# The Aalto University homepage of Teijo Arponen

My research interests are from quite a broad selection, both Pure and Applied Mathematics.

Typically I classify myself as a numerical analyst with a specialty in numerical solution of ODEs (Ordinary Differential Equations). But my true passion is combining different areas of mathematics.

Firstly, links related to my position at TKK (previously known as "HUT"):

Secondly, links to people who I consider having been my mentors and can give further information about my work:

Thirdly, general links to some interesting research groups on geometrical numerical integration (an incomplete list):

## Research topics, latest first:

1. (Singularities of) Multibody systems. This is continuation to my PhD topic (for which see below). Collaborators in alphabetical order: Villesamuli Normi, Samuli Piipponen, and Jukka Tuomela.
2. Mathematical modelling using Hamiltonian systems with thermostats. Collaboration with Ben Leimkuhler. As an application we simulated ferromagnetic materials as spin lattices. This is a PDE which is space-discretized to get a large ODE. The algorithm has also been parallelized using OpenMP (with C). Some snapshots with periodic boundary conditions:
3. Using tensors in geometric numerical integration. The principle behind geometric numerical integration is that some geometrical structure (a priori known) is conserved (up to round-off error) during the numerical discretization. This has been essential for ability of integration over long time intervals. The most successful approaches use either symplecticness or time-reversibility. It is interesting to note, when one looks more closely, all of the abovementioned structures are such that in addition to their geometrical meaning there exists also an equivalent algebraic representation of them. This is essential: we can find lots of geometrical structures, but without an algebraic representation they are of little use in algorithms. We need to work with geometry at the background, algebra on the foreground! My tensor approach starts from the other end: they are algebraic objects, not all of them are important but those which have geometrical meaning. Full text article (via ScienceDirect).
4. The topic of my PhD: Structural Analysis and Numerical Solution of Differential-Algebraic Equations. The approach is based on their involutive (or complete) form. Introduction and summary are here. (Unfortunately, copyright restrictions prevent me from giving the whole Thesis.) Advisor Jukka Tuomela.
5. Matrix approach to univariate polynomials. Part I. Part II.
6. "Two-sided search" techniques in AI (artificial intelligence). Collaboration with David Sarne, Harvard University, Boston USA. (No preprints here, please send an email request if you are interested.)

## Teaching (i.e. lecturing)

In the University of Warwick, U.K., 2004-2005:
In the Aalto University (formerly known as Helsinki University of Technology):
• S1, Basic Course in Mathematics 1 for MSc in Electrical Engineering
• K1, Basic Course in Mathematics 1 for MSc in Civil Engineering
• P2, Basic Course in Mathematics 2 for MSc in Chemical Engineering
• S2, Basic Course in Mathematics 2 for MSc in Electrical Engineering
• V2, Basic Course in Mathematics 2 for MSc in Engineering (multidisciplinary)
• KP3-II, Basic Course in Mathematics 3 for MSc in Chemical Engineering
• L4, Basic Course in Mathematics 4 for MSc with Extended Curriculum in Science
• Numerical and Symbolic Computation (Maple and Matlab)
• Numerical Linear Algebra (direct and iterative methods)
• Multibody Systems and Numerics
• Finite Difference Methods,
• Numerics of Hamiltonian Systems

• The topics of the "basic courses" include complex numbers, linear algebra, univariate and multivariate calculus, numerical integration, ordinary differential equations, analytic geometry in 2D and 3D, series, curvilinear coordinates.

This page is maintained by Teijo.Arponen at aalto.fi.
Last updated April 12th, 2012.