Alternating Projections

**Alternating Projections : an Application to Road Design**:

- Réf
- **ABG-110286**
- Stage master 2 / Ingénieur- Durée 4 mois- Salaire net mensuel 500- 17/01/2023- Université de Perpignan Via-Domitia- Lieu de travail- Perpignan Occitanie France- Champs scientifiques- Mathématiques
- Mots clés- Alternating projections, intrepid operator, relaxed operator-
- 17/02/2023**Établissement recruteur**:
**Site web**:
LAMPS is a multidisciplinary laboratory which regroups researchers in mathematics, physics and computer science.

**Description**:
The road design problem consists in automatically computing a road between several specific stations according to certain constraints.

The road design problems is divided in two subproblems : a two dimensional one which is mainly focused on the turns of the road (horizontal problem); the other in dimension one devoted to the elevation of the road (vertical problem). The second subproblem is the one we want to study. Usually, the road design vertical problem arrives with a first guess called a ground profile. The road design vertical problem becomes a road alignment problem and the proposed piecewise affine solution must be as close as possible to the ground profile.To follow the ground profile provides constraints to satisfy.

First, the road has to cross some specific stations or roads. These crossing point constraints lead to piecewise affine interpolations. The other constraints come from specifications imposed by civil engineers and specify lower and upper bounds over slopes and differences between two successive slopes. A simple approach consists in computing a first piecewise affine solution as an interpolation of crossing points. This is the step after where the projection methods are important. The first piecewise affine solution may not satisfy the slope constraints and we have to project the piecewise affine interpolation onto the slope constraints set. In simple cases, the slope constraints set are closed and convex.
- The objectives of the internship is to mathematically and numerically understand the improvments obtained by the use of intrepid and relaxed operators in comparaison with Dijkstra's algorithm.
- A strong background in Hilbert and convex analysis;
- Programming skills in Python, Matlab or C are recommended but not mandatory.

**Profil**:
**Prise de fonction**:

- 01/03/2023