Veerle Ledoux Home Page

Veerle Ledoux

Contact Information

Postdoctoral fellow of FWO-Flanders
Numerical Mathematics Group
Dept. of Applied Mathematics and Computer Science
Ghent University

Krijgslaan 281-S9
B9000 Ghent
Belgium

E-Mail: Veerle.Ledoux[at]UGent.be
Tel : 09 264 47 63
Fax : 09 264 49 95

Research

Numerical analysis for ODEs
- Highly oscillatory ordinary differential equations and their numerical solution
- e.g. Sturm-Liouville and Schrödinger problems. Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behavior of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. The challenges are to do this more cheaply.
- Coefficient approximation methods (Piecewise perturbation methods)
- Integral series methods: Magnus, Neumann
Computation of quantum mechanical bound-states
Mathematical software

Matlab software package for automatic solution of Sturm-Liouville and Schrodinger problems: MATSLISE
Some Matlab and Maple codes are available here
Automatic software codes for systems of coupled Schrödinger equations
Automatic software for oscillatory integrals

Computational spectral theory

Shooting and matching, Evans function;
Grassmannian and Riccati numerical methods;
Multi-dimensional stability.

Numerical Integration

Publications

Papers

V. LEDOUX, M. VAN DAELE and G. VANDEN BERGHE, CP methods of higher order for Sturm-Liouville and Schrodinger equations, Comp. Phys. Commun., 162 (2004), 151-165.
V. LEDOUX, M. VAN DAELE and G. VANDEN BERGHE, CP methods and the evaluation of negative Coulomb Whittaker functions, J. Comp. Appl. Math., 183 (2005), 168-176.
V. LEDOUX, M. VAN DAELE and G. VANDEN BERGHE, Matslise: A Matlab package for the Numerical Solution of Sturm-Liouville and Schrodinger equations, ACM Transactions on Mathematical Software, 31 (2005), 532-554.
V. LEDOUX, M. VAN DAELE and G. VANDEN BERGHE, The numerical solution of Sturm-Liouville and Schrodinger problems using Piecewise Perturbation methods, Proceedings: 3rd International Conference on Advanced Computational Methods in Engineering, Ghent 2005.
V. LEDOUX, M. VAN DAELE and G. VANDEN BERGHE, High order CP methods for the numerical solution of Schrodinger problems, Extended abstracts of ICNAAM 2005.
V. LEDOUX, M. VAN DAELE and G. VANDEN BERGHE, CPM{P,N} methods extended for the solution of coupled channel Schrodinger equations, Comp. Phys. Commun., 174 (2006), 357-370.
V. LEDOUX, M. VAN DAELE and G. VANDEN BERGHE, Piecewise Constant Perturbation Methods for the Multichannel Schrodinger Equation, Lecture Notes in Computer Science, Computational Science. ICCS 2006: 6th International Conference, Reading, UK, May 28-31, 2006, Proceedings, Part IV, 716 - 723.
V. LEDOUX, M. RIZEA, L. IXARU, G. VANDEN BERGHE and M. VAN DAELE, Solution of the Schrodinger equation by a high order perturbation method based on a linear reference potential, Comp. Phys. Commun. 175 (2006) 424-439.
