Professor Labahn's research interests span a variety of areas in mathematical computation particularly in the area of symbolic computation. Since 1993 he has been the Director of the Symbolic Computation Group, a research group best known for its creation of the MAPLE computer algebra system.
During the past twenty years Professor Labahn has been heavily involved in both algorithm and software development in the Maple system, particularly in the areas of graphical, numeric and symbolic computation. He has been the primary author or designer of the plots and plottools packages for graphical computation, DEtools (now with various other co-authors) for manipulation of differential equations, along with the author of algorithms for elliptic integration and for finding closed form solutions of linear differential equations. Additional software development includes packages for algebraic computation for numeric polynomial algebra and symbolic computation for matrix polynomial algebra.
Computational algorithms research in computer algebra often encounters the problem that a mathematical algorithm ignores the actual domain where one must do the particular computation. For example, algorithms for exact arithmetic need to worry about the growth of the size of coefficients during intermediate computations while algorithms for inexact arithmetic need to worry about errors in their intermediate computations. In the first case fast algorithms are often not fast at all while in the second case fast algorithms often end up with answers that are far from correct. One of the main goals of Professor Labahn's work with algebraic algorithms is to make these computations effective, that is, fast in exact domains and fast and correct when meant for inexact domains. His main co-author in this work is Bernhard Beckermann from the Universite de Lille, France.
Presently he is working on projects for the algebraic reduction of matrices of differential and recurrence operators. In particular, these reductions can be used for finding closed form solutions for differential-algebraic systems.
Professor Labahn also has interests in applications of mathematical computation, in particular in the area of computational finance. This particular work centers on finding fast, stable numerical methods for solving financial option problems. Technically this typically involves modelling the option pricing problem as a nonlinear PDE, usually in the form of Hamilton-Jacobi-Bellman (HJB) equations and then developing a discretization which provably converges to the correct financially viable solution. This work is done jointly with Peter Forsyth.
Professor Labahn also heads a research project to integrate mathematical computation with pen-based input for the new PC tablets, called the MathBrush project. There are a number of difficult problems in such a project. These include the obvious issue of correctly recognizing symbols in mathematical expressions rather than in text (where the problem is difficult but still much easier than with mathematical input), inferring semantics of syntactically correct mathematical expressions (so that they can eventually be computed if desired) and editing of mathematical expressions using pen-based input. The goal is to create an environment for doing mathematics using a natural rich set of mathematical notations which significantly extends the ability of a user to do mathematics with a pen and paper model. At present MathBrush works on PC tablets with new research being done for computation on newer devices such as Ipads and Blackberry playbooks.
Degrees and awards
BSc, MSc, PhD (Alberta)
Marsland Faculty Fellow, University of Waterloo (2004-2007); NSERC Synergy Innovation Award for Maplesoft & Symbolic Computation Group (2004)
Industrial and sabbatical experience
Professor Labahn has been a visiting professor at both the University of Lille and the University of Limoges in France. In addition he spent three months at INRIA, the leading research organization in France. His primary industrial interaction has been with MapleSoft Inc. where he has been involved with both software design and implementation.
M. Giesbecht, G. Labahn and W-s Lee. Symbolic-numeric Sparse Interpolation of Multivariate Polynomials. Journal of Symbolic Computation, 44(8):943-959, 2009
P.A. Forsyth and G. Labahn. Numerical Methods for Controlled Hamilton-Jacobi-Bellman PDEs in Finance. Journal of Computational Finance, 11(2):1-44, 2008.
B. Beckermann, G. Golub, and G. Labahn. On the Numerical Condition of a Generalized Hankel Eigenvalue Problem. Numerische Mathematik, 106(1):41-68, 2007.
B. Beckermann, G. Labahn, and G. Villard. Normal Forms for General Polynomial Matrices. Journal of Symbolic Computation, 41(6):708-737, 2006.
Y. d'Halluin, P.A. Forsyth, and G. Labahn. A Semi-Lagrangian approach for American Asian Options under Jump Diffusion. SIAM Journal of Scientific Computation, 27:315-345, 2005.
Y. d'Halluin, P.A. Forsyth, and G. Labahn. A Penalty Method for American Options with Jump Diffusion Processes. Numerische Mathematik, 97(2):321-352, 2004.
B. Beckermann and G. Labahn. Fraction-free Computation of Matrix Rational Interpolants and Matrix GCD's. SIAM J. Matrix Analysis and Applications, 22(1):114-144, 2000.
B. Beckermann and G. Labahn. When are two numerical polynomials relatively prime? J. of Symbolic Computation 26, pp. 677-689, 1998.