## Jürgen Garloff |

- Prof. Dr. Jürgen Garloff
- Institut für Angewandte Forschung
- Hochschule für Technik, Wirtschaft und Gestaltung Konstanz / University of Applied Sciences
- Postfach 100543
- D-78405 Konstanz
- Germany
- Phone: +49-7531-206-406 (office)
- +49-7533-934975 (home)
- Fax: +49-7531-206-559
- email: garloff at htwg-konstanz dot de
- URL: http://www-home.htwg-konstanz.de/~garloff/

Jürgen Garloff is a Professor emeritus for Mathematics at the Institute for
Applied Research of the Hochschule für Technik, Wirtschaft und Gestaltung -
University of Applied Sciences in Constance, Germany. He studied sociology, philosophy,
economics, business administration, and mathematics at the Universities of Cologne and
Heidelberg, from where he received his diploma in mathematics under the supervision of
Prof. Romberg. 1976-87 he worked as a research associate at the Institute for Applied
Mathematics of the University of Freiburg i. Br.; there he received his doctorate in
mathematics and the Dr. habil. degree. 1987-89 he held an appointment as research
scientist for computational fluid dynamics in industry. 1989-90 he was a professor for
mathematics and computer science at the University of Applied Sciences in Esslingen. He
is also a professor (apl.) at the Department of Mathematics and Statistics at the
University of Constance. He serves on the editorial boards of *Reliable
Computing*, *International Journal of Nonlinear Science (IJNS)* and
*International Journal of Polynomials and Applications*, on the advisory board of
*International Journal of Fuzzy Computation and Modeling* and is a member of the
International Linear Algebra Society (ILAS).

Literaturübersicht
zu den Vorlesungen MAT1 und MAT2 (MS Word file) *(version Mar 2007)*

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Auszug für AIN2:

Download latest
version as PDF file *(version Nov 2011)*

Auszug für AIT2:

Download latest
version as PDF file *(version Jan 2013)*

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AIT:

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*Global Optimization:*use of a new relaxation technique for multivariate polynomials.*Robust Control:*robust stability with emphasis on nonlinear parametric dependencies, development of software.*Matrix Analysis:*focusing on total positivity.*Polynomials*with emphasis on the zero distribution of the Hadamard (i.e. coefficientwise) product of polynomials.*Scientific Computing with Result Verification:*Solution of systems of linear equations with not sharply defined coefficients; solution of systems of algebraic equations; tight enclosures for the range and graph of polynomials; interpolation of interval-valued data for algebraic and trigonometric polynomials, splines, and exponential sums.

The project aims to treat the following problems:

1. Risk analysis of existing buildings. In the structural re-analysis of existing buildings, material values and geometric parameters are often only known to lay within certain bounds. The regions specified by these bounds can be taken into account if methods from interval computations are employed. The resulting enclosing intervals provide the engineer with insight into the behaviour of the structure and the present safety margin.

2. Quality assurance of the obtained numerical results by rigorous treatment of all rounding and discretization errors when the finite element method is used. Often it is believed that rounding errors have a negligible influence on the results of the computation. The same applies to the discretization error by which the approximate presentation of the stiffness matrices of beams with respect to theory of second order effects are effected. However, rounding errors can cause disastrous results, in particular when the systems of linear equations to be solved are ill-conditioned, caused by, e.g., large differences in the stiffness parameters of the system. In the presence of different orders of magnitude, often the influence of the discretization error is also non-negligible. It is intended that the use of interval arithmetic may help to cover all rounding and discretization errors so that the results of the finite element computations can be guaranteed. As a result, the risk of damage to buildings or even the probability of failure of structures can be minimized. See papers nos. 54, 55, 58-60, 66.