V. LEDOUX, L. IXARU, M. RIZEA, M. VAN DAELE and G. VANDEN BERGHE Solution of the Schrodinger equation over an infinite integration interval by perturbation methods, revisited, Comp. Phys. Comm. 175 (2006) 612-619
V. LEDOUX, M. VAN DAELE and G. VANDEN BERGHE, Some improvements to the MATSLISE package, Extended abstracts of ICNAAM 2006.
V. LEDOUX, M. VAN DAELE and G. VANDEN BERGHE, A numerical procedure to solve the multichannel Schrodinger eigenvalue problem, Comp. Phys. Commun. 176 (2007) 191-199.
M. RIZEA, V. LEDOUX, M. VAN DAELE, G. VANDEN BERGHE and N. CARJAN, Finite difference approach for the two-dimensional Schrodinger equation with application to scission-neutron emission, Comp. Phys. Comm. 179 (2008) 466-478.
V. LEDOUX, M. VAN DAELE and G. VANDEN BERGHE, Using a (higher-order) Magnus method to solve the Sturm-Liouville problem , Lecture Notes in Computer Science. ICCS 2008: International Conference on Computation Science, Krakow, Poland, May 23-25, 2008, Proceedings, Part I, vol. 5101, 1032-1041.
V. LEDOUX, M.RIZEA, M. VAN DAELE, G. VANDEN BERGHE and I. SILISTEANU, Eigenvalue problem for a coupled channel Schrodinger equation with application to the description of deformed nuclear systems, J. Comp. Appl. Math. 228 (2009) 197-211.
V. LEDOUX, S. MALHAM, J. NIESEN and V. THUMMLER, Computing stability of multi-dimensional travelling waves, SIAM J. Applied. Dynamical Systems 8 (2009) 480-507. (arxiv link)
V. LEDOUX, M. VAN DAELE, and G. VANDEN BERGHE, Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics, Comp. Phys. Commun. 180 (2009)
V. LEDOUX, M. VAN DAELE, and G. VANDEN BERGHE, Efficient numerical solution of the 1D Schrödinger eigenvalue problem using Magnus integrators, IMA Journal of Numerical Analysis 30 (2010) 751-776. (doi:10.1093/imanum/drn062)
V. LEDOUX, S. MALHAM and V. THUMMLER, Grassmannian spectral shooting, Mathematics of computation 79 (2010) 1585-1619 (arxiv link)
V. LEDOUX, M. VAN DAELE, Solution of Sturm-Liouville problems using modified Neumann schemes, SIAM Journal on Scientific Computing 32 (2010) 563-584.
V. LEDOUX, M. VAN DAELE, Solving Sturm-Liouville problems by piecewise perturbation methods, revisited, Computer Physics Communications 181 (2010) 1335-1345.
V. LEDOUX, M. VAN DAELE, On CP, LP and other piecewise perturbation methods for the numerical solution of the Schrödinger equation, Zeitschrift für angewandte Mathematik und Physik 62 (2011) 993-1011.
V. LEDOUX, M. VAN DAELE, Interpolatory quadrature rules for oscillatory integrals, Journal of Scientific Computing, 53 (2012) 586-607.
V. LEDOUX, M. VAN DAELE, Automatic computation of quantum-mechanical bound states and wavefunctions, Computer Physics Communications 184 (2013) 1287-1296.

