Paid Internship

Le descriptif de l’offre ci-dessous est en Anglais_

**Type de contrat**: Stage

**Niveau de diplôme exigé**: Bac + 4 ou équivalent

**Fonction**: Stagiaire de la recherche

**Niveau d'expérience souhaité**: Jeune diplômé

**Contexte et atouts du poste**:
The Makutu project team specializes in large-scale simulations applied to the reconstruction of complex media, used to gain a better understanding of the internal dynamics of environments that are difficult or even impossible to probe. To this end, it develops advanced numerical methods that are integrated into open-source platforms deployed on state-of-the-art HPC environments.

**Mission confiée**:
Seismic imaging is a critical technology for subsurface characterization in exploration geophysics, CO₂ storage monitoring, and underground hazard assessment. The industry increasingly relies on **full waveform inversion (FWI)** and **reverse time migration (RTM)**, which require repeated and highly accurate solutions of the acoustic (and elastic) wave equation on large-scale 2D/3D domains.

Finite difference solvers dominate current industrial codes, but their limitations are well known:

- Poor performance on complex geometries and topography.
- High numerical dispersion for high-frequency modeling.
- Difficulty in coupling with adaptive meshes.

High-order **spectral finite element methods (SFEM)** offer superior accuracy per degree of freedom and are naturally suited to **HPC architectures** (CPU/GPU clusters). Two main Galerkin formulations exist:

- **Continuous Galerkin (CG-SFEM)**: Memory-efficient and fast in homogeneous regions.
- **Discontinuous Galerkin (DG-SFEM)**: Robust near discontinuities, complex geology, and allows local adaptivity, but at higher computational cost.

A **hybrid CG/DG approach** could combine the efficiency of CG with the robustness of DG, yielding a competitive industrial solver that reduces time-to-solution while maintaining seismic accuracy requirements.

This internship is a great opportunity to develop the following skills:

- Advanced knowledge of **numerical methods for wave propagation**.
- Experience with **high-order Galerkin spectral methods** (CG and DG).
- Hands-on **HPC programming** (MPI, OpenMP, CUDA).
- Understanding of **seismic workflows** (modeling, FWI, RTM).

This will be facilitated by collaborating with members of Makutu team.

The expected deliverables are:

- **Prototype hybrid solver** (validated, scalable, with documented source code).
- **Benchmark report** comparing CG, DG, and hybrid CG/DG on seismic test cases.
- **HPC scalability study** with recommendations for evolution.
- **Final internship report & presentation** with emphasis on industrial impact.

**Principales activités**:
The internship will deliver a **proof-of-concept hybrid CG/DG spectral finite element solver** for the acoustic wave equation. The developments will extend the SFEM wave equation platform based on CG SFEM formulation to DG SFEM formulation. The different tasks will be:

- **Prototype implementation** of a hybrid CG/DG-SFEM solver targeting 3D seismic wave propagation in complex media
- **Quantitative evaluation** of accuracy, dispersion, and stability versus existing solvers.
- **HPC performance benchmarking** (MPI, GPU acceleration) with focus on scalability to large meshes.
- **Integration scenarios** for seismic imaging workflows (forward modeling, FWI/RTM kernels).
- **Mathematical & Numerical Formulation**:Develop the variational form of the acoustic wave equation; Implement DG-SFEM solvers (high-order Lagrange basis with Gauss-Lobatto quadrature); Design CG/DG interface treatment.
- **Prototype Development**:Build the solver in the wave equation platform developed by Makutu. This platform is developed in C++ and uses Kokkos to expose the computational kernels to GPUS
- **Benchmarking and Validation**:Validate against reference solutions; Test seismic benchmarks; Compare with pure CG SFEM implementation.

The expected impact is:

- **Reduced computational cost**: Hybrid approach may lower CPU/GPU requirements for large-scale seismic modeling.
- **Improved imaging accuracy**: High-order spectral methods reduce numerical dispersion, crucial for high-frequency content in FWI.
- **HPC readiness**: The solver design will be optimized for modern heterogeneous clusters (CPU/GPU).
- **Innovation opportunity**: Potential integration into next-generation industrial seismic imaging platforms.

Key references:

- Komatitsch, D., & Tromp, J. (2002). Spectral-element simulations of global seismic wave propagation—I. Validation. _J. Int._
- De Basabe, J. D., & Sen, M. K. (2007). Grid dispersion and stability criteria of some common finite‐element methods for acoustic and elastic wave equations. _Geophysics_.
- Chaljub, E., et al. (2007). Spectral element analysis in seismology. _Advances in Geophysics_.
- Abdi, R., et al. (2021). GPU-accelerated spectral-element method for seismic wave propagation. _Computers & Geosciences_.

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