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401-2650-58L 4 Credits BSC D-MATH

Numerical Analysis Proseminar: Numerical Methods for Oscillatory Integrals

Proseminar Numerische Mathematik: Numerische Methoden für oszillatorische Integrale

Lecturers & Examiners: Prof. Dr. Ralf Hiptmair
VVZ CR n/a

Last Updated: 2026-02-05 15:23:56

Abstract

The seminar involves presentations based on mathematical journal articles devoted to numerical methods for oscillatory integrals. Both theoretical and algorithmic issues are covered.

Objective

* Knowledge of main ideas underlying numerical methods for oscillatory integrals * Ability to obtain information from mathematical research papers. * Practicing structured presentation of mathematical topics.

Content

Themen fuer Studentenvortraege (Nummern beziehen sich auf die Literaturliste): 1. Asymptotic expansions for oscillatory integrals (method of stationary phase, Laplace method): Chapter 2, Sections 3.1–3.5 and 5.1–5.4 from [14]. See also [3, Sect. 1.2.2], [18, Sect. 2.2, 2.3]. 2. Collocation methods (Levin-type methods): [11–13] 3. Expansion methods: [1, 2] 4. Filon-type methods I: [6] (except Sects. 2 & 5) and [7] 5. Asymptotic and Filon-type methods: [8, 9] 6. Moment-free methods: [15, 17] 7. Multidimensional Levin-type methods: [16] 8. Method of steepest descent: Sections 4.1–4.5 from [14]. 9. Numerical steepest descent: [4] 10. Numerical steepest descent: the multidimensional case [5]

Resources

Literature

[1] £ G. Evans, Two robust methods for irregular oscillatory integrals over a finite range, Appl. Num. Math., 14 (1994), pp. 383–395. , An expansion method for irregular oscillatory integrals, Int. J. Computer Math- [2] £ ematics, 63 (1997), pp. 137–148. [3] £ D. Huybrechs and S. Olver, Highly oscillatory quadrature, Proceedings of HOP INI Program, (2008). [4] £ D. Huybrechs and S. Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal., 44 (2006), pp. 1026–1048. , The construction of cubature rules for multivariate highly oscillatory integrals, [5] £ Math. Comp., 76 (2007), pp. 1955–1980. [6] £ A. Iserles, On the numerical quadrature of highly oscillatory integrals I: Fourier transforms, IMA J. Numer. Anal., 24 (2004), pp. 365–391. [7] £ , On the numerical quadrature of highly oscillatory integrals II: Irregular oscillators, IMA J. Numer. Anal., 25 (2005), pp. 25–44. [8] £ A. Iserles and S. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT, 44 (2004), pp. 755–772. [9] £ A. Iserles and S. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), pp. 1383– 1399. [10] £ A. Iserles, S. Nørsett, and S. Olver, Highly oscillatory quadrature: The story so far, in Proceedings of EnuMath, Santiago de Compostela, A. Bermudez de Castro, ed., Berlin, 2006, Springer, pp. 97–118. [11] £ D. Levin, Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations, Math. Comp., 38 (1982), pp. 531–538. [12] £ , Fast integration of rapidly oscillatory functions, J. Comp. Appl. Math., 67 (1996), pp. 95–101. [13] £ , Analysis of a collocation method for integrating rapidly oscillatory functions, J. Comp. Appl. Math., 78 (1997), pp. 131–138. [14] £ P. Miller, Applied Asymptotic Analysis, vol. 75 of Graduate Studies in Mathematics, AMS, Providence, RI, 2006. [15] £ S. Olver, Moment-free numerical integration of highly oscillatory functions, IMA J. Numer. Anal., 26 (2006), pp. 213–227. [16] £ , On the quadrature of multivariate highly oscillatory integrals over non-polytope domains, Numer. Math., 103 (2006), pp. 643–665. [17] £ , Moment-free numerical approximation of highly oscillatory integrals with sta- tionary points, European J. Appl. Math., 18 (2007), pp. 435–447. [18] £ , Numerical Approximation of Highly Oscillatory Integrals, phd thesis, University of Cambridge, Cambridge, UK, 2008. [19] £ S. Vandewalle, Numerical integration of highly oscillatory functions based on ana- lytic continuation. Lecture Slides, HOP Workshop, Newton Insitutute Cambridge, 2007.

General Information

Language
German
Levels
BSC
Frequency
Yearly recurring

Examination

Type
ungraded semester performance

Course Components

Type Title Time & Place Hours
seminar Proseminar Numerische Mathematik
für Studierende im 3. Semester
  • Mon 15:15-17:00 (HG E 41)
2 h weekly

Offered In