Topics in numerical mathematics

2022/2023
Programme:
Doctoral Programme Mathematics and Physics
Orientation:
Mathematics
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
ECTS:
6
Language:
slovenian, english
Course director:
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
0
Tutorial
0
Lab
0
Content (Syllabus outline)

The content consists of a selection of standard topics in postgraduate numerical mathematics. Possible themes include geometric interpolation and approximation by parametric polynomial curves and surfaces, multivariable polynomial interpolation, parametric curves and surfaces in CAGD (computer aided geometric design), wavelets in signal processing and image analysis, subdivision schemes for curves and surfaces, numerical methods for functions of matrices, iterative subspace methods and preconditioning, nonlinear eigenvalue problems, multiparameter eigenvalue problems, inverse eigenvalue problems,continuation methods, multigrid methods, spline theory, model reduction, ill-conditioned problems and regularization.The choice depends on students' research interests.

Readings

N. J. Higham: Functions of matrices, Theory and Computation, SIAM, Philadelphia, 2008.
P. C. Hansen: Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, 1998.
N.J. Higham: Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 2002.
J.P. Boyd: Chebyshev and Fourier Spectral Methods, Dover publications, Mineola, 2000.
F.W. Faierman: Linear Control Theory. The State Space Approach, John Wiley & Sons, Chichester, 1998.
W.L. Briggs, V.E. Henson, S.F. McCormick: A Multigrid tutorial, Second Edition, SIAM, Philadelphia, 2000
M.T. Chu, G.H. Golub: Inverse eigenvalue problems: theory, algorithms and applications, Numerical mathematics and Scientific Computation, Oxford University Press, New York, 2005
R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.
Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, H. van der Vorst: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, 2000.
G. Farin, J. Hoschek in M.-S. Kim: Handbook of Computer Aided Geometric Design, Elsevier, 2002.
R. T. Farouki: Pythagorean-hodograph curves: algebra and geometry inseparable, Vol. 1 of Geometry and Computing, Springer, Berlin, 2008.
N. Dyn in D. Levin: Subdivision Schemes in Geometric Modelling, Acta Numer. 11 (2002) 73-144.

Objectives and competences

The main goal of the course is to provide students with some important topics in numerical mathematics.

Intended learning outcomes

Knowledge and comprehension of presented concepts.
Ability to use acquired knowledge and skills.

Learning and teaching methods

Lectures, consultations, problem sessions

Assessment

Writen exam (homeworks), oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Bor Plestenjak:
HOCHSTENBACH, Michiel E., MUHIČ, Andrej, PLESTENJAK, Bor. On linearizations of the quadratic two-parameter eigenvalue problem. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2012, vol. 436, iss. 8, str. 2725-2743. [COBISS-SI-ID 16095065]
MUHIČ, Andrej, PLESTENJAK, Bor. On the quadratic two-parameter eigenvalue problem and its linearization. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2010, vol. 432, iss. 10, str. 2529-2542. [COBISS-SI-ID 15469913]
HOCHSTENBACH, Michiel E., KOŠIR, Tomaž, PLESTENJAK, Bor. A Jacobi-Davidson type method for the two-parameter eigenvalue problem. SIAM journal on matrix analysis and applications, ISSN 0895-4798, 2005, vol. 26, no. 2, str. 477-497. [COBISS-SI-ID 13613401]
Gašper Jaklič:
JAKLIČ, Gašper, ŽAGAR, Emil. Curvature variation minimizing cubic Hermite interpolants. Applied mathematics and computation, ISSN 0096-3003. [Print ed.], 2011, vol. 218, iss. 7, str. 3918-3924. [COBISS-SI-ID 16049241]
JAKLIČ, Gašper, ŽAGAR, Emil. Planar cubic G [sup] 1 interpolatory splines with small strain energy. Journal of Computational and Applied Mathematics, ISSN 0377-0427. [Print ed.], 2011, vol. 235, iss. 8, str. 2758-2765. [COBISS-SI-ID 15770969]
JAKLIČ, Gašper. On the dimension of bivariate spline space S [sub] 3 [sup] 1 ([triangle]). International journal of computer mathematics, ISSN 0020-7160, 2005, vol. 82, no. 11, str. 1355-1369. [COBISS-SI-ID 13801305]
Marjetka Krajnc:
JAKLIČ, Gašper, KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito, ŽAGAR, Emil. High order parametric polynomial approximation of conic sections. Constructive approximation, ISSN 0176-4276, 2013, vol. 38, iss. 1, str. 1-18. [COBISS-SI-ID 16716121]
KRAJNC, Marjetka. Interpolation scheme for planar cubic G [sup] 2 spline curves. Acta applicandae mathematicae, ISSN 0167-8019, 2011, vol. 113, no. 2, str. 129-143. [COBISS-SI-ID 16215385]
KRAJNC, Marjetka. Geometric Hermite interpolation by cubic G[sup]1 splines. Nonlinear Analysis, Theory, Methods and Applications, ISSN 0362-546X. [Print ed.], 2009, vol. 70, iss. 7, str. 2614-2626. [COBISS-SI-ID 15508569]
Emil Žagar:
JAKLIČ, Gašper, KOZAK, Jernej, VITRIH, Vito, ŽAGAR, Emil. Lagrange geometric interpolation by rational spatial cubic Bézier curves. Computer Aided Geometric Design, ISSN 0167-8396, 2012, vol. 29, iss. 3-4, str. 175-188. [COBISS-SI-ID 16207449]
KOZAK, Jernej, ŽAGAR, Emil. On geometric interpolation by polynomial curves. SIAM journal on numerical analysis, ISSN 0036-1429, 2004, vol. 42, no. 3, str. 953-967. [COBISS-SI-ID 13398617]
ŽAGAR, Emil. On G [sup] 2 continuous spline interpolation of curves in R [sup] d. BIT, ISSN 0006-3835, 2002, vol. 42, no. 3, str. 670-688. [COBISS-SI-ID 12027993]