@article {Simula.SC.35,
	title = {Order Optimal Preconditioners for Fully Implicit Runge-Kutta Schemes Applied to the Bidomain Equations},
	journal = {Numerical Methods for Partial Differential Equations},
	volume = {27},
	year = {2011},
	note = {Listed in annual report 2010. Published online june 2010.},
	pages = {1290-1312},
	publisher = {Wiley Subscription Services, Inc., A Wiley Company},
	abstract = {The partial differential equation part of the bidomain equations is discretized in time with fully implicit Runge-Kutta methods, and the resulting block systems are preconditioned with a block diagonal preconditioner. By studying the time-stepping operator in the proper Sobolev spaces, we show that the preconditioned systems have bounded condition numbers given that the Runge-Kutta scheme is A-stable and irreducible with an invertible coefficient matrix. A new proof of order optimality of the preconditioners for the one-leg discretization in time of the bidomain equations is also presented. The theoretical results are verified by numerical experiments. Additionally, the concept of weakly positive-definite matrices is introduced and analyzed.},
	doi = {10.1002/num.20582},
	author = {Nilssen, Trygve Kastberg and Gunnar Andreas {Staff} and Mardal, Kent-Andre}
}
@article {Mardal.2006.5,
	title = {Order Optimal Preconditioners for Implicit Runge-Kutta Schemes Applied to Parabolic PDE{\textquoteright}s},
	journal = {SIAM Journal on Scientific Computing},
	volume = {29},
	number = {1},
	year = {2007},
	note = {Accepted for publication at SIAM Journal of Scientific Computing},
	pages = {361-375},
	abstract = {In this paper we show that standard preconditioners for parabolic PDEs discretized by implicit Euler or Crank-Nicolson schemes can be reused for higher-order fully implicit Runge-Kutta time discretization schemes. We prove that the suggested block diagonal preconditioners are order-optimal for A-stable and irreducible Runge-Kutta schemes with invertible coefficient matrices. The theoretical investigations are confirmed by numerical experiments.},
	author = {Mardal, Kent-Andre and Nilssen, Trygve Kastberg and Gunnar Andreas {Staff}}
}
@inproceedings {Staff.2006.2,
	title = {Block Preconditioners for Fully Implicit Runge-Kutta Schemes Applied to the Bidomain Equations},
	journal = {ECCOMAS CFD 06},
	year = {2006},
	note = {The proceeding is published electronically},
	pages = {1-15},
	publisher = {European Community on Computational Methods in Applied Sciences},
	type = {Conference},
	abstract = {Recently, the authors presented different block preconditioners for implicit Runge-Kutta discretization of the heat equation. The preconditioners were block Jacobi and block Gauss-Seidel preconditoners where the blocks reused existing preconditioners for the implicit Euler discretization of the same equation. In this paper we will introduce similar block preconditioners for the implicit Runge-Kutta discretization of the Bidomain equation. We will, by numerical experiments, show the properties of the preconditoners, and that higher-order Runge-Kutta discretization of the Bidomain equation may be superior to lower-order in some cases.},
	isbn = {90-9020970-0},
	author = {Gunnar Andreas {Staff} and Mardal, Kent-Andre and Nilssen, Trygve Kastberg},
	editor = {Wesseling, P. and O{\~n}ate, E. and P{\'e}riaux, J.}
}
@article {Staff.2006.1,
	title = {Preconditioning of Fully Implicit Runge-Kutta Schemes for Parabolic PDEs},
	journal = {Model. Identif. Control},
	volume = {27},
	number = {2},
	year = {2006},
