Research details

Current interests

Numerical Analysis of differential equations

I aim to design, analyse and implement robust methods for singularly perturbed problems. Solutions to these problems exhibit boundary and interior layers - narrow regions where the solution changes rapidly - and are known to be difficult to solve numerically. Robust methods should yield approximations whose accuracy is independent of the layer width. I have a particular interest in methods for coupled systems with solutions exhibiting multiple interacting layers of different scales, making their numerical resolution quite challenging.

Recent work includes the study of finite difference and finite element methods applied on specially design meshes.

Numerical Modelling

• Efficient computer modelling of the transport and diffusion of pollutants in advection dominated flows with Michael Hartnett (Civil Engineering) funded by SFI RFP/CMS/1205. With Dr Naresh Chadha, a postdoctoral researcher, we are investigating how existing software models can be improved using recent advanced in adaptive methods and domain decomposition.
• Handling Nonlinearity and Uncertainty in Drug Delivery Modelling with Michael Madden (IT, NUI Galway) and Petri T Piiroinen, funded by SFI RFP/CMS/1254. With a graduate student, Nhan Anh Thai, I am work on designing suitable numerical methods for models designed by Petri's team that can be incorporated into DBN software by Michael's group.

Published Journal Articles and Refereed Conference Proceedings

1. N. Madden and M. Stynes. A curious property of oscillatory FEM solutions of one-dimensional convection-diffusion problems, Applications of Mathematics 2012, 188--196. (pdf). ISBN 978-80-85823-60-8.
2. C.G. Enright, M.G. Madden, N. Madden and J.G. Laffey. Clinical Time Series Data Analysis Using Mathematical Models and DBNs, Artificial Intelligence in Medicine, Lecture Notes in Computer Science, 2011, Volume 6747/2011,159-168.
3. Naresh M. Chadha and Niall Madden, A two-weight scheme for a time-dependent advection-diffusion problem. In C. Carmelo, J. L. Gracia, and F. J. Lisbona, (eds), BAIL 2010 - Boundary and Interior Layers, Computational and Asymptotic Methods, Vol 81 of Lect. Notes Comput. Sci. Eng., 99--108. Springer, 2011. Abstract. [MR2849732]
4. Niall Madden and Kajal Kumar Mondal. Improved mathematical and numerical modelling of dispersion of a solute from a continuous source, In C. Carmelo, J. L. Gracia, and F. J. Lisbona, (eds), BAIL 2010 - Boundary and Interior Layers, Computational and Asymptotic Methods, Vol 81 of Lect. Notes Comput. Sci. Eng., 177--185. Springer, 2011. Abstract. [MR2849740]
5. T. Linß and N. Madden, Analysis of an alternating direction method applied to singularly perturbed reaction-diffusion problems, International Journal of Numerical Analysis and Modelling, Volume 7, Number 3, p507-519 (2010). [MR2644287]
6. C.G. Enright, M.G. Madden, S. Russell, N. Aleks, G.T. Manley, J. Laffey, B. Harte, A. Mulvey, N. Madden. Modelling glycaemia in ICU patients: A dynamic Bayesian network approach BIOSIGNALS 2010 - Proceedings of the 3rd International Conference on Bio-inpsired Systems and Signal Processing, Proceedings 2010, Pages 452-459.
7. T. Linß and N. Madden, Layer-adapted meshes for a system of coupled singularly perturbed reaction-diffusion problems. IMA J Numer Anal 29 (1): 86-108; doi:10.1093/imanum/drm049 (2009).
8. M. Stephens and N. Madden A Schwarz technique for a system of reaction diffusion equations with differing parameters. Proc. BAIL 2008 - Boundary and Interior Layers , Lecture Notes in Computational Science and Engineering, Vol. 69 Hegarty, A.F.; Kopteva, N.; O' Riordan, E.; Stynes, M. (Eds.). p247-255 (2009) [Zbl 1181.65107] [MR2581492]
9. Fang Liu, Niall Madden, Martin Stynes, and Aihui Zhou, A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions, IMA J Numer Anal 29 (4): 986-1007, doi:10.1093/imanum/drn048 (2009) [MR2557053]
10. M. Stephens and N. Madden, A Parameter-Uniform Schwarz Method for a Coupled System of Reaction Diffusion Equations, Journal of Computational and Applied Mathematics, Volume 230, Issue 2, Pages 360-370 (15 August 2009). [MR2532330]
11. R. Bruce Kellogg, Niall Madden and Martin Stynes, A parameter-robust numerical method for a system of reaction-diffusion equations in two dimensions, Numerical Methods for Partial Differential Equations, Volume 24 (1), pp312--334, 2008. Abstract. DOI: 10.1002/num.20265
12. T. Linß and N. Madden, Parameter uniform approximations for time-dependent reaction-diffusion problems, Numerical Methods for Partial Differential Equations, Volume 23, Issue 6, 1290--1300, 2007. Abstract.
13. J. Newell, D. Higgins, N. Madden, J. Cruickshank, J. Einbeck, K. McMillan, K., R. McDonald, Software for Calculating Blood Lactate Endurance Markers, Journal of Sports Sciences, Vol 25, Issue 12, 1403--1409, 2007.
14. J. Newell, D. Higgins, N. Madden, J. Cruickshank, J. Einbeck, K. McMillan, K., R. McDonald, al., Model free endurance markers based on the second derivative of blood lactate curves, in Statistical Solutions to Modern Problems, (A.R. Francis, K.M. Matawie, A. Oshlack, and G.K. Smyth, eds.), Proceedings of the 20th International Workshop on Statistical Modelling, Sydney, Australia, 2005, p357--364.
15. N. Kopteva, N. Madden, and M. Stynes, Grid equidistribution for reaction-diffusion problems in one dimension, Numerical Algorithms, Volume 40, No. 3, 305--322, 2005. [Abstract],
16. T. Linß and N. Madden, A finite element analysis of a coupled system of singularly perturbed reaction-diffusion equations. Applied Mathematics and Computation, Vol 148 (2004), Issue 3 pp 869-880 Abstract. PS file, "Prescreen" PDF. [Zbl 1042.65065] [MR2024550]
17. T. Linß and N. Madden, Accurate solution of a system of coupled singularly perturbed reaction-diffusion equations, Computing Vol 73 (2004), No 2, 121-133. Abstract. DOI: 10.1007/s00607-004-0065-3
18. T. Linß and N. Madden, An improved estimate for a numerical method for a system of coupled singularly perturbed reaction-diffusion equations, Computational Methods in Applied Mathematics (CMAM), Vol. 3 (2003), No.3, 417-423. Abstract. [Zbl 1040.65066] [MR2058038]
19. N. Madden and M. Stynes, A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems. IMA Journal of Numerical Analysis, Vol. 23 (2003), No. 4, pp627-644. Abstract. [Zbl 1048.65076] [MR2011343]
20. N. Madden M. Stynes and G.P. Thomas, On the development of complete flow models for wave-current interactions, in "Coastal Dynamics 97" (Conference Proceedings of the Third Coastal Dynamics Conference held in Plymouth, UK, June 1997), by Edward B. Thornton, ed. Reston, VA: ASCE, 0-7844-0321-X, 1998. Abstract.
21. N. Madden and M. Stynes, Efficient generation of oriented meshes for solving convection-diffusion problems, Int. J. Numer. Methods Engrg. 40 (1997), 565-576. Gzipped postscript file
22. N. Madden and M. Stynes, Linear enhancements of the streamline diffusion method for convection-diffusion problems, Computers Math. Applic. 32 (1996), 29-42. Abstract. doi:10.1016/S0898-1221(96)00184-8. [Zbl 0870.65100] [MR1426204]

