Phd Position F - M Efficient Numerical Solvers For Wave Propagation Problems In Multiscale Random Media And Application To Quantitative Ultrasound Medical Imaging H/F INRIA

PhD Position F/M Efficient numerical solvers for wave propagation problems in multiscale random media and application to quantitative ultrasound medical imaging
Le descriptif de l'offre ci-dessous est en Anglais
Type de contrat : CDD

Niveau de diplôme exigé : Bac +5 ou équivalent

Fonction : Doctorant

Niveau d'expérience souhaité : Jeune diplômé

Contexte et atouts du poste

Context. Ultrasound imaging is a non-invasive, portable, and cost-effective medical imaging technique widely used worldwide. Recent advances in sensor manufacturing and access to significant computational resources have shifted ultrasound imaging research towards improving the reconstruction algorithms and the underlying mathematical models. Conventional ultrasound imaging algorithms, such as sum-and-delay or Kirchhoff migration, assume that the speed of sound in the medium is constant and known. However, this assumption fails in medical applications, for example in the presence of adipose tissue, leading to significant image distortion.Anactive research directionis therefore concerned with the development of quantitative ultrasound imaging algorithms that not only provide an image of the medium's reflectivity but also reconstruct the local speed of sound within the medium, thereby optimizing image quality and possible diagnostics.

In this context, our main goal is to develop numerical simulation tools to solve the direct problem, namely simulate numerically the propagation of acoustic waves in the time-harmonic regime, modelled by the heterogeneous Helmholtz equation with variable coefficients. Numerical simulations are indeed required to validate the asymptotic mathematical models ofwave propagation in biological tissues.They are alsowidely used to generate synthetic data to test the reconstruction algorithms without the need to perform expensive experiments.

Challenges. In our ultrasound imaging application, the typical wavelength of incident waves is approximately 0.5 mm (frequency ~3 MHz, assuming a sound speed of 1500 m/s). The computational domain size is on the order of 100 mm, which corresponds to hundreds of wavelengths in each direction,making the problem high-frequency. In addition, the echoes measured insoft tissuescome from numerous unresolved scatterers, a phenomenon known as speckle. These small, randomly distributed reflectors in the medium are on the scale of biological cells (5 µm to 50 µm), making them much smaller than the wavelength. This introduces an additional, much finer scale and makes theproblem multi-scale. High-frequency and multi-scale features imply that thedirect simulation of the full problem is very challenging. To put things into perspective, preliminary 2D tests at real-world scales required nearly 500 million degrees of freedom and utilized 80% of the nodes (~1,000 CPUs) on the CLEPS cluster at Inria Paris. The same problem in 3D is far beyond current capabilities and exceeds the state-of-the-art in direct numerical simulation capacity.

After discretization, the models lead to large-scale, indefinite, and non-Hermitian linear systems. These systems cannot be inverted using classical methods and require the implementation of preconditioned iterative solvers, such as domain decomposition methods, which can run on the parallel architectures of supercomputers. To achieve efficient methods, at least two resolution levels must be implemented. One of the main challenges we want to address is constructing an appropriate coarse space for the second level, capable of capturing the relevant features of the solution across the different scales. While several methods have been proposed for constructing coarse spaces, the case of indefinite wave-type problems remains particularly challenging.Defining the coarse space is not the only challenge, in factsolving the global coarse problem quickly becomes the limiting factor and we intend to leverage the properties of the problem to propose innovative methods to reduce this computational bottleneck.

Position.Funding is available for one PhD position (3 years), possibly starting with a research internship of up to 6 months. Starting date is flexible, but not earlier than 1st of February 2026. Thisposition is located within the Inria team-project ALPINES, a joint research group between the Inria Research center of Paris and the Laboratoire Jacques-Louis Lions at Sorbonne University. The projectwill be jointly supervised by Laure Giovangigli (Inria team project Poems) and Pierre Millien (Institut Langevin) with expertise in medical imaging algorithms and the underlying mathematical models, together with Emile Parolin (Inria team-projectAlpines)with knowledgein high-performance computing and large scale numerical simulations.

Mission confiée

With the help of the supervisors, themain tasks of the successful candidate will involve:

- conducting an extensive literature review on domain decomposition methods for solving high-frequency and multi-scale heterogeneous Helmholtz problem, as well as on reconstruction algorithms used in real-time ultrasound medical imaging;
- proposing new approaches to construct coarse-spaces suitable for the problem at hand, and designing efficient methods for solving the coarse space problem in two-level domain decomposition methods;
- proving well-posedness and convergence of the algorithms proposed using tools from analysis of PDEs, numerical analysis and numerical linear algebra;
- implementing and conducting numerical testing to validate on real-world test cases the proposed approach;
- performing reconstruction of sound-of-speed maps from synthetic numerical data.

Principales activités

Main activities:

- Research in applied mathematics (analysis of PDEs, numerical analysis, scientific computing);
- Implementation and testing of the proposed algorithms;
- Redaction of scientific articles;
- Participation to research seminars, national and international conferences.

Additional activities:

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

Compétences

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

Languages : English and/or French.

Avantages

- 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
- Compulsory health insurance as from May 2026