	note = {An earlier version of this paper var publisehd in SIMS05. 46th Conference on Simulation and Modeling, Trondheim, October 13-14, edited by J{\o}rn Amundsen, Helge I. Andersson, Elena Celledoni, Tommy Gravdahl, Finn A. Michelsen, Henrik R. Nagel, Thorvald Natvig. Tapir Academic Press, pages 315-324, 2005.},
	pages = {109-123},
	abstract = {Recently, the authors introduced a preconditioner for the linear systems that arise from fully implicit Runge-Kutta time stepping schemes applied to parabolic PDEs \cite{mns}. The preconditioner was a block Jacobi preconditioner, where each of the blocks were based on standard preconditioners for low-order time discretizations like implicit Euler or Crank-Nicolson. It was proven that the preconditioner is optimal with respect to the timestep and the discretization parameter in space. In this paper we will improve the convergence by considering other preconditioners like the upper and the lower block Gauss-Seidel preconditioners, both in a left and right preconditioning setting. Finally, we improve the condition number by using a generalized Gauss-Seidel preconditioner.},
	author = {Gunnar Andreas {Staff} and Mardal, Kent-Andre and Nilssen, Trygve Kastberg}
}
@article {Mardal.2006.2,
	title = {Reuse of Standard Preconditioners for Higher-Order Time Discretizations of Parabolic PDEs},
	journal = {Journal of Numerical Mathematics},
	volume = {14},
	number = {2},
	year = {2006},
	pages = {103-122},
	author = {Mardal, Kent-Andre and Nilssen, Trygve Kastberg}
}
@inproceedings {Staff.2005.2,
	title = {Preconditioning of Fully Implicit Runge-Kutta Schemes for Parabolic PDEs},
	journal = {SIMS05. 46th Conference on Simulation and Modeling, Trondheim, October 13-14},
	year = {2005},
	pages = {315-324},
	publisher = {Tapir Academic Press},
	type = {1},
	abstract = {Recently, the authors introduced a preconditioner for the linear systems that arise from fully implicit Runge-Kutta time stepping schemes applied to parabolic PDEs \cite{mns}. The preconditioner was a block Jacobi preconditioner, where each of the blocks were based on standard preconditioners for low-order time discretizations like implicit Euler or Crank-Nicolson. It was proven that the preconditioner is optimal with respect to the timestep and the discretization parameter in space. In this paper we will improve the convergence by considering other preconditio ners like the upper and the lower block Gauss-Seidel preconditioners, both in a left and right preconditioning setting . Finally, we improve the condition number by using a generalized Gauss-Seidel p reconditioner.},
	author = {Gunnar Andreas {Staff} and Mardal, Kent-Andre and Nilssen, Trygve Kastberg},
	editor = {Amundsen, J{\o}rn and Helge I. {Andersson} and Celledoni, Elena and Gravdahl, Tommy and Finn A. {Michelsen} and Henrik R. {Nagel} and Natvig, Thorvald}
}
@techreport {25344,
	title = {RK methods biodomain},
	year = {2005},
	publisher = {Simula Research Laboratory},
	author = {Nilssen, Trygve Kastberg}
}
@techreport {25343,
	title = {RK methods parabolic},
	year = {2005},
	publisher = {Simula Research Laboratory},
	author = {Nilssen, Trygve Kastberg}
}
@techreport {Nilssen.2005.1,
	title = {Weakly Positve Definite Matrices},
	number = {2005-07},
	year = {2005},
	publisher = {Simula Research Laboratory},
	abstract = {A weakly positive definite matrix is defined to be a matrix which can be written as a product of two positive definite matrices. This paper proves that a matrix is weakly positive definite if and only if the real eigenvalues are positive. The main components of the proof are Schur decomposition and mathematical induction.},
	author = {Nilssen, Trygve Kastberg}
}
@techreport {SC.7.Nilssen.2004,
	title = {Resolution Requirements for a 1D Monodomain Model for Simulating},
