Finite difference methods; finite element methods; partial
differential equations; coupled systems; singularly perturbed
problems; boundary and interior layers; numerical linear algebra; ADI;
domain decomposition; multigrid methods.
-
Scott P. MacLachlan, Niall Madden, Thái Anh Nhan.
A Boundary-Layer Preconditioner for Singularly Perturbed
Convection Diffusion. Aug
2021. arxiv.org/abs/2108.13468
Published Journal Articles and Refereed Conference Proceedings
-
Niall Madden
and Martin
Stynes.
A weighted and balanced FEM for singularly
perturbed reaction-diffusion problems.
Calcolo 58, 28 (2021). DOI: 10.1007/s10092-021-00421-w.
Preprint
at
ResearchGate.
-
Róisín Hill and Niall Madden.
Generating layer-adapted meshes using mesh partial differential
equations.
Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 559-588.
DOI: 10.4208/nmtma.OA-2020-0187. Preprint
and and FEniCS code available
from osf.io/dpexh/.
-
Nanda Poddar, Kajal Kumar Mondal and Niall Madden.
Layer-adapted meshes for solute dispersion in a steady flow
through an annulus with wall absorption: Application to a
catheterized artery.
Korea-Australia Rheology Journal 33, 11-24 (2021).
DOI:
10.1007/s13367-021-0002-4.
-
Thái Anh Nhan and
Niall Madden.
An analysis of diagonal and incomplete Cholesky
preconditioners for singularly perturbed problems
on layer-adapted meshes.
J. Appl. Math. Comput. 65, 245-272
(2021). DOI: 10.1007/s12190-020-01390-z
- H. Jahandari, S. MacLachlan, R. Haynes, and N. Madden.
Finite element modelling of geophysical electromagnetic data
with goal-oriented hr-adaptivity.
Computational Geosciences volume 24, p 1257-1283 (March 2020)
-
J. H. Adler, S. MacLachlan, and N. Madden.
First-order
system least squares finite-elements for singularly perturbed
reaction-diffusion equations.
In Ivan Lirkov and Svetozar Margenov, editors, Large-Scale
Scientific
Computing, Vol
11958 of Lecture Notes in Computer Science, pages
3-14. Springer, Cham, 2020.
-
Stephen Russell and Niall Madden.
Analysis of a Galerkin finite element method applied to a
singularly perturbed reaction-diffusion problem in three
dimensions. Int. J. Numer. Anal. Mod., 17 (2020), pp. 297-315.
Abstract.
-
Thái Anh Nhan, Scott MacLachlan and
Niall Madden
Boundary layer preconditioners for finite-element
discretizations of singularly perturbed reaction-diffusion
problems
Numerical Algorithms (2018) 79: 281.
https://doi.org/10.1007/s11075-017-0437-3
Abstract.
- José Luis
Gracia,
Niall Madden and
Thái Anh Nhan.
Applying a patched mesh method to
efficiently solve a singularly perturbed reaction-diffusion
problem
.
In: Bock H., Phu H., Rannacher R., Schlöder
J. (eds) Modeling, Simulation and Optimization of Complex
Processes HPSC 2015. Springer (2017).
Abstract
-
Russell, Stephen, and Niall Madden.
An introduction to the analysis and implementation of sparse
grid finite element methods. Computational Methods in Applied
Mathematics 17, no. 2 (2017):
299-322. DOI: doi.org/10.1515/cmam-2016-0042. Source code is available from
https://github.com/niallmadden/SparseGrids/
-
P. Panaseti, A. Zouvani, N. Madden and
C. Xenophontos,
A C1-conforming hp Finite Element Method for fourth order singularly perturbed
boundary value problems, Applied Numerical Mathematics
Volume 104 (2016), Pages 81-97.
doi:10.1016/j.apnum.2016.02.002.
- James
Adler, Scott MacLachlan, and Niall Madden.
A first-order system Petrov-Galerkin
discretization for a reaction-diffusion problem on a fitted
mesh. IMA J. Numer. Anal. (2016) 36 (3): 1281-1309.
doi:
10.1093/imanum/drv045.
