, Jürgen
T1 - Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials
JF - Applied Mathematics and Computation
N2 - 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.
KW - Bernstein coefficient
KW - Simplicial Bernstein representation
KW - Range enclosure
KW - Simplicial subdivision
KW - Polynomial evaluation
Y1 - 2017
SN - 0096-3003
SS - 0096-3003
U6 - https://dx.doi.org/10.1016/j.amc.2017.07.026
DO - https://dx.doi.org/10.1016/j.amc.2017.07.026
N1 - Volltextzugriff für Angehörige der Hochschule Konstanz möglich
IS - 315
SP - 246
EP - 256
ER -
TY - CHAP
U1 - Konferenzveröffentlichung
A1 - Titi, Jihad
A1 - Garloff, Jürgen
T1 - Efficient methods for computation of the simplicial Bernstein coefficients
T2 - Pacific Institute for the Mathematical Sciences Young Researchers Conference (PIMS YRC 2017), University of Saskatchewan, Saskatoon, Canada, 2017, June 5-8
Y1 - 2017
UR - https://math.usask.ca/pimsyrc2017/programme.pdf
SP - 12
EP - 12
ER -
TY - CHAP
U1 - Konferenzveröffentlichung
A1 - Titi, Jihad
A1 - Garloff, Jürgen
T1 - Efficient methods for computation of tensorial Bernstein coefficients
T2 - The Prairie Discrete Math Workshop (PDMW 2017), 2–5 June, 2017, Living Skies Conference Centre Lumsden, Saskatchewan, Canada
Y1 - 2017
SP - 9
EP - 9
ER -
TY - GEN
U1 - Sonstiges
A1 - Titi, Jihad
A1 - Garloff, Jürgen
T1 - Fast determination of the tensorial and simplicial Bernstein enclosure
T2 - Meeting of the International Linear Algebra Society (ILAS 2017: Connections), Ames, Iowa State University, July 24-28 2017
N2 - Contributed Talks
Y1 - 2017
SP - 92
EP - 93
ER -
TY - JOUR
U1 - Zeitschriftenartikel, wissenschaftlich - begutachtet (reviewed)
A1 - Titi, Jihad
A1 - Garloff, Jürgen
T1 - Matrix methods for the tensorial Bernstein form
JF - Applied Mathematics and Computation
N2 - 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.
KW - Bernstein coefficient
KW - Tensorial Bernstein form
KW - Range enclosure
KW - Subdivision
KW - Interval polynomial
Y1 - 2019
SN - 0096-3003
SS - 0096-3003
U6 - https://dx.doi.org/10.1016/j.amc.2018.08.049
DO - https://dx.doi.org/10.1016/j.amc.2018.08.049
VL - 346
SP - 254
EP - 271
ER -
TY - THES
U1 - Dissertation / Habilitation
A1 - Titi, Jihad
T1 - Matrix methods for the tensorial and simplicial Bernstein forms with application to global optimization
Y2 - 2019
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bsz:352-2-k106crqmste71
UN - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bsz:352-2-k106crqmste71
ER -
TY - CHAP
U1 - Konferenzveröffentlichung
A1 - Garloff, Jürgen
A1 - Titi, Jihad
T1 - Bounds for the range of a complex polynomial over a rectangular region
T2 - Book of Abstracts of the 3rd International Conference and Summer School 'Numerical Computations: Theory and Algorithms', June 15-21, 2019, Crotone, Italy
N2 - Abstract
Y1 - 2019
UR - si.dimes.unical.it/~yaro/numta2019/pdf/Book_NUMTA2019.pdf
SN - 9788874581016
SB - 9788874581016
SP - 185
EP - 185
S1 - 1
PB - University of Calabria
CY - Rende, Italy
ER -
TY - CHAP
U1 - Konferenzveröffentlichung
A1 - Titi, Jihad
A1 - Garloff, Jürgen
T1 - 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
T2 - Computer Algebra in Scientific Computing, 22nd International Workshop, CASC 2020, September 14–18, 2020, Linz, Austria, Proceedings
N2 - 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.
KW - Multivariate polynomial
KW - Bernstein polynomial
KW - Bernstein coefficient
Y1 - 2020
SN - 978-3-030-60026-6
SB - 978-3-030-60026-6
U6 - https://dx.doi.org/10.1007/978-3-030-60026-6_34
DO - https://dx.doi.org/10.1007/978-3-030-60026-6_34
N1 - Volltextzugriff für Angehörige im Campusnetz der Hochschule Konstanz möglich.
SP - 583
EP - 599
PB - Springer
CY - Cham
ER -