Cerfacs ALGO Team Scientific Activity Report for the
year 2025

Luciano Drozda Dantas Martins, Luc Giraud, Selime Gürol, Carola Kruse,
Paul Mycek, Anthony Trevor Weaver

► To cite this version:

Luciano Drozda Dantas Martins, Luc Giraud, Selime Gürol, Carola Kruse, Paul Mycek, et al. Cerfacs ALGO Team Scientific Activity Report for the year 2025. TR/PA/26/6, Cerfacs - Toulouse, France. 2026. hal-05519493

HAL Id: hal-05519493

https://hal.science/hal-05519493v1

Submitted on 19 Feb 2026

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Cerfacs ALGO Team Scientific Activity Report

2025

TR/RA/26/6

<figure><img src="image_2.png" /></figure>
# Contents

<table>
  <tbody>
    <tr>
      <td>1</td>
      <td>Team members</td>
      <td>3</td>
    </tr>
    <tr>
      <td>1.1</td>
      <td>Senior researchers</td>
      <td>3</td>
    </tr>
    <tr>
      <td>1.2</td>
      <td>Postdoctoral associates</td>
      <td>3</td>
    </tr>
    <tr>
      <td>1.3</td>
      <td>PhD students</td>
      <td>3</td>
    </tr>
    <tr>
      <td>1.4</td>
      <td>Engineers</td>
      <td>3</td>
    </tr>
    <tr>
      <td>1.5</td>
      <td>Interns and apprentices</td>
      <td>3</td>
    </tr>
    <tr>
      <td>1.6</td>
      <td>External consultants</td>
      <td>4</td>
    </tr>
    <tr>
      <td>1.7</td>
      <td>Visitors</td>
      <td>4</td>
    </tr>
    <tr>
      <td>2</td>
      <td>Overall objectives</td>
      <td>5</td>
    </tr>
    <tr>
      <td>3</td>
      <td>Application domains</td>
      <td>5</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Highlights of the year</td>
      <td>6</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Software development, platforms, open data</td>
      <td>7</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Global diffusion</td>
      <td>7</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>GMGPolar</td>
      <td>7</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>JADE</td>
      <td>7</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>JuCUDA</td>
      <td>8</td>
    </tr>
    <tr>
      <td>5.5</td>
      <td>NEMOVAR</td>
      <td>8</td>
    </tr>
    <tr>
      <td>5.6</td>
      <td>OOPS</td>
      <td>8</td>
    </tr>
    <tr>
      <td>5.7</td>
      <td>PhyDLL</td>
      <td>9</td>
    </tr>
    <tr>
      <td>5.8</td>
      <td>PYTHIA-DA</td>
      <td>9</td>
    </tr>
    <tr>
      <td>6</td>
      <td>New results</td>
      <td>9</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Toward Nonlinear Quantum Numerical Methods for Engineering Applications</td>
      <td>9</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>On some orthogonalization schemes in tensor train format</td>
      <td>10</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>A note on TT-GMRES for the solution of parametric linear systems</td>
      <td>10</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Solving eigenvalue problems in high dimensions using contour integration and Tensor Train format</td>
      <td>10</td>
    </tr>
    <tr>
      <td>6.5</td>
      <td>A Tensor train solver for the magnetic moment method</td>
      <td>11</td>
    </tr>
    <tr>
      <td>6.6</td>
      <td>Generalized Golub-Kahan bidiagonalization for nonsymmetric saddle-point systems</td>
      <td>11</td>
    </tr>
    <tr>
      <td>6.7</td>
      <td>A note on the partial convergence management for the solution of symmetric linear systems with multiple right-hand sides</td>
      <td>11</td>
    </tr>
    <tr>
      <td>6.8</td>
      <td>A scalable and parameter-robust preconditioner for a second gradient of dilation regularization applied to a mechanics problem</td>
      <td>12</td>
    </tr>
    <tr>
      <td>6.9</td>
      <td>Neural network preconditioning: a case study for the solution of the parametric Helmholtz equation</td>
      <td>12</td>
    </tr>
    <tr>
      <td>6.10</td>
      <td>Memory-and compute-optimized geometric multigrid GMGPolar for curvilinear coordinate representations -Applications to fusion plasma</td>
      <td>12</td>
    </tr>
    <tr>
      <td>6.11</td>
      <td>Complexity analysis and scalability of a matrix-free extrapolated geometric multigrid solver for curvilinear coordinates representations from fusion plasma applications</td>
      <td>12</td>
    </tr>
    <tr>
      <td>6.12</td>
      <td>Fault-tolerant numerical iterative algorithms at scale</td>
      <td>13</td>
    </tr>
    <tr>
      <td>6.13</td>
      <td>A filtered multilevel Monte Carlo method for estimating the expectation of cell-centered discretized random fields</td>
      <td>13</td>
    </tr>
    <tr>
      <td>6.14</td>
      <td>Multilevel Monte Carlo methods for ensemble variational data assimilation</td>
      <td>13</td>
    </tr>
    <tr>
      <td>6.15</td>
      <td>Mixed precision sketching for least-squares problems and its application in GMRES-Based iterative refinement</td>
      <td>14</td>
    </tr>
    <tr>
      <td>6.16</td>
      <td>An introduction to solving the least-squares problem in variational data assimilation</td>
      <td>14</td>
    </tr>
    <tr>
      <td>6.17</td>
      <td>A comparison of mixed precision iterative refinement approaches for least-squares problems</td>
      <td>14</td>
    </tr>
    <tr>
      <td>6.18</td>
      <td>Block alpha-circulant preconditioners for all-at-once diffusion-based covariance operators</td>
      <td>15</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>6.19</td>
      <td>Convergence analysis of overlapping domain decomposition preconditioners for nonlinear problems</td>
      <td>15</td>
    </tr>
    <tr>
      <td>6.20</td>
      <td>Improvements to the Met Office's global ocean-sea ice forecasting system including model and data assimilation changes</td>
      <td>15</td>
    </tr>
    <tr>
      <td>6.21</td>
      <td>On the impact of observation error correlations in data assimilation, with application to along-track altimeter data</td>
      <td>16</td>
    </tr>
    <tr>
      <td>6.22</td>
      <td>Impact of ensemble-based hybrid background-error covariances in ECMWF's next-generation ocean reanalysis system*</td>
      <td>16</td>
    </tr>
    <tr>
      <td>6.23</td>
      <td>Robustness and reliability of state-space, frame-based modeling for thermoacoustics</td>
      <td>17</td>
    </tr>
    <tr>
      <td>6.24</td>
      <td>Code generation and performance engineering for matrix-free finite element methods on hybrid tetrahedral grids</td>
      <td>17</td>
    </tr>
    <tr>
      <td>6.25</td>
      <td>Training loss matters: exploring new fit-for-purpose loss combinations for statistical downscaling</td>
      <td>17</td>
    </tr>
    <tr>
      <td><b>7</b></td>
      <td><b>Partnerships and cooperations</b></td>
      <td><b>18</b></td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>European Initiatives</td>
      <td>18</td>
    </tr>
    <tr>
      <td>7.1.1</td>
      <td>EoCoE-III</td>
      <td>18</td>
    </tr>
    <tr>
      <td>7.1.2</td>
      <td>ERGO2 (Copernicus Climate Change Service)</td>
      <td>19</td>
    </tr>
    <tr>
      <td>7.1.3</td>
      <td>NEMOVAR software development (Memorandum of Understanding)</td>
      <td>19</td>
    </tr>
    <tr>
      <td>7.1.4</td>
      <td>PVDAP</td>
      <td>19</td>
    </tr>
    <tr>
      <td>7.1.5</td>
      <td>ROSAS</td>
      <td>20</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>National Partnerships</td>
      <td>20</td>
    </tr>
    <tr>
      <td>7.2.1</td>
      <td>CECI</td>
      <td>20</td>
    </tr>
    <tr>
      <td>7.2.2</td>
      <td>Concace</td>
      <td>21</td>
    </tr>
    <tr>
      <td>7.3</td>
      <td>Collaborative projects</td>
      <td>21</td>
    </tr>
    <tr>
      <td>7.3.1</td>
      <td>Maison du Quantique Occitanie</td>
      <td>21</td>
    </tr>
    <tr>
      <td>7.3.2</td>
      <td>ANITI - EXPLEARTH Chair</td>
      <td>21</td>
    </tr>
    <tr>
      <td>7.3.3</td>
      <td>ANITI - RELEO Chair</td>
      <td>22</td>
    </tr>
    <tr>
      <td>7.3.4</td>
      <td>ANR PHLUSIM</td>
      <td>23</td>
    </tr>
    <tr>
      <td>7.3.5</td>
      <td>IRT Saint Exupéry - R-Evol project</td>
      <td>23</td>
    </tr>
    <tr>
      <td>7.3.6</td>
      <td>LEFE MANU: ADA</td>
      <td>23</td>
    </tr>
    <tr>
      <td>7.3.7</td>
      <td>LEFE MANU: DOGODA</td>
      <td>24</td>
    </tr>
    <tr>
      <td>7.3.8</td>
      <td>SANTANA</td>
      <td>24</td>
    </tr>
    <tr>
      <td><b>8</b></td>
      <td><b>Dissemination</b></td>
      <td><b>24</b></td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>Promoting scientific activities</td>
      <td>24</td>
    </tr>
    <tr>
      <td>8.1.1</td>
      <td>Scientific events: organisation</td>
      <td>24</td>
    </tr>
    <tr>
      <td>8.1.2</td>
      <td>Scientific events: selection</td>
      <td>25</td>
    </tr>
    <tr>
      <td>8.1.3</td>
      <td>Journal</td>
      <td>25</td>
    </tr>
    <tr>
      <td>8.1.4</td>
      <td>Invited talks</td>
      <td>25</td>
    </tr>
    <tr>
      <td>8.1.5</td>
      <td>Scientific expertise</td>
      <td>25</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>Teaching - Supervision - Juries - Educational and pedagogical outreach</td>
      <td>25</td>
    </tr>
    <tr>
      <td>8.2.1</td>
      <td>Supervision</td>
      <td>26</td>
    </tr>
    <tr>
      <td>8.2.2</td>
      <td>Thesis committees</td>
      <td>27</td>
    </tr>
    <tr>
      <td>8.2.3</td>
      <td>Juries</td>
      <td>27</td>
    </tr>
    <tr>
      <td><b>9</b></td>
      <td><b>Scientific publications of the year</b></td>
      <td><b>28</b></td>
    </tr>
    <tr>
      <td>9.1</td>
      <td>Journal articles</td>
      <td>28</td>
    </tr>
    <tr>
      <td>9.2</td>
      <td>Conferences</td>
      <td>29</td>
    </tr>
    <tr>
      <td>9.3</td>
      <td>Doctoral dissertations and habilitation theses</td>
      <td>30</td>
    </tr>
    <tr>
      <td>9.4</td>
      <td>Reports & preprints</td>
      <td>30</td>
    </tr>
  </tbody>
</table>
# 1 Team members