Project Partners:

- Prof. Rafi L. Muhanna, PhD, Director of the Center for Reliable Engineering and Computing at Georgia Tech, Savannah, USA

- Prof. Dr. Evgenija D. Popova, Bulgarian Academy of Sciences, Institute of Mathematics and Computer Science, Sofia, Bulgaria

- Sofistik AG in 85764 Oberschleissheim, Germany

- Prof. Dr.-Ing. Horst Werkle, HTWG Konstanz, Dean of the Faculty for Civil Engineering, Konstanz, Germany

Funded by the State of Baden-Württemberg.

Convex lower bound functions for multivariate polynomials can be constructed in a
natural way if we expand the given polynomial *p* into Bernstein polynomials. The
coefficients of this expansion, the so-called Bernstein coefficients, can easily be
computed from the coefficients of *p*. A fundamental property of the Bernstein
expansion is its *convex hull property* which states that the graph of *p*
over a box is contained in the convex hull of the control points associated with its
Bernstein coefficients. Based on this property, convex lower bound functions of
increasing complexity can be constructed. We use these underestimating functions for a
relaxation of the given constrained global optimization problem.

We concentrate on the case in which the objective function and the functions defining the constraints are multivariate polynomials, which we bound from below by affine functions. If we solve the global optimization problem, by substituting these affine underestimating functions, then we obtain a lower bound for the global minimum value. This approach is integrated into a branch and bound framework.

In the case of general functions, we use Taylor expansion and enclose the remainder in an interval resulting again in affine lower bound functions. Due to rounding errors the branch and bound algorithm may compute lower and upper bounds for the global minimum points which do not enclose all of these points. But with the tools of interval arithmetic it is possible to compute rigorous results. See paper nos. 43-45, 47, 48, 51, 52, 62, 65; for software and recent results for sparse polynomials see here.

Funded by the German Research Council (DFG).

1. Optimal Bounds for Interval Interpolation with Polynomials and Functions

2. Investigations on Interval Interpolation (Dissertation), *Freiburger
Intervall-Berichte* __80/5__ (1980) (in German)

3. On the Interval Performance of the Fast Fourier Transformation, *Z. Angew.
Math. Mech.* __60__, T291-T292 (1980) (in German)

4. Totally Nonnegative Interval Matrices, in 'Interval Mathematics 1980', K. Nickel,
Ed., Academic Press*,* New York,London,Toronto, pp. 317-327 (1980)

5. Criteria for Sign Regularity of Sets of Matrices, *Linear Algebra and its
Applications* __44__, 153-160 (1982)

6. Majorization between the Diagonal Elements and the Eigenvalues of an Oscillating
Matrix, *Linear Algebra and its Applications* __47__, 181-184 (1982)

7. Intervals of P-Matrices and Related Matrices, *Linear Algebra and its
Applications* __58__, 33-41 (1984) (with S. Bialas)

8. An Inverse Eigenvalue Problem for Totally Nonnegative Matrices, *Linear and
Multilinear Algebra* __17__, 19-23 (1985)

9.
Convex Combinations of Stable Polynomials, *Journal of the Franklin
Institute* __319__, 373-377 (1985) (with S. Bialas)

10. Stability of Polynomials under Coefficient Perturbation, *IEEE Transactions on
Automatic Control* __AC-30__, 310-313 (1985) (with S. Bialas)

11. On Power-Boundedness of Interval Matrices, *J. of Computational and Applied
Mathematics* __14__, 353-360 (1986)

12. Bounds for the Eigenvalues of the Solution of the Discrete Riccati and Lyapunov
Equations and the Continuous Lyapunov Equation, *International Journal of Control*
__43__, 423-431 (1986)

13. Convergent Bounds
for the Range of Multivariate Polynomials, in 'Interval Mathematics 1985', K.
Nickel, Ed., *Lecture Notes in Computer Science,* vol. 212, pp. 37-56, Springer,
Berlin, Heidelberg, New York (1986)

14. Optimal Inclusion of a Solution Set, *SIAM Journal of Numerical Analysis*
__23__, 217-226 (1986) (with R. Krawczyk)

15. New Integration Formulas which Use Nodes outside the Integration Interval,
*Journal of the Franklin Institute* __321__, 115-126 (1986) (with W. Solak and
Z. Szydelko)

