Associate Professor  Antti  Rasila
Mathematics with Computer Science,Guangdong Technion – Israel Institute of Technology in China,  Finland

Title: Numerical Conformal Mappings on Surfaces

Abstract:

We consider numerical computation of conformal mappings on simply and multiply connected domains on plane domains and surfaces by using the conjugate function method. This method is based on conformal moduli of rings and quadrilaterals, which are originating from the theory of quasiconformal mappings. See [1, 2, 3].

Besides being frequently applied in the geometric function theory, and these concepts are also very useful in numerical analysis. In particular, study of these quantities allow transforming the problem of finding Riemann mappings between domains into a PDE problem that can be solved by using PDE methods, including stochastic methods [5]. This approach is also workable on Riemannian surfaces [4].

In this presentation, we recall background and basic properties of these quantities and discuss their numerical as well as computation of mappings on surfaces by using finite element methods and a recent stochastic approach that works in the planar case. This presentation is based on joint work with H. Hakula, Q. Han, and T. Sottinen.

References

[1] H. Hakula,  A. Rasila, M. Vuorinen: On moduli of rings and quadrilaterals: algorithms and experiments. SIAM J. Sci. Comput. 33 (1) (2011), 279-302.

[2] H. Hakula, T. Quach, A. Rasila: Conjugate Function Method for Numerical Conformal Mappings. J. Comput. Appl. Math.  237 (1) (2013), 340-353.

[3] H. Hakula, T. Quach, A. Rasila: Conjugate Function Method and Conformal Mappings in Multiply Connected Domains. SIAM J. Sci. Comp.  41 (2019), A1753--A1776. DOI: 10.1137/17M1124164

[4] H. Hakula, A. Rasila: Laplace-Beltrami equation and numerical conformal mappings on surfaces. SIAM J. Sci. Comp. 47 (2025) DOI: 10.1137/24M1656840

[5] Q. Han, A. Rasila, T. Sottinen: Efficient simulation of mixed boundary value problems and conformal mappings. Appl. Math. Comput. 488 (2025), 129119 DOI: 10.1016/j.amc.2024.129119

Biography:

Antti Rasila obtained his PhD degree in mathematics in 2005 from the University of Helsinki, from where he also holds M.Sc. degree in mathematics. Before joining GTIIT, he worked from 2006 to 2018 at Aalto University, Helsinki Metropolitan Area, Finland. His fields of interest include complex analysis (in particular conformal invariants, harmonic mappings and quasiconformal mappings), potential theory (PDEs), numerical analysis, and in e-learning in mathematics education. He is also interested in applications of mathematics, computers and programming.

Dr. Rasila is an author 105 peer-reviewed publications and three textbooks. He is a docent at University of Vaasa (Finland) and Aalto University (Finland). Previously he had held positons of Xiaoxiang Scholar Chair Lecture Professor (Hunan Normal University, 2012-2015), Guest Professor at Hengyang Normal University (China, 2015-2018).  From the beginning of 2015 Dr. Rasila has acted as the coordinator of the international e-assessment material bank Abacus (http://abacus.aalto.fi/). He is was a member of the board of the Finnish Mathematical Society (2017-2019).