Unrefereed conference proceedings

23. T. Linßand N. Madden, Uniform convergence of a finite difference scheme for a system of coupled reaction-diffusion equations. Proceedings in Applied Mathematics and Mechanics 3 (2003) 1, 567-568.
24. N. Kopteva, N. Madden and M. Stynes, On equidistribution for reaction-diffusion problems, in Numerical Methods for Problems with Layer Phenomena (A. R. Ansari, A. F. Hegarty & G. I. Shishkin, eds.), Proceedings of the 3rd Annual Workshop, Department of Mathematics & Statistics, University of Limerick, Ireland, February 2004, 46-51.
25. N. Madden, M. Stynes, and G.P. Thomas, On the application of robust numerical methods to a complete-flow wave-current model, Boundary and Interior Layers (BAIL), Toulouse, July 2004. Abstract. Also available from here.
26. R. Bruce Kellogg, N. Madden, and Martin Stynes, On the numerical solution of a system of M reaction-diffusion equations two dimensions, In: I. Farago, P. Vabishchevich, L. Vulkov (eds.), Finite Difference Methods: Theory and Applications. Pro. IVth Int.. Conf. FDM: T&A (Lozenetz, Bulgaria, 26-29 August 2006), Rousse 2007, 252-257.

Submitted

27. Naresh M. Chadha and Niall Madden, An optimal time-stepping algorithm for unsteady advection-diffusion problems, Submitted for publication, Aug 2011. Abstract

Theses

• Numerical Methods for the Accurate Solution of Singularly Perturbed Convection-Diffusion Problems, M.Sc., Dept of Mathematics, University College Cork, Ireland. 1996. Supervisor: Prof. Martin Stynes.
• Numerical Methods for Wave-Current interactions, Ph.D., Dept of Mathematics, NUI Cork, Ireland, 2000. Supervisor: Prof. Martin Stynes.

Other

• ENTCS Volume 74. The Second Irish Conference on the Mathematical Foundations of Computer Science and Information Technology (MFCSIT 2002) Galway, Ireland, 18th and 19th of July 2002. Online Publication Date: October 2003 Guest Editors: Sharon Flynn, Ted Hurley, Micheal Mac an Airchinnigh, Niall Madden, Michael McGettrick, Michel Schellekens and Anthony Seda.
• A list of research presentations that I've given at conferences, workshops, seminars, colloquia, etc., since 2000.

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