	number = {2004-07},
	year = {2004},
	publisher = {Simula Research Laboratory},
	author = {Nilssen, Trygve Kastberg and Ingebrigtsen, L. and Hanslien, Monica and Glenn T. {Lines} and Tveito, Aslak}
}
@misc {SC.8.Nilssen.2004,
	title = {Reuse of Standard Preconditioners for Higher-Order Time Discretizations of Parabolic PDEs},
	howpublished = {Talk at the Eigth Copper Mountain Conference on Iterative Methods, Copper Mountain, CO, USA},
	year = {2004},
	author = {Nilssen, Trygve Kastberg and Mardal, Kent-Andre}
}
@article {SC.4.Ingebrigtsen.2003,
	title = {On the Accuracy of Operator Splitting for a Fluid-Structure Interaction Problem},
	journal = {International Journal of Nonlinear Sciences and Numerical Simulation},
	volume = {4},
	year = {2003},
	pages = {209-218},
	author = {Ingebrigtsen, L. and Nilssen, Trygve Kastberg and Hans Petter {Langtangen} and Tveito, Aslak}
}
@misc {SC.8.Nilssen.2003,
	title = {Grid Size Requirements for Simulation of Electrical Activity in the Heart},
	howpublished = {Talk at the Conference on Computational Science and Engineering, San Diego, CA, USA},
	year = {2003},
	author = {Nilssen, Trygve Kastberg and Ingebrigtsen, L.}
}
@inbook {SC.5.Heimsund.2003,
	title = {Level Set Methods for a Parameter Identification Problem},
	booktitle = {Analysis and optimization of differential systems},
	year = {2003},
	pages = {189-200},
	publisher = {Kluwer Academic Publishers},
	organization = {Kluwer Academic Publishers},
	author = {B. O. {Heimsund} and T. F. {Chan} and Nilssen, Trygve Kastberg and Tai, Xue-Cheng},
	editor = {Barbu, V. and Lasiecka, I. and Tiba, D. and Varsan, C.}
}
@article {SC.4.Nilssen.2003,
	title = {Permeability Estimation With the Augmented Lagrangian Method for a Nonlinear Diffusion Equation},
	journal = {Computational Geosciences},
	volume = {7},
	year = {2003},
	pages = {27-47},
	author = {Nilssen, Trygve Kastberg and Mannseth, T. and Tai, Xue-Cheng}
}
@article {SC.4.Lyche.2002,
	title = {Preconditioned Iterative Methods for Scattered Data Interpolation},
	journal = {Advances in Computational Mathematics},
	volume = {17},
	year = {2002},
	pages = {237-256},
	author = {Lyche, T. and Nilssen, Trygve Kastberg and Winther, Ragnar}
}
@phdthesis {SC.3.Nilssen.2001,
	title = {Parameter Estimation With the Augmented Lagrangian Method and a Study of Some Fourth Order Problems},
	year = {2001},
	note = {Dr.Scient. thesis in Applied Mathematics},
	school = {Department of Mathematics, University of Bergen},
	type = {phd},
	author = {Nilssen, Trygve Kastberg}
}
@misc {SC.8.Nilssen.2001,
	title = {Permeability Estimation With the Augmented Lagrangian Method for Nonlinear Diffusion Equations},
	howpublished = {Talk at the International Parallel CFD 2001 Conference, Egmond aan Zee, The Netherlands},
	year = {2001},
	author = {Nilssen, Trygve Kastberg and Mannseth, T. and Tai, Xue-Cheng}
}
@article {SC.4.Nilssen.2001,
	title = {A Robust Nonconforming H2-Element},
	journal = {Mathematics of Computation},
	volume = {70},
	year = {2001},
	pages = {489-505},
	author = {Nilssen, Trygve Kastberg and Tai, Xue-Cheng and Winther, Ragnar}
}
@misc {SC.8.Nilssen.2000.a,
	title = {Methods for Efficient Recovering of Parameters},
	howpublished = {Talk at The 2000 Nordic computational differential equations circus, Bergen, Norway},
	year = {2000},
	author = {Nilssen, Trygve Kastberg}
}
@misc {SC.8.Nilssen.2000.b,
	title = {Preconditioned Iterative Methods for Scattered Data Interpolation},
	howpublished = {Talk at the Mathematical Methods for Curves and Surfaces, Oslo, Norway},
	year = {2000},
	author = {Nilssen, Trygve Kastberg and Winther, Ragnar}
}