## 1.1 Senior researchers

* Luciano Drozda

* Selime Gürol

* Carola Kruse

* Paul Mycek (HDR)

* Anthony Weaver (HDR)

## 1.2 Postdoctoral associates

* Anthony Larroque, funded by EuroHPC Inno4scale

* Fernando Gonzalez, co-funded by IRIT and CIR Cerfacs

* Rachid El Montassir, funded by TotalEnergies IMPACT

* Ieva Daužickaitė, funded by CIR Cerfacs

* Olivier Goux, funded by EU Copernicus (C3S) contract (ERGO2 project)

* Julien Zylberman (until Dec. 2025, Senior since)

## 1.3 PhD students

* Olivier Goux (until Feb. 2025), funded by C3S (ERGO2)

* Oussama Mouhtal, funded by CIR Cerfacs and Polytechnique Montréal

* Aurélien Theret, funded by ANR PHLUSIM

* Sai Aakash Dasari, funded by H2020 EoCoE-III

* Théo Briquet (Inria), funded through Concace

* Mehdi El Ettaouchi (CIFRE EDF)

## 1.4 Engineers

* Youssef Achari-Berrada

* Nathan Aubert

## 1.5 Interns and apprentices

* Lina Kaddouri-Wisniewski, Master 2 (6 months), Telecom-Paris, *Analysis and optimization of scalar weights and allocation for statistical multilevel estimation*.

* Samuel Duarte dos Santos, Master 2 (6 months), IPSA.

* Emile Rouot, Master 1 (2 months), ENSEEIHT, *Automatic Differentiation and High Performance Computing: Towards hybrid Physics-AI solvers on GPU*.

* Thomas Lasserre, Master 1 (2 months), ENSIIE, *Automatic Differentiation and High Performance Computing: Towards hybrid Physics-AI solvers on GPU*.

* Yahia Ait Alla, Master 1 (2 months), ENSEEIHT, *Online learning of Homogeneous Isotropic Turbulence dynamics*.

* Eliot Martin, Master 2 (6 months), ISAE-Supaéro, *Randomized algorithms for data assimilation*.
## 1.6 External consultants

* Luc Giraud [Inria, scientific advisor]

* Ulrich Rüde

* Andrea Piacentini

## 1.7 Visitors

* Hélène Henon (Inria), 20–24 October 2025.

* PVDAP Network meeting, A. Lawless (University of Reading), J. Scott (University of Reading/RAL), J. Tabeart (University of Eindhoven), 12–16 May 2025.
## 2 Overall objectives

The Algo team at Cerfacs develops robust, efficient, and scalable numerical methods and designs their parallel implementation on modern scientific computing architectures. The diversity of backgrounds in applied mathematics within the team enables the Algo team to address a broad range of scientific topics. A long-standing and key research area is numerical linear algebra, with a particular emphasis on preconditioning and subspace methods for the solution of large-scale linear systems including saddle point problems, using multigrid approaches, randomized techniques, and mixed-precision computing, among others.

Data Assimilation is also an important research topic within the team, where expertise in numerical linear algebra and scientific computing is leveraged to develop methods and algorithms for variational data assimilation, ensemble-based approaches, and hybrid formulations. These developments are primarily motivated by large-scale, high-dimensional problems arising in Earth system modelling and forecasting.

The team's activities further extend to uncertainty quantification in high-dimensional settings, including the estimation and representation of error covariances, as well as multi-level and multi-fidelity approaches. The Algo team is also active in scientific machine learning (SciML), with contributions to differentiable solvers, AI-based numerical methods, and hybrid strategies combining data assimilation and machine learning.

More recently, the team has initiated research activities in quantum computing algorithms for fault-tolerant quantum machines, with a particular focus on algorithms for the solution of partial differential equations.

Much of the research is carried out in collaboration with other Cerfacs teams, as well as with research groups in academia, industry, and operational forecasting centres at the local, national, and international levels. The Algo team members contribute to the research activities of CECI (UMR with Cerfacs, CNRS and IRD) and the joint Inria team-project Concace between Airbus CR&T, Cerfacs and Inria.

## 3 Application domains

* *Data assimilation for ocean forecast initialization and retrospective analysis.* Research developments for this application primarily focus on improving the NEMOVAR DA software (see Section 5.5). In particular, Algo provides specialized expertise in covariance modelling and linear solvers. The development of implicit diffusion-based correlation models is a central focus of this work. Recent highlights include: (1) the development of a hybrid ensemble-based background-error covariance model and its application to ECMWF's global ocean reanalysis system; (2) various algorithmic refinements that have led to improvements to the UK Met Office's global ocean analysis system; and (3) development of an diffusion-based correlation model for representing observation errors. In addition, some new approaches for diffusion-based correlation modelling have emerged from fundamental research.

* *Data assimilation for weather forecast initialization.* Algo conducts research on core algorithmic aspects of variational data assimilation, with a focus on iterative minimization methods, preconditioning strategies, and multi-level methods for background-error covariance estimation. Many of the developments are integrated into the generic OOPS software (see Section 5.6), which is used operationally by major weather forecasting centres. In addition, specific implementations of some of these algorithms have been incorporated into the ocean DA software NEMOVAR.

* *Monitoring climate essential variables.* Essential Variables (EVs) are key indicators to adequately describe and monitor the evolution of Earth's changing climate. Accordingly, the regular and accurate mapping of variables such as Leaf Area Index, Land Surface Temperature, Evapotranspiration or Soil Moisture are becoming increasingly important for environmental applications. Satellite imagery is an efficient tool to effectively predict EVs trends, define present conditions,
and inform about their future evolution. Specifically, Sentinel satellite missions offer opportuni-
ties to provide accurate, timely and easily accessible information of the Earth surface. However,
extracting useful information from raw satellite images acquired by multimodal sensors, which
are nonstationary, not dense, multi-variate and have temporal gaps is challenging. The retrieval
of EV from noisy satellite observations is traditionally seen as an inverse modelling problem. Algo
contributes to the ANITI-RELEO project by developing advanced methods to predict EVs' future
states, combining machine learning techniques with data assimilation approaches to effectively
exploit noisy and sparse satellite observations.

*   **Fusion simulation.** In the gyrokinetic plasma simulations within the GyselaX framework, a high-dimensional kinetic model is coupled to a 2D Poisson-like equation for the electrostatic potential on poloidal tokamak cross sections. While these cross sections can initially be treated as circular domains, deformed geometries have been shown to provide more realistic and physically relevant results. Geometric multigrid methods tailored to curvilinear coordinates are, however, less commonly used in this context. Within the Algo team, we develop a specialized geometric multigrid solver using optimized line smoothers to enable parallel scalability. We employ a finite-difference discretization of the energy potential of the Poisson equation, leading to a symmetric system and enabling a matrix-free implementation with low memory usage.

*   **Aerodynamics.** The Algo team contributes to the further development and performance improvement of the CODA solver, which is primarily used for the simulation of aerodynamic phenomena. Profiling studies identified the linear solver stage within the Newton–Krylov framework as a computationally expensive part. Algo focuses on accelerating this linear solution phase, with the objective of significantly reducing the overall time to solution of the CFD simulations. The Algo team also contributes to the outreach of Cerfacs CFD solver AVBP in the era of AI by co-developing the Physics Deep Learning coupLer (PhyDLL) library, in collaboration with the COOP Cerfacs team (see Section 5.7). Using PhyDLL, the Algo team works on the development of data-driven subgrid-scale models and wall models for Large Eddy Simulations (LES) which outperform traditional models in industry-grade simulation settings. Algo also works on proposing data-driven numerical schemes for LES which would be robust to mesh and solution irregularities. This activity is supported by the implementation of scalable differentiable solvers using the automatic differentiation tool JADE (see Section 5.3) and the GPU porting tool Ju-CUDA (see Section 5.4), both developed within the Algo team.

4 Highlights of the year

* Launch of quantum computing research on algorithm design for physics simulations with the recruitment of: Julien Zylberman, initially as a postdoctoral scientist and recently promoted to permanent researcher; and Nathan Aubert, who joined as a research engineer and will begin his PhD on quantum algorithms in 2026.

* Anthony Weaver was appointed **ECMWF Fellow** in May 2025. The purpose of the Fellowship is “to foster and formalise links with individuals who are carrying out pioneering scientific and technical research in areas relevant to the strategic goals of ECMWF”.

* Olivier Goux was the recipient of one of five prizes from the **ISAE-SUPAERO Foundation** for the best PhD theses in 2025.

* The HCERES evaluation of the **CECI** team took place in November 2025 and resulted in very positive feedback. Selime Gürol was jointly responsible for the Methods and Data thematic area. S. Gürol, P. Mycek, A. Weaver and L. Drozda contributed to the preparation of the CECI activity report for the 2019–2025 period and the oral presentation in November.

* The Inria evaluation of the **Concace** joint team-project is ongoing with very positive reports on the scientific contributions from international referees from MIT, Lawrance Berkeley National
Lab and Safran Tech. This enables the Concace steering committee to approve the continuation of the project for the next 4 years once the complete Inria evaluation process will be eventually completed (early Summer 2026).

# 5 Software development, platforms, open data

## 5.1 Global diffusion

**Keywords:** correlation operator, diffusion

**Scientific Description:** This code consists of a Python implementation of a diffusion-based correlation operator in two dimensions. Beyond the core operator, it contains multiple wrappers and utility routines to facilitate quick experiments on different kinds of domains (rectangle, sphere, global ocean, etc). Experimental methods for the estimation of the normalisation factors based on multilevel Monte Carlo (MLMC) and convolutional neural networks (CNN) are included.

**Publications:** The diffusion operator is based on Weaver and Mirouze (2013). The CNN normalisation experiments are described in Skrunes et al. (2023). The MLMC normalisation experiments are described in Briant et al. (2025).

**Contact:** Olivier Goux

**Participants:** Olivier Goux, Jérémy Briant, Folke Skrunes

**URL:** The code is currently hosted at https://github.com/ogoux/global_diffusion.

## 5.2 GMGPolar

**Name:** Geometric Multigrid for curvilinear domains

**Keywords:** Geometric multigrid, curvilinear domains, plasma fusion

**Scientific Description:** GMGPolar offers a taylored Geometric Multigrid solver for geometries described by curvilinear coordinates (in its simplest case, by polar coordinates). GMGPolar uses an implicit extrapolation scheme to raise the convergence order. For nonstandard finite elements, this result has been shown theoretically and for nine-point finite difference stars, we observed up to convergence order four in all numerical simulations. This extrapolation scheme can be used likewise for other domains described by tensor- or product-format meshes in 2D. For curvilinear domains, suited smoothing schemes have been developed and implemented. GMGPolar offers a matrix and matrix-free version for most of its operators. Its original application is on cross-sectional views of realistic Tokamak geometries for fusion plasma simulations.

**Publications:** GMGPolar v2

**Contact:** Carola Kruse

**Participants:** Carola Kruse, Ulrich Ruede

**URL:** GMGPolar repository

## 5.3 JADE

**Name:** Julia Automatic Differentiation Engine

**Keywords:** Automatic Differentiation, Julia programming language, Source-to-source transformation

**Scientific Description:** JADE is a source-to-source transformation tool that provides the adjoint code of any scalar program written using the Julia programming language. It aims at helping to design differentiable solvers at ease.
**Contact:** Luciano Drozda

**Participants:** Luciano Drozda

**URL:** https://gitlab.com/cerfacs/jade

## 5.4 JuCUDA

**Name:** Julia CUDA kernel generator

**Keywords:** CUDA, Julia programming language, Source-to-source transformation

**Scientific Description:** JuCUDA is a source-to-source transformation tool that generates a CUDA-compatible implementation of any scalar program written using the Julia programming language. Along with JADE, it aims at helping to design scalable differentiable solvers at ease, targeting hybrid Physics-AI in high-dimensional settings.

**Contact:** Luciano Drozda

**Participants:** Luciano Drozda

**URL:** https://gitlab.com/cerfacs/jucuda

## 5.5 NEMOVAR

**Name:** NEMO Variational Data Assimilation software

**Keywords:** Variational data assimilation, Ocean state estimation and prediction

**Scientific Description:** NEMOVAR is a FORTRAN-based variational data assimilation (DA) software designed for use with the NEMO ocean model. Cerfacs is the original architect of NEMOVAR and remains one its main developers. It is developed and maintained by a four-partner consortium comprising Cerfacs, ECMWF (since 2002), UK Met Office (since 2008) and Inria (since 2012). NEMOVAR encapsulates nearly three decades of scientific and technical development in ocean DA and serves as the ocean DA component for operational ocean and coupled forecasting and reanalysis activities at ECMWF and the UK Met Office, including those operated by those institutes for the Copernicus services C3S (climate) and CMS (marine).

**Publications:** Brief descriptions of NEMOVAR can be found in the ECMWF Newsletter articles Mogensen et al. (2009) and Chrust et al. (2021).

**Contact:** Anthony Weaver

**Participants:** Anthony Weaver, Olivier Goux, Andrea Piacentini

**URL:** The NEMOVAR source code has the same Open-Source license (CeCill-B) as NEMO. It is hosted on a Git server at ECMWF, currently with restricted access. Plans are underway to transfer the NEMOVAR source code to GitLab and to make it publicly available.

## 5.6 OOPS

**Name:** Object Oriented Prediction System

**Keywords:** Variational data assimilation, Numerical Weather Prediction, iterative methods, preconditioning

**Scientific Description:** OOPS provides generic data assimilation algorithms that can be used with different models and different observations. The OOPS code is available under an Apache-2 licence. OOPS is an abstract control layer that can manipulate elements of the data assimilation system without needing to know their model-specific implementation details. OOPS includes
simplified models (Quasi-Geostrophic and Lorenz-95) to enable early testing of data assimilation algorithms. OOPS is an international effort, involving major input from ECMWF, Météo-France and the HIRLAM-ALADIN community. Cerfacs contribute code development related to iterative methods and preconditioning strategies. More information can be found from ECMWF-OOPS.

**Contact:** Selime Gürol

## 5.7 PhyDLL

**Name:** Physics Deep Learning coupLer

**Keywords:** Deep Learning, Coupling, Hybrid architectures

**Scientific Description:** PhyDLL is an open-source library to couple massively parallel physical solvers to distributed deep learning inference.

