Jerry Gagelman

Jerry Gagelman
Numerical Analysis of Partial Differential Equations

DDFG Matheon

Mailing
Address

Jerry Gagelman
Technische Universität Berlin
Institut für Matematik, 3-3
Straße des 17. Juni 136
10623 Berlin

gagelman(at)math.tu-berlin.de

Office /
Telephone

MA 365
(030) 314 - 29286

Secretary

Veronika Twilling

(030)	314 - 29289 -- tel.
	314 - 29621 -- fax.

Current Research

Software

Previous Research

Current research

DFG Matheon Project A20: Numerical methods in quantum chemistry

The project is a combination of the previous Matheon projects A7 and A20. Details, publications and preprints are maintained on the websites from the older projects

See also R. Schneider and H. Yserentant for details and preprints discussing related research.

My own research focuses on the approximation of eigenvalues and eigenfunctions of elliptic operators. Recent theoretical results of Yserentant indicates that, when the anti-symmetry (i.e., Fermionic properties) of the electronic wave functions is taken into account, numerical solution of the Schroedinger equation attains computation complexity asymptotically equivalent to that of the helium atom. Thorough accounts are contained in the recently published Regularity and Approximability of Electronic Wave Functions.

With a view toward applications, a recent project explores the approximability of basis sets that arise as eigenfunctions of special operators (i.e., spectral bases). Numerical experiments discussed in [1] establish a proof-of-concept for applicability of such bases in the context of high-dimensional eigenvalue problems.

[1]	Eigenfunction expansions, with H. Yserentant (2010).	in preparation
[2]	Eigenvalue convergence for spectral bases (2010).	in preparation

Software

Numerical quadrature

Experiments in approximation theory at times require computations to a high degree of precision that is traditionally the realm of symbolic or computer-algebra systems. Increasing availability of packages for multiple-precision floating-point computations (combined with the steadily increasing availability of computing resources) are making the traditional "numerical" approach to high-precision computations a tractable alliterative.

quadpack++ was developed in the context of project A20 and recently released open source. It facilitates multiple-precision adaptive quadrature à la QUADPACK using C++ templates for the floating-point type.

quadpack++ source repository.

Hermite approximation

The development of quadpack++ proceeded in tandem with the development of a code base for numerical experiments with Hermite approximation that are described in these notes. The codes are GPL licensed and are available via email inquiries.

Ritz-Galerking laboratory

This code base evolved over a period time, not so much as library, but more as a design approach to Ritz-Galerkin type solvers. The documentation page. describes this in some detail. The codes are GPL licensed and freely available for distribution.

Download source codes [2010-08-20].
Documentation.

Previous research

My background is in model theory. My dissertation (University of Illinois at Urbana-Champaign, 2004) examines the model theory of geometric surgical theories. Such theories have a well behaved dimension function on the class of definable sets that is induced via an algebraic pregometry. The main results, concerning U-rank and Morley rank in stable structures that are interpretable in geometric structures, are collected in the following two papers.

[3]	J. Gagelman, Stability in geometric theories, Annals of Pure and Applied Logic 132 (2005) pp 313--326.	pdf
	Abstract. The class of geometric surgical theories (which includes all o-minimal theories) is examined. The main theorem is that every stable theory that is interpretable in a geometric surgical theory is superstable of finite U-rank.
[4]	J. Gagelman, A note on superstable groups, Journal of Symbolic Logic 70 (2005) pp. 661--663.	pdf
	Abstract. It is proved that all groups of finite U-rank that have the descending chain condition on definable subgroups are totally transcendental. A corollary is that any stable group that is definable in an o-minimal structure has finite Morley rank.