uu.seUppsala University Publications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Approximation of pluricomplex Green functions based on Monte Carlo integration
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
(English)Manuscript (preprint) (Other academic)
Keywords [en]
Pluricomplex Green functions, pluriregular sets, Bernstein-Markov property, orthogonal polynomials, Monte Carlo simulation
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-355808OAI: oai:DiVA.org:uu-355808DiVA, id: diva2:1231059
Available from: 2018-07-05 Created: 2018-07-05 Last updated: 2018-07-06
In thesis
1. Approximation of pluricomplex Green functions: A probabilistic approach
Open this publication in new window or tab >>Approximation of pluricomplex Green functions: A probabilistic approach
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This PhD thesis focuses on probabilistic methods of approximation of pluricomplex Green functions and is based on four papers.

The thesis begins with a general introduction to the use of pluricomplex Green functions in multidimensional complex analysis and a review of their main properties. This is followed by short description of the main results obtained in the enclosed papers.

In Paper I, we study properties of the metric space of pluriregular sets, that is zero sets of continuous pluricomplex Green functions. The best understood non-trivial examples of such sets are composite Julia sets, obtained by iteration of finite families of polynomial mappings in several complex variables. We prove that the so-called chaos game is applicable in the case of such sets. We also visualize some composite Julia sets using escape time functions and Monte Carlo simulation.

In Paper II, we extend results in Paper I to the case of infinite compact families of proper polynomials mappings. With composition as the semigroup operation, we generate families of infinite iterated function systems with compact attractors. We show that such attractors can be approximated probabilistically in a manner of the classic chaos game.

In Paper III, we study numerical approximation and visualisation of pluricomplex Green functions based on the Monte-Carlo integration. Unlike alternative methods that rely on locating a sequence of carefully chosen finite sets of points satisfying some optimal conditions for approximation purposes, our approach is simpler and more direct by relying on generation of pseudorandom points. We examine numerically the errors of approximation for some simple geometric shapes for which the pluricomplex Green functions are known. If the pluricomplex Green functions are not known, the errors in Monte Carlo integration can be expressed with the aid of statistics in terms of confidence intervals.

Finally, in Paper IV, we study how perturbations of an orthonomalization procedure influence the resulting approximate Bergman functions. To this end we consider the concept of near orthonormality of a finite set of vectors in an inner product space, understood as closeness of the Gram matrix of those vectors to the identity matrix. We provide estimates for the errors resulting from using nearly orthogonal bases instead of orthogonal ones. The motivation for this work comes from Paper III: when Gram matrices are calculated via Monte Carlo integration, the outcomes of standard orthogonalisation algorithms are nearly orthonormal bases.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2018. p. 47
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 109
Keywords
pluricomplex Green function, pluriregular sets, Bernstein-Markov property, Bergman function, nearly orthonormal polynomials, orthogonal polynomials, Monte Carlo simulation, composite Julia sets, Julia sets, iterated function systems, the chaos game.
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-355810 (URN)978-91-506-2714-5 (ISBN)
Public defence
2018-09-21, Polhemsalen, 10134, Ångströmslaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:00 (English)
Opponent
Supervisors
Available from: 2018-08-31 Created: 2018-07-06 Last updated: 2018-08-31

Open Access in DiVA

No full text in DiVA

Authority records BETA

Alghamdi, AzzaKlimek, Maciej

Search in DiVA

By author/editor
Alghamdi, AzzaKlimek, Maciej
By organisation
Department of Mathematics
Computational Mathematics

Search outside of DiVA

GoogleGoogle Scholar

urn-nbn

Altmetric score

urn-nbn
Total: 19 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf