PhD Position F/M Stable approximation of solutions of wave propagation problems in harmonic regime

March 31, 2023
Offerd Salary:Negotiation
Working address:N/A
Contract Type:Other
Working Time:Negotigation
Working type:N/A
Ref info:N/A

2023-05782 - PhD Position F/M Stable approximation of solutions of wave propagation problems in harmonic regime

Contract type : Fixed-term contract

Level of qualifications required : Graduate degree or equivalent

Fonction : PhD Position

Level of experience : Recently graduated


This thesis will take place within the Inria team-project ALPINES, a joint research group between the Inria center of Paris and the Laboratoire Jacques- Louis Lions at Sorbonne University. The team's activities focus on high performance scientific computing, more precisely the design, theoretical analysis and implementation of algorithms for the numerical simulation of complex physical phenomena. The numerical methods developed aim to exploit modern architectures of supercomputers and are validated on real-life industrial test cases.

During this thesis, we will be interested in time-harmonic wave propagation problems, as modeled by the Helmholtz equation for example. Among the existing numerical schemes, Trefftz methods use particular solutions of the equation considered locally in each mesh cell. This allows to reduce the number of degrees of freedom to reach a given precision, compared to standard approaches. Moreover, many properties of the solutions of the equation are naturally preserved by the approximation. The counterpart is a high numerical instability which is due to the inherent redundancy between the elements of the approximation sets. The main objective of the thesis is to build sets of particular solutions that provide a good quality of approximation without suffering from numerical instability.

For time-harmonic acoustic wave propagation problems, plane waves form a family of particular solutions with many advantageous properties. However, these waves lead to strong numerical instability. It has recently been shown that the enrichment of the approximation sets by evanescent plane waves allows to stabilize the algorithms. The theoretical analysis of the approximation properties of these new sets is however reduced so far to simple geometries (the unit ball) and observed empirically in more complex geometries. In addition, the construction in practice of sets of adequate evanescent plane waves rests on incomplete theoretical bases.


Assignments : A first direction of research will consist in extending the available theoretical analysis to domains with more complex geometries. On the model of recent work, this axis will exploit tools from approximation theory and harmonic analysis in order to first prove new approximation results of solutions of time-harmonic acoustic wave propagation problems using evanescent plane waves, with a theoretical guarantee of stability. Several generalizations will then be considered, for example to other types of particular solutions, such as fundamental solutions, or other partial differential equations (Maxwell equations, linear elasticity).

Secondly, the new approximation sets constructed during the first axis will be exploited in numerical schemes such as discontinuous Galerkin methods. The numerical analysis of the method will be conducted and the implementation can be carried out in the FreeFem++ software, developed in particular within the Inria project-team ALPINES. The challenge is to propose high-performance algorithms, able to leverage modern supercomputer architectures to solve large-scale industrial test cases. To this end, it will be necessary to propose and study the convergence of iterative linear solvers, for instance in the case of a coupling of the numerical scheme with domain decomposition methods.

For a better knowledge of the proposed research subject : A survey on Trefftz methods is available here, and a recent work which follows the proposed approach is available here.

Collaboration : The thesis will supervised by Emile Parolin (Inria) and Bruno Després (LJLL, Sorbonne Université). Additionally, he recruited person will work in close connection with the members of the Inria project-team ALPINES and more generally with the members of the Inria center of Paris and the members of the Laboratoire Jacques-Louis Lions at Sorbonne University. Scientific discussions with Andrea Moiola from the University of Pavia (Italy) are also planned.

Main activities

Main activities :

  • Research in applied mathematics;
  • Implementation of the proposed algorithms;
  • Redaction of scientific articles;
  • Participation to seminars and conferences.
  • Additional activities :

  • Participation in the social and administrative life of Inria Paris and Laboratoire Jacques-Louis Lions;
  • Participation (optional) in the courses of Laboratoire Jacques-Louis Lions.
  • Skills

    Technical skills and level required : Master degree in mathematics or applied mathematics (partial differential equations, numerical analysis, approximation theory)

    Languages : English and/or French

    Benefits package
  • Subsidized meals
  • Partial reimbursement of public transport costs
  • Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
  • Possibility of teleworking and flexible organization of working hours
  • Professional equipment available (videoconferencing, loan of computer equipment, etc.)
  • Social, cultural and sports events and activities
  • General Information
  • Theme/Domain : Numerical schemes and simulations Scientific computing (BAP E)

  • Town/city : Paris

  • Inria Center : Centre Inria de Paris
  • Starting date : 2023-06-01
  • Duration of contract : 3 years
  • Deadline to apply : 2023-03-31
  • Contacts
  • Inria Team : ALPINES
  • PhD Supervisor : Parolin Emile / [email protected]
  • About Inria

    Inria is the French national research institute dedicated to digital science and technology. It employs 2,600 people. Its 200 agile project teams, generally run jointly with academic partners, include more than 3,500 scientists and engineers working to meet the challenges of digital technology, often at the interface with other disciplines. The Institute also employs numerous talents in over forty different professions. 900 research support staff contribute to the preparation and development of scientific and entrepreneurial projects that have a worldwide impact.

    Instruction to apply

    Defence Security : This position is likely to be situated in a restricted area (ZRR), as defined in Decree No. 2011-1425 relating to the protection of national scientific and technical potential (PPST).Authorisation to enter an area is granted by the director of the unit, following a favourable Ministerial decision, as defined in the decree of 3 July 2012 relating to the PPST. An unfavourable Ministerial decision in respect of a position situated in a ZRR would result in the cancellation of the appointment.

    Recruitment Policy : As part of its diversity policy, all Inria positions are accessible to people with disabilities.

    Warning : you must enter your e-mail address in order to save your application to Inria. Applications must be submitted online on the Inria website. Processing of applications sent from other channels is not guaranteed.

    From this employer

    Recent blogs

    Recent news