The general theme of this project is to study numerical
methods for systems of partial differential equations which depend on
one or more critical parameters. Typically, we are interested in
systems which change type as a critical perturbation parameter tend to
zero. Our goal is to construct numerical methods with convergence
properties which are uniform with respect to the perturbation
parameter. We will also study the influence of the
parameter on iterative solvers. The project is divided into three
sub-projects
with one of the PIs of the project managing the progress of the
associated
research.
The program will have three principle investigators (PIs) serving as a program committee:
Professor Magne
Espedal, University of Bergen, Department of Mathematics; Magne.Espedal@mi.uib.no;
Professor Xue-Cheng Tail,
University of Bergen, Department of Mathematics; Xue-Cheng.Tai@mi.uib.no;
Professor Ragnar Winther,
University of Oslo, Department of Mathematics and Department of
Informatics, rwinther@ifi.uio.no.
Professor Winther will act as the administrative coordinator of the project and be responsible for the correspondence with the Research Council. The formation of the present group is based on informal contacts between the members over many years. These contacts has lead to some joint papers between Espedal and Tai and between Tai and Winther. The main motivation for making a joint project is to create a research group with a balanced background in mathematical theory, numerical computations and applications. It is also the intention that there will be educational benefits from the cooperation between the groups in Oslo and Bergen.
This project is supported by The Research Council of Norway (Norges forskningsraad) under project BeMatA.
The following candidates are taken up for the positions in connection with this project.
We listed some of the publications that have been done
in connections with this project. Some of them are available online and
can be downloaded from: http://www.mi.uib.no/~resme,
http://www.mi.uib.no/~tai or http://www.ifi.uio.no/~rwinther
T. K. Nilssen, X.-C. Tai and R. Winther,
A
robust nonconforming $H^2$-element,
Math. Comp. ,
vol. 70, pp. 489-505, 2001.
D. Arnold and R. Winther
Mixed finite elements for elasticity
URL: http://www.ifi.uio.no/~rwinther/mixedelas.ps
Tom Lyche, Trygve K. Nilssen and Ragnar Winther,
Preconditioned iterative methods for scattered data interpolation,
Advances in Computational Mathematics 17 (2002), pp. 237--256.
Kent A. Mardal, Xue--Cheng Tai and Ragnar
Winther,
A
robust
finite element method for Darcy-stokes flow,
SIAM J.
Numer. Anal.. , vol. 40, no. 5, pp. 1605--1631, 2002.
Xue-Cheng Tai,
Convergence rate analysis of domain decomposition methods for obstacle
problems, East-West J. Numer. Math. 9 (2001), pp. 233--252.
Xue-Cheng Tai,
Rate
of convergence for some constraint decomposition methods for nonlinear
variational inequalities,
Numer. Math., 2003, 93:755-786.
Xue-Cheng Tai and Paul Tseng,
Convergence rate analysis of an asynchronous space decomposition method
for convex minimization, Math. Comp. 71 (2002), pp. 1105--1135.
Xue-Cheng Tai and Jinchao Xu,
Global and uniform convergence of subspace correction methods for some
convex optimization problems, Math. Comp. 71 (2002), pp. 105--124.
Bjorn-ove Heimsund, Xue-Cheng Tai and Junping Wang,
Superconvergence
for the gradient of finite element approximations by $L^2$-projections
,
SIAM J. Numer.
Anal.. , vol. 40, no. 4, pp. 1263--1280, 2002.
T. K. Nilssen and T. Mannseth and X.-C. Tai,
Permeability estimation with the augmented Lagrangian method for
a
nonlinear diffusion equation,
Comput. Geosci. , 7:27-47, 2003.
T. K. Nilssen and X.-C. Tai,
Parameter estimation with the augmented Lagrangian method for a
parabolic equation. Report-153, Department of Mathematics, University
of Bergen, 2001.
B. Heimsund, T.F. Chan, T.K. Nilssen and
X.-C. Tai,
Level Set Methods and Augmented Lagrangian for a Parameter
Identification Problem, in book: "Analysis and optimization
of differential systems", V.Barbu, I.Lasiecka, D.Tiba and C.Varsan
editors, Kluwer Academic Publishers, Boston/Dordrecht/London, p.
189-200, 2003.
M. Lysaker, A. Lundervold and X.-C. Tai,
Noise Removal Using Fourth-Order Partial Differential Equations with
Applications to Medical Magnetic Resonance Images in Space and Time.
To appear in IEEE Transection on image processing. (Also UCLA, Applied
Mathematicsd, CAM-report-02-44).
A. Tveito and R. Winther, Einf\"{u}hrung in
partielle
differentialgleichungen; Ein numerischer zugang. Springer Verlag 2002,
ISBN 3-540-42404-0 (German translation of English edition published in
1998.)
R. Kozlov and B. Owren,
On order reduction in operator splitting methods. Submitted to
Journal on Comp. and Appl. Math.
V DORODNITSYN and R KOZLOV
A Heat Transfer with a Source: the Complete Set of Invariant Difference Schemes. Journal of Nonlinear Mathematical Physics, vol. 10, no. 1, pp. 16-50.
Tony F. Chan and Xue-Cheng Tai
Level Set And Total Variation Regularization For Elliptic Inverse
Problems With Discontinuous Coefficients,
Accepted and to appear in: Journal of Computational Physics.
Lori Badea, Xue-Cheng Tai and Junping Wang
Convergence rate analysis of a multiplicative Schwarz method for
variational inequalities.
SIAM J.
Numer. Anal.., vol. 41, no. 3, pp. 1052-1073.
Marius Lysaker, Stanley Osher, Xue-Cheng Tai,
Noise Removal Using Smoothed Normals and Surface Fitting,
January 2003. UCLA, Applied
Mathematicsd, CAM-report-03-03.
Xue-Cheng Tai
Some new domain decomposition and multigrid methods for variational
inequalities,
Proceedings of the 14th international domain decomposition conference,
Maxico.
Xue-Cheng Tai,
Domain decomposition and multigrid methods for obstacle problems,
in "Computationa Science-ICCS 2002", Lecture notes in Computer
science, Springer, P. Sloot, C. J.
Tan, J. Dongarra and A. Hoekstra eds, pp. 345-352.