**Publications:** 10.1016/j.compfluid.2024.106306

**Contact:** Luciano Drozda

**Participants:** Jeffrey Legaux (COOP Cerfacs team), Luciano Drozda

**URL:** https://gitlab.com/cerfacs/phydll

## 5.8 PYTHIA-DA

**Name:** Python-based variational-ensemble data assimilation training framework

**Keywords:** Data assimilation, nonlinear least-squares, 4DVar, ensemble Kalman filter, Lorenz-96

**Scientific Description:** Variational and ensemble data assimilation methods are implemented using the Lorenz-96 model in Python. The variational data assimilation problem is solved with a Gauss-Newton method, where the resulting linear least-squares subproblems are solved using a preconditioned conjugate gradient algorithm. The ensemble data assimilation approach is based on the stochastic Ensemble Kalman Filter, incorporating localization and inflation techniques for regularization. This code is used in data assimilation training courses at CERFACS and ISAE-SUPAERO.

**Publications:** For further details on variational data assimilation approach see to Courtier et al. (1994) and Fisher et al. (2008). For further details on ensemble Kalman filter algorithms see Ash et al. (2016).

**Contact:** S. Gürol

**Participants:** S. Gürol, O. Goux, M. Destouches (Météo-France), Rachid El-Montassir, B. Ménétrier (Met-Norway), A. Fillion, E. Lumet

**URL:** https://gitlab.com/cerfacs/datraining

# 6 New results

## 6.1 Toward Nonlinear Quantum Numerical Methods for Engineering Applications

The objective of this novel research activity is to develop quantum numerical schemes for physics and engineering applications, with a particular focus on fluid dynamics simulations. A major challenge arises from the implementation of nonlinear processes on digital quantum computers, as elementary quantum gates are inherently linear. To overcome this limitation, we investigate the implementation of a nonlinear iterative scheme (NLIS) on digital quantum platforms. By exploiting tensor products of parallel copies, we introduce an approach that enables polynomial nonlinearities acting on the qubit
state. Iterating this procedure leads to a full NLIS protocol. The corresponding quantum circuits have been designed, and their computational complexity has been characterized in terms of gate count (size), circuit depth (layer number), and ancilla qubit requirements (width). Future work will aim at optimizing the NLIS protocol and applying it to quantum simulations of fluid dynamics.

Future work will further improve the NLIS quantum protocol, leverage it for fluid simulations, and address the possibility of proving a quantum advantage for the simulation of nonlinear problems on quantum computers.

## 6.2 On some orthogonalization schemes in tensor train format

In the framework of tensor spaces, we consider orthogonalization algorithms to generate an orthogonal basis of a tensor subspace from a set of linearly independent tensors. All variants, except for the Householder transformation, are straightforward extensions of well-known algorithms in matrix computation to tensors. In particular, we experimentally study the loss of orthogonality of six orthogonalization methods: Classical and Modified Gram-Schmidt with (CGS2, MGS2) and without (CGS, MGS) re-orthogonalization, the Cholesky-QR, and the Householder transformation. To overcome the curse of dimensionality, we represent tensors with a low-rank approximation using the Tensor Train (TT) formalism. Additionally, we introduce recompression steps in the standard algorithm outline through the TT-round method at a prescribed accuracy. After describing the structure and properties of the algorithms, we illustrate their loss of orthogonality with numerical experiments. Although no formal proof exists at this time, we observe very clearly that the well-established properties verified over decades of research by the round error analysis community in matrix computation appear to extend to the case of low-rank tensors, with the unit round-off replaced by the TT-round accuracy. The computational analysis for each orthogonalization scheme in terms of memory requirements and computational complexity, measured as a function of the number of TT-round operations, which happens to be the most computationally expensive operation, completes the study.

For more details on this work we refer to [8].

## 6.3 A note on TT-GMRES for the solution of parametric linear systems

We study the solution of linear systems with tensor product structure using the Generalized Minimal RESidual (GMRES) algorithm. To manage the computational complexity of high dimensional problems our approach relies on low-rank tensor representation, focusing specifically on the Tensor Train format. We implement and experimentally study the TT-GMRES algorithm. Our analysis bridges the heuristic methods proposed for TT-GMRES by Dolgov in [Russian J. Numer. Anal. Math. Modelling, 28 (2013), pp. 149–172] and the theoretical framework of inexact GMRES by Simoncine and Szyld [SIAM J. Sci. Comput. 25 (2003), pp. 454–477]. This approach is particularly relevant in a scenario where a ($d+1$)-dimensional problem arises from concatenating a sequence of $d$-dimensional problems, as in the case of a parametric linear operator or parametric right-hand-side formulation. Thus, we provide backward error bounds that link the accuracy of the computed ($d+1$)-dimensional solution to the numerical quality of the extracted $d$-dimensional solutions. This facilitates the prescription of a convergence threshold ensuring that the $d$-dimensional solutions extracted from the ($d+1$)-dimensional result have the desired accuracy once the solver converges. We illustrate these results with academic examples across varying dimensions and sizes. Our experiments indicate that the TT-GMRES retains the theoretical rounding error properties observed in matrix-based GMRES.

For more details on this work we refer to [7].

## 6.4 Solving eigenvalue problems in high dimensions using contour integration and Tensor Train format

In high-dimensional settings, solving eigenvalue problems is hindered by the curse of dimensionality, particularly when only a subset of eigenpairs within a prescribed spectral interval is sought. In this work, we investigate an adaptation of the FEAST algorithm, originally developed for symmetric eigenproblems based on contour integration, to computations where both operators and vectors are represented in the Tensor Train (TT) format. This representation drastically reduces memory and
computational demands. We introduce an adaptive scheme for determining the projection subspace dimension by incorporating a rank-revealing Modified Gram-Schmidt procedure with pivoting tailored to TT-vectors. A perturbation-based analysis provides explicit bounds on the attainable residual accuracy, from which we derive a robust stopping criterion for the proposed TT-FEAST algorithm. Moreover, we design a continuation strategy that gradually refines convergence and rounding tolerances to effectively control memory growth during iterations. To demonstrate the effectiveness of TT-FEAST as a viable alternative to existing high-dimensional eigensolvers when a few eigenvalues are required, we present numerical experiments on problems up to twelve dimensions, including the Laplacian and a vibrational Hamiltonian operator.

For more details on this work we refer to [30].

## 6.5 A Tensor train solver for the magnetic moment method

In this work, we focus on enhancing the computational efficiency of the Magnetic Moment Method (MMM) using low-rank tensor representations, specifically the Tensor Train (TT) formats. By transforming the dense linear system generated by MMM into TT format, we achieve significant reductions in both computational cost and memory usage. Furthermore, when the problem structure allows—such as in cases with regular grids and binary-compatible sizes—the Quantized Tensor Train (QTT) format is employed to exploit additional compression through quantization, leading to even larger performance gains. The proposed methods are tested on several problems involving a ferromagnetic part with a regular mesh. For simple cases, our approach demonstrates superior performance compared to traditional techniques. As the problem complexity increases, the TT- and QTT-based methods remaincompetitive, maintaining efficiency while addressing the added computational challenges.

This work is part of Amine Zekri's PhD thesis and is carried out in the context of the TensoVim ANR project. For more details on this work we refer to [18].

## 6.6 Generalized Golub-Kahan bidiagonalization for nonsymmetric saddle-point systems

The generalized Golub-Kahan bidiagonalization has been used to solve saddle-point systems where the leading block is symmetric and positive definite. We extend this iterative method for the case where the symmetry condition no longer holds. We do so by relying on the known connection the algorithm has with the conjugate gradient method and following the line of reasoning that adapts the latter into the full orthogonalization method. We propose appropriate stopping criteria based on the residual and an estimate of the energy norm for the error associated with the primal variable. Numerical comparison with GMRES highlights the advantages of our proposed strategy regarding its low memory requirements and the associated implications.

For more details on this work we refer to [10].

## 6.7 A note on the partial convergence management for the solution of symmetric linear systems with multiple right-hand sides

We consider the solution of large sparse symmetric linear systems with multiple right-hand sides available simultaneously. Based on the partial convergence detection and management, described in IB-BGMRES [Linear Algebra Appl., 419 (2006), pp. 265-285] and the breakdown-free idea discussed in [BIT Numer. Math., 57 (2017), pp. 379-403], the block conjugate residual and block conjugate gradient methods with partial convergence management are proposed. It enables to select the directions to use for extending the search space from one iteration to the next by choosing the directions that contribute the most to the residual norms. We illustrate the numerical and computational benefits of these two novel block conjugate direction variants on a set of simple academic examples enabling reproducible experiments.

For more details on this work we refer to [34].
## 6.8 A scalable and parameter-robust preconditioner for a second gradient of dilation regularization applied to a mechanics problem

We propose a rigorous analysis of the second gradient of dilation regularization as it is used in many geomechanical applications. In this method, a new primal unknown, modeling the trace of the displacement gradient, is introduced leading to a saddle-point system. The resulting balance equations depend on parameters that range on several orders of magnitude. We prove the well-posedness of the continuous and discrete problems using parameter dependent norms. This allows the definition of a robust block preconditioner for the discrete problem with respect to the parameter variations and to mesh refinement. Numerical results confirm our theoretical findings.

For more details on this work we refer to [35].

## 6.9 Neural network preconditioning: a case study for the solution of the parametric Helmholtz equation

This work presents a hybrid numerical approach for solving linear systems arising from the discretization of the two-dimensional parametric Helmholtz equation. A convolutional neural network based on the U-Net architecture is trained in an unsupervised manner to approximate the inverse of the discretized Helmholtz operator, using a loss function involving the residual norm of the linear system. The trained network is used as a nonlinear preconditioner within the Flexible GMRES (FGMRES) algorithm. Numerical experiments show that while the neural network is not accurate enough to act as a standalone solver, it significantly improves the convergence of FGMRES when employed as a preconditioner. The neural preconditioner demonstrates robust performance and generalization capabilities with respect to variations in the velocity field and the domain size. Comparisons with classical algebraic preconditioners based on sparsified LU factorizations indicate superior efficiency of the neural approach under equivalent conditions. We believe that the proposed method is not tied to a specific neural architecture and can be extended to other parametric PDEs.

For more details on this work we refer to [33].

## 6.10 Memory-and compute-optimized geometric multigrid GMGPolar for curvilinear coordinate representations -Applications to fusion plasma

Tokamak fusion reactors are actively studied as a means of realizing energy production from plasma fusion. However, due to the substantial cost and time required to construct fusion reactors and run physical experiments, numerical experiments are indispensable for understanding plasma physics inside tokamaks, supporting the design and engineering phase, and optimizing future reactor designs. Geometric multigrid methods are optimal solvers for many problems that arise from the discretization of partial differential equations. It has been shown that the multigrid solver GMGPolar solves the 2D gyrokinetic Poisson equation in linear complexity and with only small memory requirements compared to other state-of-the-art solvers. In this paper, we present a completely refactored and object-oriented version of GMGPolar which offers two different matrix-free implementations. Among other things, we leverage the Sherman-Morrison formula to solve cyclic tridiagonal systems from circular line solvers without additional fill-in and we apply reordering to optimize cache access of circular and radial smoothing operations. With the Give approach, memory requirements are further reduced and speedups of four to seven are obtained for usual test cases. For the Take approach, speedups of 16 to 18 can be attained.

For more details on this work we refer to [36].

## 6.11 Complexity analysis and scalability of a matrix-free extrapolated geometric multigrid solver for curvilinear coordinates representations from fusion plasma applications

Tokamak fusion reactors are promising alternatives for future energy production. Gyrokinetic simulations are important tools to understand physical processes inside tokamaks and to improve the design of future plants. In gyrokinetic codes such as Gysela, these simulations involve at each time step the
solution of a gyrokinetic Poisson equation defined on disk-like cross sections. The authors of [14,15] proposed to discretize a simplified differential equation using symmetric finite differences derived from the resulting energy functional and to use an implicitly extrapolated geometric multigrid scheme tailored to problems in curvilinear coordinates. In this article, we extend the discretization to a more realistic partial differential equation and demonstrate the optimal linear complexity of the proposed solver, in terms of computation and memory. We provide a general framework to analyze floating point operations and memory usage of matrix-free approaches for stencil-based operators. Finally, we give an efficient matrix-free implementation for the considered solver exploiting a task-based multithreaded parallelism which takes advantage of the disk-shaped geometry of the problem. We demonstrate the parallel efficiency for the solution of problems of size up to 50 million unknowns.

For more details on this work we refer to [14].

## 6.12 Fault-tolerant numerical iterative algorithms at scale

This work investigates how to protect numerical iterative algorithms from all types of errors that can strike at scale: fail-stop errors (a.k.a. failures) and silent errors, striking both as computation errors and memory bit-flips. We combine various techniques: detectors for computation errors, checksums for memory errors, and checkpoint/restart for failures. The objective is to minimize the expected time per iteration of the algorithm. We design a hierarchical pattern that combines and interleaves all these fault-tolerance mechanisms, and we determine the optimal periodic pattern that achieves this objective. We instantiate these results for the performance analysis of the Preconditioned Conjugate Gradient (PCG) algorithm: we report several scenarios where the optimal pattern dramatically decreases the overhead due to error mitigation.

For more details on this work we refer to [17].

## 6.13 A filtered multilevel Monte Carlo method for estimating the expectation of cell-centered discretized random fields

We investigate the use of multilevel Monte Carlo (MLMC) methods for estimating the expectation of discretized random fields. Specifically, we consider a setting in which the input and output vectors of numerical simulators have inconsistent dimensions across the multilevel hierarchy. This requires the introduction of grid transfer operators borrowed from multigrid methods. By adapting mathematical tools from multigrid methods, we perform a theoretical spectral analysis of the MLMC estimator of the expectation of discretized random fields, in the specific case of linear, symmetric and circulant simulators. We then propose filtered MLMC (F-MLMC) estimators based on a filtering mechanism similar to the smoothing process of multigrid methods, and we show that the filtering operators improve the estimation of both the small- and large-scale components of the variance, resulting in a reduction of the total variance of the estimator. Next, the conclusions of the spectral analysis are experimentally verified with a one-dimensional illustration. Finally, the proposed F-MLMC estimator is applied to the problem of estimating the discretized variance field of a diffusion-based covariance operator, which amounts to estimating the expectation of a discretized random field. The numerical experiments support the conclusions of the theoretical analysis even with non-linear simulators, and demonstrate the improvements brought by the F-MLMC estimator compared to both a crude MC and an unfiltered MLMC estimator.

For more details on this work we refer to [2].

## 6.14 Multilevel Monte Carlo methods for ensemble variational data assimilation

Ensemble variational data assimilation relies on ensembles of forecasts to estimate the background error covariance matrix **B**. The ensemble can be provided by an Ensemble of Data Assimilations (EDA), which runs independent perturbed data assimilation and forecast steps. The accuracy of the ensemble estimator of **B** is strongly limited by the small ensemble size that is needed to keep the EDA computationally affordable. We investigate here the potential of the multilevel Monte Carlo (MLMC) method, a type of multifidelity Monte Carlo method, to improve the accuracy of the standard Monte-Carlo estimator of **B** while keeping the computational cost of ensemble generation comparable. MLMC
exploits the availability of a range of discretization grids, thus shifting part of the computational work
from the original assimilation grid to coarser ones. MLMC differs from the mere averaging of statistical
estimators, as it ensures that no bias from the coarse resolution grids is introduced in the estimation.
The implications for ensemble variational data assimilation systems based on EDAs are discussed.
Numerical experiments with a quasi-geostrophic model demonstrate the potential of the approach, as
MLMC yields more accurate background error covariances and reduced analysis error. The challenges
involved in cycling a multilevel variational data assimilation system are identified and discussed.

For more details on this work we refer to [9].

## 6.15 Mixed precision sketching for least-squares problems and its application in GMRES-Based iterative refinement

Sketching-based preconditioners have been shown to accelerate the solution of dense least-squares problems with coefficient matrices having substantially more rows than columns. The cost of generating these preconditioners can be reduced by employing low precision floating-point formats for all or part of the computations. We perform finite precision analysis of a mixed precision algorithm that computes the *R*-factor of a QR factorization of the sketched coefficient matrix. Two precisions can be chosen and the analysis allows understanding how to set these precisions to exploit the potential benefits of low precision formats and still guarantee an effective preconditioner. If the nature of the least-squares problem requires a solution with a small forward error, then mixed precision iterative refinement (IR) may be needed. For ill-conditioned problems the GMRES-based IR approach can be used, but good preconditioner is crucial to ensure convergence. We theoretically show when the sketching-based preconditioner can guarantee that the GMRES-based IR reduces the relative forward error of the least-squares solution and the residual to the level of the working precision unit roundoff. Small numerical examples illustrate the analysis.

For more details on this work we refer to [4].

