Paid Internship

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

**Type de contrat**: CDD

**Niveau de diplôme exigé**: Bac + 5 ou équivalent

**Fonction**: Stagiaire de la recherche

**Contexte et atouts du poste**:
The objective of NumPEx is to develop an exascale software stack, and the ExaMA project is dedicated to the mathematical algorithms to achieve this. The Inria MAKUTU project specilalizes in inverse problems for wave propagation problems and the theoretical and numerical methods needed for their solution

**Mission confiée**:
In an inverse problem, we seek the (unknown) model parameters from (known) measurements of the system’s solution. The direct system—in our case a partial differential equation with its boundary and initial conditions— can be considered as an operator from parameter space into data/observation space.

The inverse problem, from data space into parmater space, is ill-posed [1, 2].

The deterministic approach for solving the inverse problem consists of minimizing a cost function that expresses the error, or mismatch, between the model predictions and the measured observations, or data. We usually add a
regularization term (Tikhonov approach) to penalize solutions that are too oscillatory. This nonlinear minimization is usually performed by a method from the quasi-Newton family [3].

Bayes’ Theorem provides a framework for the solution of a more general inverse problem, where we seek the posterior probability distribution of a function, given prior knowledge of its distribution and measurements of an observed quantity that depends on the unknown, or hidden function. Mathematically, the theorem quantifies a posterior probability distribution (ppd) as a function of an a priori distribution that captures any previous knowledge about the parameters, and a likelihood function that is obtained by solving the direct mode for given parameter values.

The resulting a posteriori law is both the solution of the inverse problem and provides a complete quantification of the uncertainty in this solution. The numerator, being the product of two positive terms, less than one, implies that the result of Bayesian inversion produces a reduction of the uncertainty that gives probability distributions with smaller variance. The posterior distribution is usually simulated using Markov Chain Monte Carlo (MCMC) methods [2, 4].

**Principales activités**:
In this internship, we want to study, both theoretically and numerically, the solution of an inverse problem for wave propagation. A simple case is described by the acoustic wave equation, in a one-dimensional domain,
with varying sound speed function over the domain. This case can by easily generalized to more realistic wave propagation problems.

1. Based on [5], formulate the Bayesian inverse problem (BIP) for the wave equation.
2. Give all details of the existence, and convergence properites of the BIP solution.
3. Propose a MCMC method [2, 4] for solving the BIP.
An alternative approach for solving the BIP, is to use an ensemble Kalman filter (EnKF) [1, 2]. This method is known as ensemble Kalman inversion, and was first formulated in [6].
4. Based on [6], formulate the EKI approach for the wave equation.
5. Give all details of the existence, and convergence properites of the EKI solution.
6. Propose a EnKF method [2] for solving the EKI problem.

Both MCMC and EKI are methods that can be readily parallelized. This is the ultimate goal of the ExaMA project: to develop a highly parallelized library for the soution of inverse problems in wave propagation.

References
[1] M. Asch, M. Bocquet, M. Nodet. Data Assimilation: Methods, Algorithms and Applications. SIAM. 2016.
[2] M. Asch. A Toolbox for Digital Twins: from Model-Based to Data-Driven.
SIAM. 2022.
[3] J. Nocedal, S. Wright. Numerical Optimization. Springer. 2006.
[4] A. Gelman, J. Carlin, H. Stern, D. Dunson, A. Vehtari, D. Rubin. Bayesian
Data Analysis, 3rd edition. CRC Press, 2014.
[5] A. Stuart. (2010). Inverse problems: A Bayesian perspective. Acta Numerica,
19, 451-559. doi:10.1017/S0962492910000061.
[6] M. A.Iglesias, K. J. H. Law and A. M. Stuart. Ensemble Kalman methods
for inverse problems. Inverse Problems 29 (2013).

**Compétences**:
The internship will be co-supervised by Prof. Mark Asch, who has a long experience in data assimilation and inverse problems.

**Avantages**:

- Subsidized meals
- Partial reimbursement of public transport costs
- 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
- Access to vocational training
- Social security coverage

**Informations générales**:

- **Thème/Domaine**: Sciences de la planète, de l'environnement et de l'énergie
Calcul Scientifique (BAP E)
- **Ville**: PAU
- **Centre Inria**: Centre Inria de l'université de Bordeaux
- **Date de prise de fonction souhaitée**: 2025-04-01
- **Durée de contrat