Section: Research Program
Software tools of the team
In addition to the abovementioned research activities, Apics develops and maintains a number of longterm software tools that either implement and illustrate effectiveness of the algorithms theoretically developed by the team or serve as tools to help further research by team members. We present briefly the most important of them.
DedaleHF
Participant : Fabien Seyfert [corresponding participant] .
http://wwwsop.inria.fr/apics/Dedale/
DedaleHF is a software dedicated to solve exhaustively the coupling matrix synthesis problem in reasonable time for the filtering community. Given a coupling topology, the coupling matrix synthesis problem (C.M. problem for short) consists in finding all possible electromagnetic coupling values between resonators that yield a realization of given filter characteristics. Solving the latter problem is crucial during the design step of a filter in order to derive its physical dimensions as well as during the tuning process where coupling values need to be extracted from frequency measurements.
DedaleHF consists in two parts: a database of coupling topologies as well as a dedicated predictorcorrector code. Roughly speaking each reference file of the database contains, for a given coupling topology, the complete solution to the C.M. problem associated to particular filtering characteristics. The latter is then used as a starting point for a predictorcorrector integration method that computes the solution to the C.M. corresponding to the userspecified filter characteristics. The reference files are computed offline using Gröbner basis techniques or numerical techniques based on the exploration of a monodromy group. The use of such continuation techniques, combined with an efficient implementation of the integrator, drastically reduces the computational time.
DedaleHF has been licensed to, and is currently used by TASEspaña.
FindSources3D
Participants : Juliette Leblond [corresponding participant] , JeanPaul Marmorat, Nicolas Schnitzler.
This work is conducted in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena EPI.
FindSources3D is a Matlab software program dedicated to the resolution of inverse source problems in electroencephalography (EEG) (see http://wwwsop.inria.fr/apics/FindSources3D/en/index.html ). From pointwise measurements of the electrical potential taken by electrodes on the scalp, FindSources3D estimates pointwise dipolar current sources within the brain in a spherical model. Following the scheme described in Section 4.2 , see also Sections 6.2 and 7.1.1 , after a first data transmission “cortical mapping” step, it makes use of best rational approximation on 2D planar crosssections and of the software RARL2 (see Section 3.4.4 ) in order to locate singularities [7] . From those planar singularities, the 3D sources are estimated in a last step.
The program is currently being tested by BESA company (Munich). Our purpose is to distribute FindSources3D to teams in partnerhospitals (like la Timone, Marseille). It has a CeCILL license.
PrestoHF
Participants : JeanPaul Marmorat, Martine Olivi, Fabien Seyfert [corresponding participant] .
PrestoHF is a toolbox dedicated to lowpass parameter identification for microwave filters https://project.inria.fr/prestohf/ . In order to allow the industrial transfer of our methods, a Matlabbased toolbox has been developed, dedicated to the problem of identification of lowpass microwave filter parameters. It allows one to run the following algorithmic steps, either individually or in a single stroke:

Determination of delay components caused by the access devices (automatic reference plane adjustment),

Automatic determination of an analytic completion, bounded in modulus for each channel,
For the matrixvalued rational approximation step, PrestoHF relies on RARL2. Constrained realizations are computed using the DedaleHF software. As a toolbox, PrestoHF has a modular structure, which allows one for example to include some building blocks in an already existing software.
The delay compensation algorithm is based on the following assumption: far off the passband, one can reasonably expect a good approximation of the rational components of S11 and S22 by the first few terms of their Taylor expansion at infinity, a small degree polynomial in 1/s. Using this idea, a sequence of quadratic convex optimization problems are solved, in order to obtain appropriate compensations. In order to check the previous assumption, one has to measure the filter on a larger band, typically three times the pass band.
This toolbox has been licensed to, and is currently used by Thales Alenia Space in Toulouse and Madrid, Thales airborne systems and Flextronics (two licenses). XLIM (University of Limoges) is a heavy user of PrestoHF among the academic filtering community and some free license agreements have been granted to the microwave department of the University of Erlangen (Germany) and the Royal Military College (Kingston, Canada).
RARL2
Participants : JeanPaul Marmorat, Martine Olivi [corresponding participant] .
RARL2 (Réalisation interne et Approximation Rationnelle L2) is a software for rational approximation (see Section 3.3.2.2 ) http://wwwsop.inria.fr/apics/RARL2/rarl2.html .
RARL2 computes a stable rational L2approximation of specified order to a given L2stable (L2 on the unit circle, analytic in the complement of the unit disk) matrixvalued function. This can be the transfer function of a multivariable discretetime stable system. RARL2 takes as input either:

discretized (uniformly distributed) values on the circle. In this case, a leastsquare criterion is used instead of the ${L}^{2}$ norm.
It thus performs model reduction in the first or the second case, and leans on frequency data identification in the third. For bandlimited frequency data, it could be necessary to infer the behavior of the system outside the bandwidth before performing rational approximation (see Section 3.2.2 ).
An appropriate Möbius transformation allows to use the software for continuoustime systems as well.
The method is a steepestdescent algorithm. A parametrization of MIMO systems is used, which ensures that the stability constraint on the approximant is met. The implementation, in Matlab, is based on statespace representations.
RARL2 performs the rational approximation step in the software tools PRESTOHF (see Section 3.4.3 ) and FindSources3D (see Section 3.4.2 ). It is distributed under a particular license, allowing unlimited usage for academic research purposes. It was released to the universities of Delft and Maastricht (the Netherlands), Cork (Ireland), Brussels (Belgium), Macao (China) and BITSPilani Hyderabad Campus (India).
Sollya
Participant : Sylvain Chevillard [corresponding participant] .
This software is developed in collaboration with Christoph Lauter (LIP6) and Mioara Joldeş (LAAS).
Preliminary remark: The coming of Sylvain Chevillard in the team in 2010 resulted in Apics hosting a research activity in certified computing, centered on the software Sollya. On the one hand, Sollya is an Inria software which still requires some tuning to a growing community of users. On the other hand, approximationtheoretic methods at work in Sollya are potentially useful for certified solutions to constrained analytic problems described in Section 3.3.1 . However, developing Sollya is not a longterm objective of Apics.
Sollya is an interactive tool where the developers of mathematical floatingpoint libraries (libm) can experiment before actually developing code. The environment is safe with respect to floatingpoint errors, i.e. the user precisely knows when rounding errors or approximation errors happen, and rigorous bounds are always provided for these errors.
Among other features, it offers a fast Remez algorithm for computing polynomial approximations of real functions and also an algorithm for finding good polynomial approximants with floatingpoint coefficients to any real function. As well, it provides algorithms for the certification of numerical codes, such as Taylor Models, interval arithmetic or certified supremum norms.
It is available as a free software under the CeCILLC license at http://sollya.gforge.inria.fr/ .