## 6.16 An introduction to solving the least-squares problem in variational data assimilation

Variational data assimilation is a technique for combining measured data with dynamical models. It is a key component of Earth system state estimation and is commonly used in weather and ocean forecasting. The approach involves a large-scale generalized nonlinear least-squares problem. Solving the resulting sequence of sparse linear subproblems requires the use of sophisticated numerical linear algebra methods. In practical applications, the computational demands severely limit the number of iterations of a Krylov subspace solver that can be performed and so high-quality preconditioners are vital. In this paper, we introduce variational data assimilation from a numerical linear algebra perspective and review current solution techniques, with a focus on the challenges that arise in large-scale geophysical systems.

For more details on this work we refer to [31].

## 6.17 A comparison of mixed precision iterative refinement approaches for least-squares problems

Various approaches to iterative refinement (IR) for least-squares problems have been proposed in the literature and it may not be clear which approach is suitable for a given problem. We consider three approaches to IR for least-squares problems when two precisions are used and review their theoretical guarantees, known shortcomings and when the method can be expected to recognize that the correct solution has been found, and extend uniform precision analysis for an IR approach based on the seminormal equations to the two-precision case. We focus on the situation where it is desired to refine the solution to the working precision level. It is shown that the IR methods exhibit different sensitivities to the conditioning of the problem and the size of the least-squares residual, which should be taken into account when choosing the IR approach. We also discuss a new approach that is based on solving multiple least-squares problems.

For more details on this work we refer to [5].
## 6.18 Block alpha-circulant preconditioners for all-at-once diffusion-based covariance operators

Covariance matrices are central to data assimilation and inverse methods derived from statistical estimation theory. Previous work has considered the application of an all-at-once diffusion-based representation of a covariance matrix operator in order to exploit inherent parallelism in the underlying problem. In this paper, we provide practical methods to apply block $\alpha$-circulant preconditioners to the all-at-once system for the case where the main diffusion operation matrix cannot be readily diagonalized using a discrete Fourier transform. Our new framework applies the block $\alpha$-circulant preconditioner approximately by solving an inner block diagonal problem via a choice of inner iterative approaches. Our first method applies Chebyshev semi-iteration to a symmetric positive definite matrix, shifted by a complex scaling of the identity. We extend theoretical results for Chebyshev semi-iteration in the symmetric positive definite setting, to obtain computable bounds on the asymptotic convergence factor for each of the complex sub-problems. The second approach transforms the complex sub-problem into a (generalized) saddle point system with real coefficients. Numerical experiments reveal that in the case of unlimited computational resources, both methods can match the iteration counts of the ‘best-case’ block $\alpha$-circulant preconditioner. We also provide a practical adaptation to the nested Chebyshev approach, which improves performance in the case of a limited computational budget. Using an appropriate choice of $\alpha$ our new approaches are robust and efficient in terms of outer iterations and matrix-vector products.

For more details on this work we refer to [37]

## 6.19 Convergence analysis of overlapping domain decomposition preconditioners for nonlinear problems

Numerical simulations of nonlinear partial differential equations often involve solving large nonlinear systems, for which Newton's method is widely employed due to its fast convergence near the solution. However, its performance can deteriorate in the presence of strong nonlinearities or poor initial guesses. Nonlinear overlapping domain decomposition methods, such as RASPEN and Substructured RASPEN (SRASPEN), have proven effective in addressing these challenges. Because SRASPEN reduces the problem size by restricting computations to a substructure, it does not update the solution outside the substructure, so that no natural initial guesses for the nonlinear local solution exists that might lead to additional inner subdomain nonlinear iterations or even prevent the local solvers to converge. In this study, we analyze the convergence of RASPEN. We show how domain decomposition improves the convergence rate of the Newton's method by highlighting the key role of the substructure on the global error contraction. Moreover, our analysis provides insight into an inexpensive modification to SRASPEN that mitigates the lack of iterations outside the substructure. The proposed variant significantly reduces computational cost while improving overall efficiency compared to existing techniques in the literature. Numerical experiments confirm the computational performance and robustness of the improved SRASPEN, establishing it as a reliable approach for solving large-scale nonlinear systems.

This work is part of Ettaouchi El Mehdi's PhD thesis and is carried out in collaboration with Nicolas Tardieu (EDF). For more details on this work we refer to [12].

## 6.20 Improvements to the Met Office's global ocean-sea ice forecasting system including model and data assimilation changes

The Forecast Ocean Assimilation Model (FOAM) is the Met Office's operational, coupled ocean-sea ice system, which produces analyses and short-range forecasts at global and regional scales each day for various stakeholders, including defence, marine navigation and science users. This paper describes and evaluates the impacts of recent model and data assimilation (DA) updates on global FOAM when compared to its current operational version. The model updates include the use of the TEOS10 formulation for the seawater equation of state, with improved ocean model settings in the Southern Ocean and the implementation of a new sea ice model. Updates to the DA include an increase in the number of DA minimisation iterations, an improved specification of observation errors for sea surface
temperature and sea level anomaly (SLA), and optimisations of the DA computational efficiency. Large-scale DA corrections for temperature have also been removed to prevent an inconsistent projection of the SLA DA signal onto large-scale temperature at depth. For 1-year runs at 1/12° resolution, the new FOAM system shows a 40% improvement in observation-minus-background (OmB) statistics for SLA and subsurface temperatures relative to the current system in eddy-rich regions, which result in a similar level of improvement for ocean currents. To evaluate potential impacts on the pre-Argo period, 1-year experiments at 1/4° resolution are run withholding profiles of temperature and salinity observations in both new and current FOAM systems. Limited to the assimilation of only surface data, OmB statistics for SLA, temperature and salinity in the new FOAM system can reach improvements up to 90 % in the Southern Hemisphere relative to the current system, resulting in more temporally consistent ocean transport and heat content results. Therefore, it is expected that the model and DA updates will lead to more potential for use of FOAM reanalyses in climate studies, particularly in the pre-Argo period, and will provide improved ocean-sea ice initial conditions to FOAM as well as to the Met Office short-range and seasonal coupled ocean-atmosphere-land-sea ice forecasting systems.

For more details on this work we refer to [15]

## 6.21 On the impact of observation error correlations in data assimilation, with application to along-track altimeter data

Data assimilation involves estimating the state of a system by combining observations from various sources with a background estimate of the state. The weights given to the observations and background state depend on their specified error covariance matrices. Observation errors are often assumed to be uncorrelated even though this assumption is inaccurate for many modern data-sets such as those from satellite observing systems. As methods allowing for a more realistic representation of observation-error correlations are emerging, our aim in this article is to provide insight on their expected impact in data assimilation. First, we use a simple idealised system to analyse the effect of observation-error correlations on the spectral characteristics of the solution. Next, we assess the relevance of these results in a more realistic setting in which simulated along-track (nadir) altimeter observations with correlated errors are assimilated in a global ocean model using a three-dimensional variational assimilation (3D-Var) method. Correlated observation errors are modelled in the 3D-Var system using a diffusion operator. When the correlation length scale of observation error is small compared to that of background error, inflating the observation-error variances can mitigate most of the negative effects from neglecting the observation-error correlations. Accounting for observation-error correlations in this situation still outperforms variance inflation since it allows small-scale information in the observations to be more effectively extracted and does not affect the convergence of the minimization. Conversely, when the correlation length scale of observation error is large compared to that of background error, the effect of observation-error correlations cannot be properly approximated with variance inflation. However, the correlation model needs to be constructed carefully to ensure the minimization problem is adequately conditioned so that a robust solution can be obtained. Practical ways to achieve this are discussed.

For more details on this work we refer to [13]

## 6.22 Impact of ensemble-based hybrid background-error covariances in ECMWF's next-generation ocean reanalysis system*

An ensemble of data assimilations (EDA) can provide valuable information on analysis and short-range forecast uncertainties. The present European Centre for Medium-Range Weather Forecasts (ECMWF) operational ocean analysis and reanalysis system, called the Ocean Reanalysis System 5 (ORAS5), produces an ensemble but does not exploit it for the specification of the background-error covariance matrix, a key component of the data assimilation system. In this article, we describe EDA developments for the ocean, which take advantage of the short-range forecast ensemble for specifying, in two distinct ways, parameters of a covariance model representation of. First, we generate a climatological ensemble over an extended period to produce seasonally varying climatological estimates of background-error variances and horizontal correlation length-scales. Second, in each assimilation
cycle, we diagnose flow-dependent variances from the ensemble and blend them with the climatological estimates to form hybrid variances. We also use the ensemble to diagnose flow-dependent vertical correlation length-scales. We demonstrate for the Argo-rich period that this new, ensemble-based hybrid formulation of results in a significant reduction of background errors compared with the parameterized formulation of used in ORAS5, with the globally averaged reduction ranging between 5% and 15% for subsurface temperature and salinity and up to and for sea-surface temperature and sea-surface height, respectively. The new ocean EDA system will be employed in ORAS6, ECMWF's next-generation ocean reanalysis system.

For more details on this work we refer to [6].

*Article chosen as one of the *Editor's highlights* in the *Quarterly Journal of the Royal Meteorological Society*.

## 6.23 Robustness and reliability of state-space, frame-based modeling for thermoacoustics

The Galerkin modal expansion is a well-known method used to develop reduced order models for thermoacoustics. A known issue is the appearance of Gibbs-type oscillations on velocity fluctuations at the interface between subdomains and at boundary conditions. Recent work of Laurent et al. (2019) and Laurent et al. (2021) have shown that it is possible to overcome this issue by using an over-completed frame, instead of a Galerkin modal basis. However, the low-order modeling based on this frame modal expansion may generate spurious modes. In this paper, the origin of these non-physical modes is identified and a method is proposed to automatically remove them from the outcome. By preventing any interaction between the physical and non-physical components, the proposed methodology drastically improves the robustness and reliability of the frame modal expansion modeling for thermoacoustics.

For more details on this work we refer to [3].

## 6.24 Code generation and performance engineering for matrix-free finite element methods on hybrid tetrahedral grids

This paper introduces a code generator designed for extreme-scalable, matrix-free finite element operators on hybrid tetrahedral grids. It optimizes the local evaluation of bilinear forms through various techniques including tabulation, relocation of loop invariants, and interelement vectorization—implemented as transformations of an abstract syntax tree. A key contribution is the generator's ability to optimize not only the local assembly, but also the loop over the underlying grid. To that end, the developed “cubes” loop pattern leverages local structure resulting from grid refinement and significantly enhances arithmetic intensity. Its effectiveness is established during a memory and layer condition analysis. The paper demonstrates the generator's capabilities through a comprehensive, educational cycle of performance analysis, bottleneck identification, and emission of dedicated optimizations. For three differential operators, the set of most effective optimizations is determined from a larger pool. They result in matrix-free operators with a kernel throughput of 1.3 to 2.1 GDoF/s, achieving up to 62% peak performance on a 36-core Intel Ice Lake socket. We make performance comparisons with a sparse matrix-vector multiplication and operators generated by the Firedrake project at increasing levels of refinement. Finally, the solution of the curl-curl problem with more than a trillion (1012) degrees of freedom on 21,504 processes in less than 50 s showcases the generated operators' performance and extreme scalability as part of a full multigrid solver.

For more details on this work we refer to [1].

## 6.25 Training loss matters: exploring new fit-for-purpose loss combinations for statistical downscaling

Deep learning-based statistical downscaling is increasingly adopted in weather and climate science. However, the choice of training loss function, which is central to guiding model optimisation, remains largely unexplored, with most studies relying on standard losses, such as the mean squared error and the negative log-likelihood Bernoulli-Gamma. In this work, we benchmark a wide range of loss functions for statistical downscaling, including standard losses and others novel to this field, such as wavelet
and gradient-based formulations. We also investigate the benefits of combining complementary losses. The experiments cover two representative downscaling tasks with distinct statistical and physical behaviours: 10m zonal wind speed and precipitation. We found that the best levels of the considered performance aspects are reached combining different losses. Most of these combinations have not been used before. Besides, we observed that no loss combination dominates across all tasks or evaluation aspects; instead, the choice and combination of losses strongly influence accuracy, performance on extreme events, and visual rendering of the model. These findings highlight the importance of designing fit-for-purpose training losses, and also provide practical guidance for choosing effective ones. These results apply not only to deep learning-based statistical downscaling but can also be considered in related applications sharing similar challenges, such as weather forecasting.

For more details on this work we refer to [16].

# 7 Partnerships and cooperations

## 7.1 European Initiatives

### 7.1.1 EoCoE-III

**Title:** Energy oriented Centre of Excellence for Exascale HPC applications

**Web page:** https://www.eurohpc-ju.europa.eu

**Duration:** 2024–2026

**Funding:** H2020

**Coordinator:** CEA

**Cerfacs coordinator:** Carola Kruse

**Partners:** Agenzia Nazionale per le Nuove Tecnologie, l'Energia e lo Sviluppo Economico Sostenibile (Italy), Barcelona Supercomputing Center - Centro Nacional de Supercomputacion (Spain), Centre Europeen de Recherche et de Formation Avancee en Calcul Scientifique (France), Centre National de la Recherche Scientifique CNRS (France), Commissariat a l'Energie Atomique et aux Energies Alternatives (France), Consiglio Nazionale delle Ricerche (Italy), Forschungszentrum Julich GmbH (Germany), Fraunhofer Gesellschaft zur Foerderung der Angewandten Forschung E.V. (Germany), Inria, Max-Planck-Gesellschaft zur Forderung der Wissenschaften EV (Germany), Rheinisch-Westfaelische Technische Hochschule Aachen (Germany), Universita degli Studi di Roma Torvergata (Italy), Universita degli Studi di Trento (Italy), Universite Libre de Bruxelles (Belgium), Universite Paris-Sud (France)

**Summary:** The Concace team (Inria, Cerfacs) participates in the Energy-oriented Centre of Excellence (EoCoE-III), starting in January 2024. The project applies cutting-edge exascale computational methods in its mission to accelerate the transition to the production, storage and management of clean, decarbonized energy. EoCoE-III is anchored in the High Performance Computing (HPC) community and targets research institutes and key commercial players who develop and enable energy-relevant numerical models to be run on exascale supercomputers, demonstrating their benefits for the net-zero energy transition. The project will draw on the experience of two successful previous projects EoCoE-I and -II, where a large set of diverse computer applications from four such energy domains achieved significant efficiency gains thanks to a multidisciplinary expertise in applied mathematics and supercomputing. EoCoE-III channels its efforts into 5 exascale lighthouse applications in the low-carbon sectors of Energy Materials, Water, Wind and Fusion. This multidisciplinary effort will harness innovations in computer science and mathematical algorithms within a tightly integrated co-design approach to overcome performance bottlenecks and to anticipate HPC hardware developments. A world-class consortium of 16 complementary partners forms a unique network of expertise in energy science, scientific computing and HPC, including 3 leading European supercomputing centres.
### 7.1.2 ERGO2 (Copernicus Climate Change Service)

**Title:** Advancing ocean data assimilation methodology for climate applications

**Web page:** https://www.ismar.cnr.it/web-content/ergo2/

**Duration:** 2022–2025

**Funding:** EU Copernicus programme

**Coordinator:** Inria

**Contact:** Anthony Weaver

**Participants:** Cerfacs, Inria, CNR (Italy)

**Summary:** A C3S-funded project to advance ocean data assimilation methodology for climate applications in support of ECMWF's Copernicus Services. The project built on the development already carried out under a previous C3S project (ERGO), which implemented the components needed to enable an Ensemble of Data Assimilations (EDA) for ocean reanalyses. The main goal of ERGO2 was to further integrate these individual components and to develop additional data assimilation tools so the next generation of C3S climate reanalysis can benefit from better exploitation of available observations. Specifically, Cerfacs contributed to workpackages aimed at enhancing the representation of background-, model- and observation-error covariances in NEMOVAR.

### 7.1.3 NEMOVAR software development (Memorandum of Understanding)

**Title:** Nucleus for European Modelling of the Ocean - Variational data assimilation

**Duration:** 2002–present

**Partners:** Cerfacs, ECMWF, Inria, UK Met Office

**Contact:** Anthony Weaver

**Summary:** The MoU is an agreement that covers the joint development and use of the NEMOVAR software.

### 7.1.4 PVDAP

**Title:** Network on Preconditioning for Variational Data Assimilation Problems

**Duration:** 2023–2025

**Funding:** partially funded by the Isaac Newton Institute network support grant and EPSRC (Ref: EP/V521929/1).

**Web page:** https://www.strath.ac.uk/science/mathematicsstatistics

**Coordinator:** University of Strathclyde

**Contact:** Selime Gürol

**Participants:** S. Gürol, I. Daužickaitė, M. Freitag (Postdam University), A. Ramage (University of Strathclyde), A. Lawless (University of Reading), J. Scott (University of Reading/RAL), J. Tabeart (University of Eindhoven)
**Summary:** The aim of this research network is to bring together experts with diverse experiences in various aspects of preconditioning VDA problems to share expertise, identify key challenges and work together to begin to tackle some of the core issues in this area. In today's world, the increasing availability of large datasets is leading to a fast growing range of data-driven applications which means that there is a pressing need to face the associated computational challenges. The ultimate goal is to develop a theoretical framework underpinning a range of preconditioners for different VDA problems that can in time be translated from traditional physics-based problems such as weather forecasting to modern applications in areas such as wearable technology and future cities. This is a two-year project (October 2023 to September 2025) partially funded by an Isaac Newton Institute network support grant and EPSRC.