16. Solution of Linear Equations Having a Toeplitz Interval Matrix as Coefficient
Matrix, *Opuscula Mathematica* __2__, 33-45 (1986)

17. The Spectra of Matrices Having Sums of Principal Minors with Alternating Sign,
*SIAM J. Algebraic and Discrete Methods* __8__, 106-107 (1987) (now: *SIAM J.
Matrix Analysis and Applications*) (with V. Hattenbach)

18. Boundary Implications for Stability Properties: Present Status, in 'Reliability in Computing', R.E. Moore, Ed., Academic Press, Boston, San Diego, New York, pp. 391-402, (1988) (with N.K. Bose)

19. Block Methods for the Solution of Linear Interval Equations, *SIAM J. Matrix
Analysis and Applications* __11__, 89-106 (1990)

20. Stability Test of a Polynomial with Coefficients Depending Polynomially on
Parameters, *Wiss. Zeitschrift TH Leipzig* __15__, 415-419 (1991)

21. An Improved Bernstein Algorithm for Robust Stability and Performance Analysis, Proceedings of the Singapore Intern. Conf. on Intelligent Control and Instrumentation, Singapore, Feb. 18-21, 1992, IEEE, pp. 1204-1210 (1992) (with S. Malan, M. Milanese, and M. Taragna)

22. B^3 Algorithm for Robust Performance Analysis in Presence of Mixed Parametric and Dynamic Perturbations, Proceedings of the 31st. Control and Decision Conference, Tucson, Arizona, USA, Dec. 16-18, 1992, IEEE Control System Society, pp. 128-133 (1992) (with S. Malan, M. Milanese, and M. Taragna)

23. The Bernstein Algorithm, *Interval Computations* __2__, 154-168 (1993)
(now: *Reliable Computing*)

24. Vertex Implications for Totally Nonnegative Matrices, in 'Total Positivity and its Applications', M. Gasca and C.A. Micchelli, Eds. , Kluwer Acad. Publ., Dordrecht, Boston, London, pp. 103-107 (1996)

25. Preservation of Total Nonnegativity unter the Hadamard Product and Related Topics, in 'Total Positivity and its Applications', M. Gasca and C.A. Micchelli, Eds., Kluwer Acad. Publ., Dordrecht, Boston, London, pp. 97-102 (1996), (with D. G. Wagner)

26. Hadamard Products of Stable Polynomials Are Stable, *Journal Math. Analysis
and Applications* __202__, 797-809 (1996) (with D.G. Wagner)

27. The Hadamard Factorization of Hurwitz and Schur stable Polynomials, in
'Stability Theory'*,* R. Jeltsch and M. Mansour, Eds., *Internat. Series of
Numerical Mathematics (ISNM)*, vol.121, Birkhäuser Verlag, Boston, Basel,
Berlin, pp. 19-21 (1996) (with B. Shrinivasan)

28. Robustness
Analysis of Polynomials with Polynomial Parameter Dependency Using Bernstein
Expansion, *IEEE Trans. Automat. Control* 43, 425-431 (1998) (with M.
Zettler)

29. Speeding
up an Algorithm for Checking Robust Stability of Polynomials, Proc. 2nd IFAC Symp.
*Robust Control Design*, Cs. Banyasz, Ed., Elsevier Sci., Oxford, pp. 183-188
(1998) (with B. Graf and M. Zettler)

30. Bounds
for the Range of a Bivariate Polynomial over a Triangle, *Reliable Computing*
4, 3-13 (1998) (with R. Hungerbühler)

31. Application of Bernstein Expansion to the Solution of Control Problems,
Proceedings of *MISC'99 - Workshop on Applications of Interval Analysis to Systems
and Control*, J. Vehi and M. A. Sainz, Eds., University of Girona, Girona (Spain),
pp. 421-430 (1999)

32. Robust
Schur Stability of Polynomials with Polynomial Parameter Dependency,
*Multidimensional Systems and Signal Processing* 10, 189-199 (1999) (with B.
Graf)

33. Solving
Strict Polynomial Inequalities by Bernstein Expansion, in *The Use of Symbolic
Methods in Control System Analysis and Design*, N. Munro, Ed., The Institution of
Electrical Engineers (IEE), London, pp. 339-352 (1999) (with B. Graf)

34. Software for Solving Robust Performance Problems Based on Bernstein Expansion,
Proceedings of *2nd NICONET Workshop on Numerical Control Software*, INRIA,
Rocquencourt (France), 3.12.99, pp. 35-39 (1999)

35. Computation
of the Bernstein Coefficients on Subdivided Triangles, *Reliable Computing* 6,
115-121 (2000) (with R. Hungerbühler)

36. Application
of Bernstein Expansion to the Solution of Control Problems, *Reliable
Computing* 6, 303-320 (2000)

37. Special Issue of *Reliable Computing* on Applications to Control, Signals,
and Systems (vol. 6, no. 3, 2000) (Guest Editor jointly with E. Walter)

38. Solution of
Systems of Polynomial Equations by Using Bernstein Expansion, in *Symbolic
Algebraic Methods and Verification Methods*, G. Alefeld, S. Rump, J. Rohn, and T.
Yamamoto, Eds., Springer, pp. 87-97 (2001) (with A. P. Smith)

39. Investigation
of a Subdivision Based Algorithm for Solving Systems of Polynomial Equations,
*Journal of Nonlinear Analysis: Series A Theory and Methods* 47/1, 167-178 (2001)
(with A. P. Smith)

40. Intervals of Totally Nonnegative and Related Matrices, *Proceedings in Applied
Mathematics and Mechanics* 1, 496-497 (2002)

41. Intervals of
Almost Totally Positive Matrices, *Linear Algebra and its Applications* 363,
103-108 (2003)

42. The Bernstein
Expansion and its Applications, *Journal of the American Romanian Academy*
(ARA J.), 2000-2003, No. 25-27, 80-85 (2003) (tutorial paper, MS Word format)

43. Lower Bound
Functions for Polynomials, *Journal of Computational and Applied Mathematics*
157, 207-225 (2003) (with C. Jansson and A. P. Smith)

44. Inclusion
Isotonicity of Convex-Concave Extensions for Polynomials Based on Bernstein
Expansion, *Computing* 70, 111-119 (2003) (with C. Jansson and A. P.
Smith)

45. An Improved
Method for the Computation of Affine Lower Bound Functions for Polynomials,
*Frontiers in Global Optimization*, C. A. Floudas and P. M. Pardalos, Eds., Series
Nonconvex Optimization and Its Applications, Kluwer Academic Publ., Boston, Dordrecht,
New York, London, 135-144 (2004) (with A. P. Smith)

46. Accelerating
Consistency Techniques and Prony's Method for Reliable Parameter Estimation of
Exponential Sums, *Global Optimization and Constraint Satisfaction*, C.
Jermann, A. Neumaier, and D. Sam, Eds., Lecture Notes in Computer Science, No. 3478,
Springer-Verlag, Berlin, Heidelberg, 31-45 (2005) (with L. Granvilliers and A. P.
Smith) (in LNCS, ©
Springer-Verlag)

47. A Comparison
of Methods for the Computation of Affine Lower Bound Functions for Polynomials,
*Global Optimization and Constraint Satisfaction*, C. Jermann, A. Neumaier, and D.
Sam, Eds., Lecture Notes in Computer Science, No. 3478, Springer-Verlag, Berlin,
Heidelberg, 71-85 (2005) (with A. P. Smith) (in LNCS, © Springer-Verlag)

48. Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use
in Global Optimisation, *Proceedings of GO'05 -- International Workshop on Global
Optimization*, Almeria, Spain, 18.-22.09.05, L. G. Casado, I. Garcia, E. M. T.
Hendrix, and B. Toth, Eds., 109-113 (2005) (with A. P. Smith)

