Logistics of Differential Algebraic Equations – LEDA

The LEDA project (Logistics of Algebraic Differential Equations) focuses on systems described by algebraic differential equations (DAE). The goal consists in carrying out a global logistics for the modeling, the transformation and then the efficient symbolic and/or numeric solving, of systems described by DAE. Models evolve from physical, chemical or biological phenomenons. The DAE systems arising from these models are not integrable by classical methods and their initializations are difficult.
The LEDA project involves four partners: the Institut Mathématiques de Toulouse (IMT, coordinator), the Laboratoire de Génie Chimique de Toulouse (LGC), the Laboratoire d'Informatique de Lille (LIFL) and the Laboratoire d'Informatique de l'?cole Polytechnique. The project has been built from the “projet exploratoire” PEPS LEDA funded by the CNRS during 2009. On the basis of scientific talks and of software realizations presented by each partner, and considering the international context in the domain of DAE solving, it appeared that an ANR project could be undertaken.
The investigation field of this project is threefold: modeling aspects, solving aspects and software aspects. The joint symbolic-numeric approach will provide the logistics and the working environment for modeling and solving DAE.
Considered models feature both reactive distillation processes and gene regulatory networks in biology. Addressing and solving distillation models in liquid and gas phase is one of the original points of the project. Modeling gene regulatory networks by parametric polynomial differential equations is particularly suited to differential algebra tools.
Structural difficulties arise while solving these systems, symbolically and/or numerically. The most important of them is that of the index. The symbolic approach will rely on the approaches of Jacobi and Pryce. In this context, the results of the TERA (Turbo Evaluation and Rapid Algorithms) group, provide a decisive advantage. Avoiding the too costly classical rewriting techniques, the idea consists in representing polynomials by evaluation programs. The bounds provided by Jacobi then provide a sharp estimation of the size of the numerical system to be solved. This context also permits to design numerical methods, well suited to high index DAE.
The software aspect is based on existing software, developed by the project partners, such as the BLAD (Bibliothèques Lilloises d'Algèbre Différentielles) libraries, which have been integrated in the MAPLE 14 software, or the software dedicated to the resolution of distillation models. We plan to use and improve these tools, by taking advantage of the Mathemagix software, whose development is funded by the MAGIX ANR. Mathemagix is an open source software which presents two major advantages for our project: it is not proprietary and it provides unique facilities for mixing symbolic and numeric computations. Moreover, we will use already available libraries (SUNDIALS or DAETS), which are dedicated to DAE solving, for evaluating our work and elaborating a test platform.
We expect major breakthroughs from our project.
1- The modeling of reactive distillation processes and the in silico modeling of gene regulatory networks are major scientific and economic issues. The improvement of automatic model reduction methods contributes to them.
2- The use of symbolic computation on DAE systems is a decisive step for the automatic generation of reliable, or even certified, numerical solvers, applying to a very large class of systems.