# Bram Reps

Welcome to my brief webpage. I have been working as a postdoc at the University of Antwerp, where I graduated in 2007 as a Master in Applied Mathematics. My latest research was in the field of HPC for the EXA2CT project. Before, I have been involved in the Intel ExaScience Lab at IMEC Leuven. If you cannot find what you are looking for below, feel free to contact me.

Currently I work as data engineer in Antwerp at Sentiance.

I believe humans get a lot done, not because we're smart, but because we have thumbs so we can make coffee.

(Flash Rosenberg)

**Phd-thesis:**

Iterative and multigrid methods for Helmholtz equations with absorbing boundary conditions.Abstract:

The ultimate goal of many a research is to develop an efficient solver for wave and scattering problems that are described by the Helmholtz equation defined on infinite domains. With this research we contribute our ideas and insights to this rich but still incomplete branch of applied mathematics.

Exterior complex scaling is used to enforce outgoing boundary conditions on the truncated numerical domain. In this light, we have developed an innovative multigrid inverted preconditioner for the numerical solution of indefinite Helmholtz problems with Krylov subspace methods, that is based on complex stretched grids. The multigrid method is further stabilized with non-standard smoothing components for the inversion of competitive preconditioners of which the wave number dependency is examined. The ideas are founded on new theoretical results on the spectrum of the Helmholtz operator with a constant wave number that is discretized with complex-valued mesh widths. Numerical results, on the one hand, confirm the analysis and, on the other hand, verify its usefulness for more general heterogeneous two- and three-dimensional Helmholtz models that originate from both acoustics and quantum mechanical break-up problems.

**Talks and conferences:**

- Public PhD defense, 11-01-2012 (Antwerp, Belgium):
*Iterative and multigrid methods for wave problems with complex-valued boundaries* - Seminar at KULeuven, group Numerical Analysis and Applied Mathematics (Leuven, Belgium):
*Iterative and multigrid methods for wave problems with complex-valued boundaries* - ESF OPTPDE Workshop (Würzburg, Germany) - Fast solvers for simulation, inversion, and control of wave propagation problems
- Waves 2011 (Vancouver, Canada):
*A polynomial multigrid smoother for the Helmholtz equation on complex stretched grids* - ICIAM 2011 (Vancouver, Canada):

*Multigrid methods for the indefinite Helmholtz equation*(minisymposium)

[slides] - EMG 2010 (Isola d'Ichia, Italy):

*Multigrid preconditioners for the Helmholtz equation on complex stretched grids*. - ICCAM 2010 (Leuven, Belgium):

*Multigrid preconditioners for the Helmholtz equation on complex stretched grids.* - Waves 2009 (Pau, France):

*Preconditioning the Helmholtz Equation with Complex Rotated Domains*.

**Publications:**

- B. Reps and Tobias Weinzierl,
*Complex additive geometric multilevel solvers for Helmholtz equations on spacetree*s. ACM Transactions on Mathematical Software (accepted).

[preprint: http://arxiv.org/abs/1508.03954] - B. Reps and W. Vanroose,
*Analyzing the wave number dependency of the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation with an absorbing layer*, accepted for publication in Numerical Linear Algebra with Applications.

[doi: 10.1002/nla.1806, preprint: arXiv:1105.3047v1] - W. Vanroose, B. Reps, and H. bin Zubair,
*A polynomial multigrid smoother for the iterative solution of the heterogeneous Helmholtz problem*, submitted.

[preprint: arXiv:1012.5379v1] - H. bin Zubair, B. Reps, and W. Vanroose,
*A preconditioned iterative solver for the scattering solutions of the Schrödinger equation*, in press at Communications in Computational Physics.

[preprint: arXiv:1012.4307v1] - B. Reps, W. Vanroose, and H. bin Zubair,
*On the indefinite Helmholtz equation: exterior complex stretched absorbing boundary layers, iterative analysis and preconditioning*, Journal of Computational Physics, (2010).

[doi: 10.1016/j.jcp.2010.07.022, preprint: arXiv:0912.0180v2] - Liang Tao, W. Vanroose, B. Reps, T. N. Rescigno, and C. W. Mc Curdy,
*Long-time solution of the time-dependent Schrödinger equation for an atom in an electromagnetic field using complex coordinate contours*, Phys. Rev. A 80, 063419 (2009).

[doi: 10.1103/PhysRevA.80.063419] - Master's thesis (in dutch):
*Numerical solution of the Black-Scholes PDE for option prices. Calculation of the Greeks and exotic option prices.*

**Assistant teaching:**

- Multilevel and Multigrid methods
- Iterative methods for linear systems
- Applications of differential equations
- Computational finance

e: Bram.Reps@ua.ac.be • t: +32 3 265 38 95 • f:+32 3 265 37 77

Middelheimcampus M.G.306, Middelheimlaan 1, 2020 Antwerpen

UA - Dept. Wiskunde-informatica