49. Parametermengenschätzung bei Exponentialsummen, *horizonte* 27, 8-11,
(2005) (with I. Idriss and A. P. Smith)

50. Guaranteed
Parameter Set Estimation for Exponential Sums: The Three-terms Case, *Reliable
Computing* 13, 351-359 (2007) (with I. Idriss and A. P. Smith)

51. Guaranteed Affine Lower
Bound Functions for Multivariate Polynomials, *Proceedings in Applied Mathematics
and Mechanics (PAMM)* 7, 1022905-1022906 (2007) (with A. P. Smith)

52. Rigorous
Affine Lower Bound Functions for Multivariate Polynomials and their Use in Global
Optimisation, Proceedings of the 1st International Conference on Applied
Operational Research, Tadbir Institute for Operational Research, Systems Design and
Financial Services, *Lecture Notes in Management Science* 1, 199-211 (2008) (with
A. P. Smith)

53. Interval
Gaussian Elimination with Pivot Tightening, *SIAM Journal of Matrix Analysis and
Applications* 30(4), 1761-1772 (2009)

54. Solving
Linear Systems with Polynomial Parameter Dependency in the Reliable Analysis of
Structural Frames, *Proceedings of the 2nd International Conference on
Uncertainty in Structural Dynamics*, 15.-17.06.09, Sheffield, UK, N. Sims and K.
Worden, Eds., 147-156 (2009) (with E. D. Popova and A. P. Smith), extended version
available here

55. Verified
Solution for a Simple Truss Structure with Uncertain Node Locations, *Proceedings
of the 18th International Conference on the Application of Computer Science and
Mathematics in Architecture and Civil Engineering*, 7.-9.07.09, Weimar, Germany, K.
Gürlebeck and C. Könke, Eds. (2009) (with A. P. Smith and H. Werkle)

56. Karl L. E.
Nickel (1924-2009), *Reliable Computing* 14, 61-65 (2010)

57. Pivot
Tightening for the Interval Cholesky Method, *Proc. in Applied Mathematics and
Mechanics (PAMM)* 10, 549-550 (2010)

58. A
Verified Monotonicity-Based Solution of a Simple Finite Element Model with Uncertain
Node Locations, *Proc. in Applied Mathematics and Mechanics (PAMM)* 10,
157-158 (2010) (with A. P. Smith and H. Werkle)

59. Verified
Solution for a Statically Determinate Truss Structure with Uncertain Node
Locations, *Journal of Civil Engineering and Architecture* 4(11), 1-10 (2010)
(with A. P. Smith and H. Werkle)

60. A Method
for the Verified Solution of Finite Element Models with Uncertain Node Locations,
*Applications of Statistics and Probability in Civil Engineering*, M. Faber, J.
Köhler, and K. Nishijima, Eds., CRC Press, Boca Raton, Fl., 506-512 (2011) (with
A. P. Smith and H. Werkle)

61. Pivot Tightening for Some Direct Methods for Solving Systems of Linear Interval
Equations, *Proceedings of the 3rd Conference of Mathematical Sciences (CMS
2011)*, 27.-28.04.11, Zarqa, Jordan, 2137-2156 (2011)

62. Bounds on the
Range of Multivariate Rational Functions, *Proc. in Applied Mathematics and
Mechanics (PAMM)* 12, 649-650 (2012), DOI: 10.1002/pamm.201210313 (with A. Schabert
and A. P. Smith)

63. Pivot Tightening
for Direct Methods for Solving Symmetric Positive Definite Systems of Linear Interval
Equations, *Computing* 94(2-4), 97-107 (2012), DOI:
10.1007/s00607-011-0159-7

64.
Special Issue on the Use of Bernstein Polynomials in Reliable Computing: A Centennial
Anniversary, *Reliable Computing* 17,
Preface, i-vii (2012) (with A. P. Smith)

