TEnsor Notation for Grid Operators

(TENGO)

Participants

• Prof. Kurt Otto, Dept. of Information Technology, Uppsala Univ.
• Docent Elisabeth Larsson, Dept. of Information Technology, Uppsala Univ.
• Dr Malin Ljungberg

Research

Tensors were first utilized to describe the elastic deformation of solids. Actually, the word tensor stems from the Latin word "tensus" meaning stretched. In the beginning of the 20th century, tensor calculus was refined by the Italian mathematicians Ricci and Levi-Cevita. Since then tensor calculus has been an invaluable tool in differential geometry, special and general relativity, and several branches of Physics. The classical way of using tensors is to let them define coordinate invariant linear operators.

In this project, a tensor notation is advocated from a different point of view. We consider discretizations (on structured grids) of PDE problems such that systems of linear equations arise. The coefficient matrices are typically large, complex, indefinite, and ill-conditioned. For these linear operators and the solvers (preconditioned Krylov subspace methods) for the corresponding systems, we use a tensor notation. That considerably facilitates the construction of the numerical algorithms as well as the design and implementation of our object-oriented software tools. The matrix notation, which has been long prevailing in the numerical linear algebra community, actually makes designing and coding unnecessarily complicated!

Publications

• A flexible solver of the Helmholtz equation for layered media. Kurt Otto and Elisabeth Larsson. In Proc. ECCOMAS CFD Conference 2006, p 8, Tech. Univ. Delft, The Netherlands, 2006. (fulltext).
• Design and usability of a PDE solver framework for curvilinear coordinates. Malin Ljungberg, Kurt Otto, and Michael Thuné. In Advances in Engineering Software, volume 37, pp 814-825, 2006. (DOI).
• Software design for finite difference schemes based on index notation. Krister Åhlander and Kurt Otto. In Future generations computer systems, volume 22, pp 102-109, 2006. (DOI).
• Curvilinear coordinates in a PDE solver framework: Validation. Malin Ljungberg and Kurt Otto. Technical report / Department of Information Technology, Uppsala University nr 2004-032, 2004. (External link).
• On software support for finite difference schemes based on index notation. Krister Åhlander and Kurt Otto. In Computational Science – ICCS 2002, volume 2331 of Lecture Notes in Computer Science, pp 711-718, Springer-Verlag, Berlin, 2002.
• A unifying framework for preconditioners based on fast transforms. Kurt Otto. Scientific Report nr 187, Dept. of Scientific Computing, Uppsala University, 1999. (External link).

Software

A result of these efforts is a solver of the Helmholtz Equation for Layered Media (HELM), which is in imminent use at the Swedish Defence Research Agency.