Jürgen Garloff      





Address

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/


Curriculum Vitae


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 and International Journal of Nonlinear Science (IJNS) and on the advisory board of International Journal of Fuzzy Computation and Modeling and is a member of the International Linear Algebra Society (ILAS).


Lecture Notes

Mathematik 1

Skriptum Diskrete Mathematik

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Skriptum Analysis Teil I

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

Übungsaufgaben zur Mathematik 1

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Mathematik 2

Skriptum Analysis Teil II

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Übungsaufgaben zur Mathematik 2

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Research Interests



Doctoral Students



Projects


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).


Publications


1. Optimal Bounds for Interval Interpolation with Polynomials and Functions ax^b, Z. Angew. Math. Mech. 59, T59-T60 (1979) (in German)

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 PerturbationOperator and Matrices 8(1), 129-137 (2014) (with M. Adm)

69. Improved Tests and Characterizations of Totally Nonnegative MatricesElectronic 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, in: A Panorama of Mathematics: Pure and Applied, Contemporary Mathematics, vol. 658, Amer. Math. Soc., Providence, RI, pp. 115-126 (2016) (with M. Adm)

74. Intervals of Special Sign Regular Matrices, Linear and Multilinear Algebra 64(7), 1424-1444 (2016), DOI: 10.1080/03081087. (with M. Adm)

75. Convergence and Inclusion Isotonicity of the Tensorial Rational Bernstein Form, in: Scientific Computing, Computer Arithmetic, and Validated Numerics, M. Nehmeier, J. Wolff von Gudenberg, and W. Tucker (Eds.), Lect. Notes in Comp. Sci. vol. 9553, Springer, pp. 171-179 (2016) (with T. Hamadneh)

76. Total Nonnegativity of the Extended Perron Complement, submitted (with M. Adm)

77. Matrix Methods for the Tensorial Bernstein Form and for the Evaluation of Multivariate Polynomials, submitted (with J. Titi)

78. Invariance of Total Nonnegativity of a Matrix under Entry-wise Perturbation and Subdirect Sum of Totally Nonnegative Matrices, submitted (with M. Adm)


Reviews

1. Review of Totally Positive Matrices by Allan Pinkus, Linear Algebra and its Applications 433, 1052-1053 (2010)

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


Conference Reports

1. Minisymposium on Applications of Interval Computations at the Third World Congress of Nonlinear Analysts, Catania, Sicily, Italy, 19-26.07.2000, Reliable Computing 7, 73-74 (2001)

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)


Contributions to Biographies

1. Professor Dr. Mieczyslaw Warmus - an early pioneer of interval computations, in J. Dutkiewicz, Mieczyslaw Warmus Life and Academic Work, Wollongong, Australia, ISBN 0-646-42666-4, pp. 130-131 (2006)

Bibliographies (A Selection)

1. A Bibliography on Interval Mathematics, Journal of Computational and Applied Mathematics 6, 67-79 (1980), (with K.-P. Schwierz)

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 .


Slides

Jürgen Garloff, Andrew P. Smith:  Fast and Tight Bound Functions for Multivariate Polynomials with Applications to the Reliable Analysis of Structural Frames SWIM 09, Lausanne, Switzerland, June 2009

Jürgen Garloff, Mohammad Adm: Sign-regular Matrices Having the Interval Property, MatTriad 2015, Coimbra, Portugal, September 2015

Last updated:  June 2016