Abstract
-
Naresh M. Chadha and Niall Madden,
An optimal time-stepping algorithm for unsteady
advection-diffusion problems
,
Journal of Computational and Applied Mathematics, 294 (2016), 57-77.
Abstract. doi:10.1016/j.cam.2015.07.029
- Stephen Russell and Niall Madden.
A multiscale sparse grid technique for a
two-dimensional convection-diffusion problem
with exponential layers.
Proc. BAIL 2014,
Vol 108 Lect. Notes Comput. Sci. Eng. Springer (2016), 245-255.
Abstract
- Thái Anh Nhan and Niall Madden.
Cholesky factorisation of linear systems coming
from finite difference approximations of
singularly perturbed problems.
Proc. BAIL 2014,
Vol 108 Lect. Notes Comput. Sci. Eng. Springer (2016), 209-220.
Abstract
- S. Russell and N. Madden. A multiscale sparse grid finite
element method for a two-dimensional singularly perturbed
reaction-diffusion problem.
Advances in Computational Mathematics
Volume 41, Issue 6, pp 987-1014. 2015.
DOI:
10.1007/s10444-014-9395-7.
-
Scott MacLachlan
and Niall Madden,
Robust solution of singularly
perturbed problems using multigrid methods.
SIAM J. Sci. Comput. 35-5 (2013), pp. A2225-A2254.
http://dx.doi.org/10.1137/120889770.
Abstract
- Christos Xenophontos, Markus Melenk, Niall Madden,
Lisa Oberbroeckling, Pandelitsa Panaseti, and Antri Zouvani.
hp Finite Element Methods for Fourth Order
Singularly Perturbed Boundary Value Problems.
Dimov, Faragó, Vulkov (Eds.): Proc. NAA
2012, Springer
Lecture Notes in Computer Science, Vol 8236, pp. 532-539, 2013
- Catherine G. Enright, Michael G. Madden, and Niall Madden.
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
[Zbl 06175673].
[MR3021575]
-
Niall Madden,
John Todd and the development of modern numerical analysis.
Irish Math. Soc. Bulletin, Number 69, Summer 2012, 11-23.
Abstract
[MR3026423].
- 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.
- 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.
-
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]
-
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]
- 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]
[Zbl 1195.65119]
- 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.
- 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).
-
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]
- 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]
-
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]
- 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
- 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.
-
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.
-
J. Newell,
J. Einbeck, N. Madden, K. McMillan, K.,
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.
- 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],
- 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]
- 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
- 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]
- 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]
- 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.
- 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
- 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]
-
Scott MacLachlan
and Niall Madden,
Robust solution of singularly
perturbed problems using multigrid methods: analysis and results in
one and two dimensions.
.
Aug 2012.
Abstract
- 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.
- 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.
- 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.
- 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.
- Stephen Russell.
Sparse grid methods for singularly
perturbed problems
Ph.D. thesis, NUI Galway 2016.
- Nhan Anh Thai. Preconditioning techniques for singularly
perturbed differential equations. Ph.D. thesis, NUI Galway 2015.
- Stephen Russell.
Sparse grid methods for the two-dimensional Poisson problem. M.Sc. thesis, NUI Galway 2012.
- Nhan Anh Thai. Numerical Solutions of Models for Glucose and
Insulin Levels in Critically Ill Patients. M.Sc. thesis, NUI Galway 2011.
- Meghan Stephens. Numerical Methods for Singularly Perturbed Differential
Equations. Ph.D. thesis, NUI Galway 2009.
- 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.
-
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.
- 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.
- 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".
- 2012--2016: Sparse grid methods for singularly perturbed problems
- with Stephen Russell (Ph.D. student)
- funded by NUI Galway.
- topic: Stephen and I are working on designing new sparse
grid methods that can be applied the problem of efficiently
computing numerical solutions to partial differential equations,
whoes solutions feature boundary layers.
- 2013--2014: Data assimilation methods for advection-diffusion problems
- with James Peter White (M.Sc student) and Prof. Michael
Hartnett (Civil Engineering, NUI Galway)
- funded by NUI Galway.
- topic: We are working on techniques for incorporating
spacially and temporal observations into the numerical solutions of
problems motivated by oceanography.