65. Bounding
the Range of a Rational Function over a Box, *Reliable Computing* 17, 34-39
(2012) (with A. Narkawicz, A. P. Smith, and C. A. Muñoz)

66. Solving
Linear Systems with Polynomial Parameter Dependency with Application to the Verified
Solution of Problems in Structural Mechanics, in *Optimization, Simulation, and
Control*, A. Chinchuluun, P. M. Pardalos, R. Enkhbat, and E. N. Pistikopoulos, Eds.,
Series *Springer Optimization and Its Applications* vol. 76, Springer, 301-318
(2013) (with E. D. Popova and A. P. Smith)

67. Intervals of Totally
Nonnegative Matrices, *Linear Algebra and its Applications*
439,3796-3806 (2013), DOI: 10.1016/j.laa.2013.10.021 (with M. Adm)

68. Invariance of Total Nonnegativity of a Tridiagonal Matrix under Element-wise Perturbation, Operator and Matrices 8(1), 129-137 (2014) (with M. Adm)

69. Improved Tests and Characterizations of Totally Nonnegative Matrices, Electronic Journal of Linear Algebra 27, 588-610 (2014) (with M. Adm)

70.
Convergence of the Simplicial Rational Bernstein Form, in *Modelling*, Computation and Optimization in Information Systems and
Management Sciences, Le Thi Hoai An, Pham Dinh Tao, and Nguyen Ngoc Thanh, Eds.,
Series Advances in Intelligent Systems and
Computing Vol. 359, Springer, 433-441 (2015) (with J. Titi and T. Hamadneh)

71.
Total Nonnegativity of Matrices Related to Polynomial Roots and Poles of Rational Functions,
*Journal of Mathematical Analysis and Applications* 434(1), 780-797 (2016) (with M. Adm and J. Titi)

72.
A Survey of Classes of Matrices Possessing the Interval Property and Related Properties,
*Reliable Computing* 22, 1-10 (2016) (with M. Adm and J. Titi)

73. Invariance of Total Positivity of a Matrix under Entry-wise Perturbation and Completion Problems, Contemporary Mathematics, Vol. 658, C. M . da Fonseca, D. V. Huynh, S. Kirkland, and V. K. Tuan, Eds., Amer. Math. Soc., Providence, RI, 2016 (with M. Adm)

74.
Intervals of Special Sign Regular Matrices, *Linear and Multilinear Algebra*, to appear, DOI: 10.1080/03081087.2015.with (1090388 M. Adm)

75. Convergence and Inclusion Isotonicity of the Tensorial Rational Bernstein Form, submitted to the proceedings of the 16th GAMM – IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2014), Warwick Tucker and Jürgen Wolff von Gudenberg, Eds., Lecture Notes in Computer Sciences, Springer, to appear (with T. Hamadneh)

76. Invariance of Total Nonnegativity of a Matrix under Entry-wise Perturbation and the Perron Complement, in preparation (with M. Adm)

2. Review of
Totally Nonnegative Matrices by Shaun M. Fallat and Charles R. Johnson, *Linear
Algebra and its Applications* 436, 3790-3792 (2012), DOI:
10.1016/j.laa.2011.11.038

2. Interval-Related Talks at the 4th International Conference on Frontiers in Global
Optimization, Santorini, Greece, June 8-12, 2003, *Reliable Computing* 10, 63-70
(2004) (with V. Kreinovich)

2. Interval Mathematics: A Bibliography, *Freiburger Intervall-Berichte,*
__85/6__ 222 pages (1985)

3. Bibliography on Interval Mathematics, Continuation, *Freiburger
Intervall-Berichte* __87/2__, 1-50 (1987)

The bibliographies nos. 2 and 3 are available under http://www.cs.utep.edu/interval-comp/ under Bibliographies on Interval and Related Methods or directly by ftp://ftp.math.utah.edu/pub/tex/bin/intarith.bib .

Last updated: January 2016