Other Publications

Ph.D.Thesis pdf. Study of Study of Special Algorithms for solving Sturm-Liouville and Schrodinger Equations. Ph.D. Thesis, Dept. of Applied Mathematics and Computer Science, Ghent University, 2007.

Talks

International Congress on Computational and Applied Mathematics (ICCAM-2004), July 26-30, 2004, Leuven. Title abstract: "A computer program for solving regular Sturm-Liouville and Schrödinger equations, based on high-order CP methods".
Workshop on Exponential Integrators, October 20-23, 2004, Innsbruck. Title abstract: "High order CP methods for the numerical solution of Sturm-Liouville and Schrödinger problems".
WOG/SRN A symposium on Software Environments for Numerical Problems, November 17-18, 2004, Gent. Title abstract: "MATSLISE: a MATLAB package for the numerical solution of Sturm-Liouville and Schrödinger equations".
SCICADE International Conference on Scientific Computation and Differential Equations May 23-27, 2005, Nagoya. Title abstract: "The numerical solution of Sturm-Liouville and Schrodinger problems using high order CP methods".
ACOMEN Third international conference on advanced computational methods in Engineering, May 30-June 2, 2005, Gent. Title abstract: "The numerical solution of Sturm-Liouville and Schrödinger problems using Piecewise Perturbation methods". NA05 Dundee, Numerical Analysis Conference 2005, June 28 - July 1, 2005, University Dundee, Scotland, UK. Title abstract: " The numerical solution of Sturm-Liouville and Schrödinger problems using Piecewise Perturbation Methods".
ICNAAM 2005, International Conference of Numerical Analysis and Applied Mathematics, September 16 - 20, 2005, Rhodes, Greece. Title abstract: " High order CP methods for the numerical solution of Schrodinger problems".
ICCS 2006, International Conference on Computational Science, May 28 - 31, 2006, Reading, UK. Title abstract: "Piecewise Constant Perturbation Methods for the Multichannel Schrodinger Equation".
ICCAM 2006, International Congress on Computational and Applied Mathematics , July 10 - 14, 2006, Leuven. Title abstract: "A Numerical Method for Calculating Eigenvalues of Regular Vector Schrodinger Problems".
NUMDIFF11, Numerical solution of differential and differential-algebraic equations, September 4 - 8, 2006, Halle, Germany. Title abstract: "The solution of singular Schrodinger problems using a piecewise perturbation method".
ICNAAM 2006, International Conference of Numerical Analysis and Applied Mathematics, September 15 - 19, 2006, Crete, Greece. Title abstract: "Some improvements to the Matslise package".
Lectures series International Francqui chair for Exact Sciences 2006-2007 (Prof. Dr. H. Van der Vorst), Seminar V: Eigenvalue Problems, March 15 2007, Antwerpen (UA). Title Guest Lecture: "The numerical solution of Sturm-Liouville and Schrodinger eigenvalue problems".
HOP programme on Highly Oscillatory Problems: Computation, Theory and Application, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK. Title of Seminar given on April 26, 2007: "An efficient numerical solution of Sturm-Liouville problems".
Isaac Newton Institute Workshop: Effective Computational Methods for Highly Oscillatory Problems: The Interplay between Mathematical Theory and Applications, July 2-6 2007, Cambridge, UK. Title abstract:"High order coefficient approximations methods for the Sturm-Liouville boundary value problem".
SCICADE 2007: International Conference on SCIentific Computation And Differential Equations. July 9-13 2007, Saint-Malo, France. Title abstract:"Efficient computation of high Sturm-Liouville eigenvalues ".
Workshop Numerical Approach of Oscillatory functions. January 16-18 2008, Ghent. Title abstract: "Efficient approximation of Sturm-Liouville problems using Neumann or Magnus expansions ".
ICCS 2008, International Conference on Computational Science, June 23 - 25, 2008, Krakow, Poland. Title abstract: "Using a (high-oder) Magnus method to solve the Sturm-Liouville problem".
GLADE 2008 conference, July 14 - 18, 2008, Auckland, New Zealand. Title abstract: "High order Magnus methods for the numerical solution of the Sturm- Liouville boundary value problem ".
SCICADE 2009, May 25 - 29, 2009, Beijing, China. Title abstract: "Grassmannian spectral shooting for Evans functions calculations in large systems ".
ICNAAM 2009, International Conference of Numerical Analysis and Applied Mathematics, September 18 - 22, 2009, Rethymno, Crete, Greece. Title abstract: " Efficient computation of Sturm-Liouville eigenvalues using modified Neumann schemes".
BIT 50, Trends in numerical computing, June 17 - 20, 2010, Lund, Sweden. Title abstract: "Using a perturbation approach to construct high order coefficient approximation methods for Sturm-Liouville problems"
ICCAM 2010, July 05 - 09, 2010, Leuven. Title abstract: "On CP, LP and other piecewise perturbation methods for the numerical solution of the Schrödinger equation"
ICNAAM 2010, International Conference of Numerical Analysis and Applied Mathematics, September 19 - 25, 2010, Rhodes, Greece. Title abstract: "How to solve Schrödinger problems by approximating the potential function "
Heriot-Watt University. November 10,2010. Title talk: "Grassmannian spectral shooting for Evans function calculations in large systems"
9th International Conference of Numerical Analysis and Applied Mathematics, Young Researchers Symposium on Numerical Methods for Differential Problems of Practical Interest, September 19 - 25, 2011, Halkidiki, Greece. Title abstract: "Algorithms and software for the Sturm-Liouville problem"
BSL Mathematical Societies Conference, June 6 - 8, 2012, Liege, Belgium. Title abstract: "Automatic solution of coupled-channel Schrödinger equations"
ICCAM 2012, July 09 - 13, 2012, Gent. Title abstract: "Interpolatory quadrature rules for oscillatory integrals"
3rd Dolomites Workshop on Constructive Approximation and Applications, September 09-14, 2012, Alba di Canazei, Italy. Title abstract: "The automatic integration of oscillatory integrals"
10th International Conference of Numerical Analysis and Applied Mathematics, Numerical methods and computational procedures for special problems in physics and chemistry2, devoted to to Prof. Liviu Ixaru to celebrate his 70th birthday, September 19 - 25, 2012, Kos, Greece. Title abstract: "On the use of CP methods for the numerical solution of the Schrodinger equation"