Research interests

Key words: finite difference; finite element; partial differential equations; coupled systems; singularly perturbed problems; boundary and interior layers; numerical linear algebra; ADI; domain decomposition; multigrid methods.

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 for linearised convection-diffusion (=advection-diffusion) and reaction-diffusion problems. See also: The Irish Research Group on Singularly Perturbed Differential Equations.

Numerical Modelling Differential equations form one of the main languages in which mathematical models and be designed and expressed. Extracting useful information from these models often involves numerical simulation. While there are many excellent classical methods and black-box tools for this, the real world is rather complicated, and so too are the models of it. So one often needs bespoke numerical schemes. This is a fruitful source of interaction between computational mathematicians and applied scientists and engineers. In recent times I've worked on the design of suitable methods for problems in wave-current interactions, dispersion of pollutants in coastal regions, and simulations of ICU patients reactions to clinical therapies.

Submitted for publication

1. P. Panaseti, A. Zouvani, N. Madden and C. Xenophontos, A C¹-conforming hp Finite Element Method for fourth order singularly perturbed boundary value problems, submitted (2013).
2. Scott MacLachlan and Niall Madden, Robust solution of singularly perturbed problems using multigrid methods. Submitted for publication, Aug 2012. Abstract
3. Naresh M. Chadha and Niall Madden, An optimal time-stepping algorithm for unsteady advection-diffusion problems, Submitted for publication, Aug 2011. Abstract

Publications

Published Journal Articles and Refereed Conference Proceedings

4. Catherine G. Enright, Michael G. Madden and Niall. Bayesian Networks for Mathematical Models: Techniques for Automatic Construction and Efficient Inference. International Journal of Approximate Reasoning, Vol. 54 (2), 2013, pp 323-342. DOI 10.1016/j.ijar.2012.10.004.
5. Niall Madden, John Todd and the development of modern numerical analysis. Irish Math. Soc. Bulletin, Number 69, Summer 2012, 11-23. Abstract
6. N. Madden and M. Stynes. A curious property of oscillatory FEM solutions of one-dimensional convection-diffusion problems. Applications of Mathematics 2012, Proc. of Conference in Honour of 60th birthday of Michal Křížek, 2--5 May 2012), J. Brandts, J. Chleboun, S. Korotov, K. Segeth, J. Šístek & T. Vejchodský (Eds.), Czech Academy of Sciences, Prague, 2012, ISBN 978-80-85823-60-8, pp.188--196.
7. 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.
8. 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] [Full text at RIAN.ie]
9. 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] [Full text at RIAN.ie]
10. 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]
11. 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.
12. 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).
13. 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]
14. 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]
15. 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]
16. 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
17. 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.
18. 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.
19. 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.
20. 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],
21. 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]
22. 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
23. 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]
24. 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]
25. 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.
26. 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
27. 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

28. 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.
29. 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.
30. 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, here, and from RIAN.ie.
31. 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.

Theses (supervised by me)

• Meghan Stephens. Numerical Methods for Singularly Perturbed Differential Equations. Ph.D. thesis, NUI Galway 2009.
• Nhan Anh Thai. Numerical Solutions of Models for Glucose and Insulin Levels in Critically Ill Patients. M.Sc. thesis, NUI Galway 2011.
• Stephen Russell. Sparse grid methods for the two-dimensional Poisson problem. . M.Sc. thesis, NUI Galway 2012.

Theses (written by me)

• 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.

Funded projects: past and present

• 2004-2008: Domain decomposition methods for singularly perturbed problems
  • with Meghan Stephens
  • funded by the Irish Research Council for Science, Engineering & Technology (IRCSET) Embark 475-2004. Awarded to Meghan Stephens.
  • topics: automatic generation of piecewise uniform meshes; schemes for parameterised problems; Schwarz-based domain decomposition methods for singularly perturbed problems.
• 2008-2012: Efficient computer modelling of the transport and diffusion of pollutants in advection dominated flows
  • funded by Science Foundation Ireland Research Frontiers grant RFP/CMS/1205. Awarded to Niall Madden (Pi) and Michael Hartnett (Co-PI).
  • with Michael Hartnett (NUI Galway Marine Modelling Centre; Ryan Institute)
    Dr Naresh Chadha (Postdoctoral researcher supported by the grant 2008-2012) Dr Kajal Kumar Mondol (postdoctoral/visiting researcher 2009-2010, supported by Govt of India BOYSCAST fellowship).
  • topics: We are investigating how existing software models can be improved using recent advanced in finite difference methods. In particular we are investigating strategies for the optimal time-stepping, as well as improved numerical modelling.
• 2008--2012: Handling Nonlinearity and Uncertainty in Drug Delivery Modelling
  • with Dr Michael Madden (Engineering and Informatics, NUI Galway) and Dr Petri T Piiroinen (Applied Mathematics, NUI Galway), Nhan Anh Thai (MSc student)
  • funded by Science Foundation Ireland Research Frontiers grant SFI RFP/CMS/1254. Awarded to Michael Madden (PI), Niall Madden (Co-PI) and Petri Piiroinen (Co-PI).
  • topic: The group is working on developing a system that models individual patients reactions to glucose and insulin infusions. Thai and I work on designing suitable numerical methods for models designed by Petri's team that can be incorporated into DBN software by Michael's group.
• 2011--2014: Mutigrid methods for singularly perturbed problems
  • with Nhan Anh Thai, NUI Galway (Ph.D student) and Dr Scott MacLachlan, Tufts University (collaborator)
  • funded by the Irish Research Council for Science, Engineering & Technology (IRCSET) EMBARK Initiative Postgraduate Research Scholarship RS/2011/179. Awarded to Nhan Anh Thai.
  • topic: In recent years there has been significant interest in the development of numerical schemes for the accurate solutions of singularly perturbed ordinary and partial differential equations. However, there have been relatively few on the accurate solution of resulting linear systems. In this project we aim to design new fast solvers, with particular emphasis on multigrid methods.
• 2011--2012: Sparse grid methods for partial differential equations
  • with Stephen Russell (M.Sc student) and Prof. Michel Destrade (Applied Mathematics, NUI Galway)
  • funded by NUI Galway (through Prof Michel Destrade).
  • topic: The numerical solution of partial differential equations is computationally expensive: as the dimension of the problem grows linearly, the computational effort required grows exponentially. Stephen and I are studying some recently devised sparse grid methods that may break this "curse of dimensionality".

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