SADDLES

The results obtained in the SADDLES project are published in

• T. Cluzeau, V. Dolean, F. Nataf, A. Quadrat, "Symbolic preconditioning techniques for linear systems of partial differential equations", to appear in the Proceedings of 20th International Conference on Domain Decomposition Methods, 2012,

and in:

• T. Cluzeau, V. Dolean, F. Nataf, A. Quadrat, "Symbolic methods for developing new domain decomposition algorithms", Inria Research Report n. 7953, 2012.

Smith factorization was already applied successfully to open problems in the design of Perfectly Matched Layers for the compressible Euler equations that model aero-acoustic phenomena:

• A new construction of perfectly matched layers for the linearized Euler equations

For domain decomposition methods, some studies have already been performed on the compressible Euler equations:

• A New Domain Decomposition Method for the Compressible Euler Equations

and on the Stokes system:

• A domain decomposition preconditioner of Neumann-Neumann type for the Stokes equations
• Deriving a new domain decomposition method for the Stokes equations using the Smith factorization

These first results are very promising but not all difficulties have been solved. Moreover the approach can be applied to other very important systems of PDEs such that the Oseen equations (linearized Navier-Stokes equations), linear elasticity, etc..

Within a constructive homological algebra approach, the factorization, reduction and decomposition problems have been studied for a large class of linear functional systems.

• T. Cluzeau, A. Quadrat, "Factoring and decomposing a class of linear functional systems", Linear Algebra and Its Applications, vol. 428, 2008, pp. 324-381.
• T. Cluzeau, A. Quadrat, "On algebraic simplifications of linear functional systems", in Topics in Time-Delay Systems: Analysis, Algorithms and Control, J.-J. Loiseau, W. Michiels, S.-I. Niculescu, R. Sipahi, Lecture Notes in Control and Information Sciences (LNCIS), vol. 388, Springer, 2009, pp. 167-178.

The algorithms developed in these papers have been implemented in a maple package called OreMorphisms. For a presentation of this package, see

• T. Cluzeau, A. Quadrat,"OreMorphisms: A homological algebraic package for factoring, reducing and decomposing linear functional systems", in Topics in Time-Delay Systems: Analysis, Algorithms and Control, J.-J. Loiseau, W. Michiels, S.-I. Niculescu, R. Sipahi, Lecture Notes in Control and Information Sciences (LNCIS), vol. 388, Springer, 2009, pp. 179-196.

This package is based on the package OreModules.

• F. Chyzak, A. Quadrat, D. Robertz, "OreModules: A symbolic package for the study of multidimensional linear systems", in Applications of Time-Delay Systems, J. Chiasson et J.-J. Loiseau, (Eds.), Lecture Notes in Control and Information Sciences (LNCIS), vol. 352, Springer, 2007, pp. 233-264.