## Section: Application Domains

### Signal processing

Due to numerous applications (e.g. sensor network, mobile robots), sources and sensors localization has intensively been studied in the literature of signal processing. The *anchor position self calibration problem* is a well-known problem
which consists in estimating the positions of both the moving sources and a set of fixed sensors (anchors) when only the distance information between the points from the different sets is available. The position self-calibration problem is a particular case of the *Multidimensional Unfolding* (MDU) problem for the Euclidean space of dimension 3. In the signal processing literature, this problem is attacked by means of optimization problems (see [65] and the references therein). Based on computer algebra methods for polynomial systems, we have recently developed a new approach for the MDU problem which yields closed-form solutions and a very efficient algorithm for the estimation of the positions [68] based only on linear algebra techniques. This first result, done in collaboration with Dagher (Inria Chile) and Zheng (DEFROST, Inria Lille), yielded a recent patent [67]. This result advocates for the study of other localization problems based on the computational polynomial techniques developed in OURAGAN.

In collaboration with *Safran Tech* (Barau, Hubert) and Dagher (Inria Chile), a symbolic-numeric study of the new *multi-carrier demodulation method* [92] has recently been initiated. *Gear fault diagnosis* is an important issue in aeronautics industry since a damage in a gearbox, which is not detected in time, can have dramatic effects on the safety of a plane. Since the vibrations of a spur gear can be modeled as a product of two periodic functions related to the gearbox kinematic,
it is proposed to recover each function from the global signal by means of an optimal reconstruction problem which, based on Fourier analysis, can be rewritten as
${\mathrm{argmin}}_{u\in {\u2102}^{n},{v}_{1},{v}_{2}\in {\u2102}^{m}}\parallel M-u\phantom{\rule{0.166667em}{0ex}}{v}_{1}^{\u2606}-D\phantom{\rule{0.166667em}{0ex}}u\phantom{\rule{0.166667em}{0ex}}{v}_{2}^{\u2606}{\parallel}_{F}$,
where $M\in {\u2102}^{n\times m}$ (resp. $D\in {\u2102}^{n\times n}$) is a given matrix with a special shape (resp. diagonal matrix), $\parallel \xb7{\parallel}_{F}$ is the Frobenius norm, and ${v}^{\u2606}$ is the Hermitian transpose of $v$. We have recently obtained closed-form solutions for the exact problem, i.e., $M=u\phantom{\rule{0.166667em}{0ex}}{v}_{1}^{\u2606}+D\phantom{\rule{0.166667em}{0ex}}u\phantom{\rule{0.166667em}{0ex}}{v}_{2}^{\u2606}$,
which is a polynomial system with parameters. This first result gives interesting new insides for the study of the non-exact case, i.e. for the above optimization problem.

Our expertise on *algebraic parameter estimation problem*, developed in the former Non-A project-team (Inria Lille), will be further developed. Following this work [84], the problem consists in estimating a set $\theta $ of parameters of a signal $x(\theta ,t)$ $-$ which satisfies a certain dynamics $-$ when the signal
$y\left(t\right)=x(\theta ,t)+\gamma \left(t\right)+\varpi \left(t\right)$ is observed, where $\gamma $ denotes a structured perturbation and $\varpi $ a noise. It has been shown that $\theta $ can sometimes be explicitly determined by means of closed-form expressions using iterated integrals of $y$. These integrals are used to filter the noise $\varpi $. Based on a combination of algebraic analysis techniques (rings of differential operators), differential elimination theory (Gröbner basis techniques for Weyl algebras), and operational calculus (Laplace transform, convolution), an algorithmic approach to algebraic parameter estimation problem has been initiated in [125] for a particular type of structured perturbations (i.e. bias) and was implemented in the `Maple` prototype `NonA` . The case of a general structured perturbation is still lacking.