### 7.1.5 ROSAS

**Title:** RObust simulation Systems exploiting Artificial Intelligence based turbulence models and high-fidelity algorithmS

**Duration:** 2025-2028

**Funding:** Horizon Europe

**Web page:** https://www.rosas-project.eu/

**Coordinator:** ONERA

**Contact:** Luciano Drozda

**Participants:** ONERA, UNIBG (Bergamo), DLR, DAV, ERDYN, BSC, CENAERO, Cerfac, KTH, Rolls Royce, SAFRAN, UCL (Université Catholique de Louvain), UNIPI (Pisa), UPM (Politecnica de Madrid)

**Summary:** ROSAS aims at paving the way for robust and accurate AI-augmented CFD tools using ML data-driven methodologies able to provide a simulation environment leading to correct first-time results without having to perform multiple trials in an incremental and expensive approach. ROSAS specific objectives are: (1) Generate HiFi databases representative of challenging physical phenomena for data-driven model development, (2) Increase numerical methods' efficiency and robustness by combining High-Order Methods (HOM) with AI algorithms, (3) Increase turbulence model accuracy for RANS and scale-resolved simulations using AI-driven methodologies, (4) Achieve Proof of Concept (PoC) for ROSAS AI-based design tools on industry-application challenges (linked to Clean Aviation).

## 7.2 National Partnerships

### 7.2.1 CECI

**Title:** Climat, Environnement, Coupling et Incertitudes

**Duration:** 1999-2013 (URA); 2014-present (UMR)

**Web page:** https://www.omp.eu/climat-environnement-couplages-et-incertitudes-ceci/

**Algo contact:** S. Gürol and A. Weaver

**Funding:** UMR CNRS

**Partners:** Cerfac, CNRS and IRD.

**Summary:** CECI is a joint research unit of Cerfac, CNRS and IRD in Toulouse, combining physicists, mathematicians, and computer scientists. It focuses on Climate Modeling (GLOBC) collaborating with other Cerfac teams. Key climate research includes variability, change detection, air-sea
interactions, low-frequency ocean variability, and high-resolution model impacts. Efforts reduce biases in coupled climate models and improve regional projections of temperature, precipitation, and extremes. CECI also develops coupling tools and data assimilation methods, providing training and distributing tools globally. Through these activities, CECI drives advances in climate science, aviation impacts, and computational modelling.

### 7.2.2 Concace

**Title:** COmposabilité Numérique and parallèle pour le CAlcul haute performanCE

**Duration:** 2022–2030

**Web page:** https://concace.gitlabpages.inria.fr/

**Algo contact:** Carola Kruse

**Funding:** joint Inria-Industry team with Airbus CR&T and Cerfacs

**Partners:** Airbus CR&T, Cerfacs and Inria.

**Summary:** To tackle challenges posed by extensive scale and dimensionality in both model- and data-driven applications, we'll leverage contemporary development tools and languages to craft high-level expressions of advanced parallel numerical algorithms. While conventional high-performance computing (HPC) strategies emphasize maximizing hardware utilization, our alternative approach will enable a richer composability, thereby harnessing the full potential of both existing and novel numerical algorithms.

## 7.3 Collaborative projects

### 7.3.1 Maison du Quantique Occitanie

**Duration:** 2025–2028

**Web page:** https://www.inp-toulouse.fr

**Algo contact:** Julien Zylberman

**Funding:** France Hybrid HPC Quantum Initiative

**Partners:** Université de Montpellier, INSA Toulouse, Toulouse INP, IRIT, ISDM, ICGM, CALMIP, ISDM-Meso, Cerfacs, SLB, Eviden-BULL), Qualitative Computing and Qraftware.

**Summary:** The Maison Quantique Occitanie (MQO) is a regional hub dedicated to the development of industrial and research projects in hybrid quantum computing. Led by the Institut National Polytechnique de Toulouse (Toulouse INP) with the support of the Occitanie / Pyrénées Méditerranée Region, it brings together a network of academic, industrial, and institutional partners throughout the region. Located at two strategic sites—the Cité de Toulouse and the Cité de l'Économie et des Métiers de Demain in Montpellier—the Maison du Quantique offers state-of-the-art quantum computing infrastructure, including direct access to the HQI hybrid HPC-quantum platform. The MQO's mission is to foster collaboration between large companies, laboratories, and start-ups in order to create a dynamic ecosystem and strengthen Occitanie's position in the global race for cutting-edge technologies.

### 7.3.2 ANITI - EXPLEARTH Chair

**Title:** EXPLainable and physics-informed Ai for Regional weaTHer prediction

**Duration:** 2025–2028

**Web page:** https://aniti.univ-toulouse.fr
**Algo contact:** Luciano Drozda

**Funding:** ANR, Industrial partners, Cerfacs

**Partners:** Météo-France, Institut de Mathématiques de Toulouse, Cerfacs

**Summary:** This chair will focus on one of the most promising perspectives for using deep learning in weather forecasting: the development of new models entirely learned from data. This approach will be leveraged for local-scale probabilistic forecasting, and its combination with existing physical models will also be explored. Another objective of the chair is to qualify the robustness and explainability of deep learning-based forecasting models. This will involve, in particular, applying certain standard Explainable AI (XAI) methods to the developed models, adapting them to the specificities of meteorological data. The integration of physical constraints into the construction and training of deep learning models will be another way to ensure their quality and strengthen user confidence in these new tools.

**Principal investigators:** Laure Raynaud (Météo-France), Laurent Risser (Institut de Mathématiques de Toulouse) **Co-chairs:** Luciano Drozda

### 7.3.3 ANITI - RELEO Chair

**Title:** REpresentation Learning for Earth Observation

**Duration:** 2024–2028

**Web page:** https://aniti.univ-toulouse.fr

**Algo contact:** Selime Gürol

**Funding:** ANR, Industrial partners

**Partners:** CS Sopra Steria, Magellium, Thales Alenia Space, Thales Services Numériques, Cerfacs, ISAE, CNES, CESBIO

**Summary:** This project will develop new self-supervised representation learning methods to produce semantically meaningful probabilistic representations from high-dimensional multi-modal EO data. The originality lies on the use of prior knowledge from physical models into DL and thus proposing advances in uncertainty estimation and interpretability. The proposed hybrid AI system will blend physical priors and DL to pre-train models that can learn (1) semantically meaningful representations related to EVs and (2) task-agnostic generic embeddings (AI-ready data) that can be used by downstream tasks. The system will process multi-modal data to capture complementary spatio-temporal patterns. Physics-guided DL methods will be designed to condition the decoding of generic embeddings to retrieve and forecast EVs and their uncertainties. To ensure the continuity of land monitoring, the system will use new data assimilation strategies combining satellite observations with pre-trained model forecasts. Continual learning will be used to update the models in response to new EO data. Non-stationary and long-term trends beyond the temporal range of the initial training will be accounted for. The project raises scientific questions regarding joint probabilistic representation learning, incorporation of physical prior information, efficient use of pre-trained models, and continuous model updating with newly acquired data and new on-orbit sensors.

**Principal investigators:** Jordi Inglada (Senior Expert CNES, CESBIO), Nicolas Dobigeon (Full Professor Toulouse-INP), Mathieu Fauvel (1st grade Researcher INRAe, CESBIO), Silvia Valero (CPJ Researcher, IRD – OMP)

**Co-chairs:** Thomas Oberlin (Professor, ISAE-SUPAERO, Université de Toulouse), Julien Michel (Research Engineer CNES, CESBIO), Selime Gürol.
### 7.3.4 ANR PHLUSIM

**Title:** Physics-based Learning for robUst fluid SIMulation

**Duration:** 2024-2028

**Web page:** https://anr.fr/en/funded-projects-and-impact

**Algo contact:** Luciano Drozda

**Funding:** ANR

**Partners:** Cerfacs, Sorbonne Université, Conservatoire national des arts et métiers (Cnam)

**Summary:** This project explores the benefits and challenges provided by using context-aware learning techniques to model fluid flows. To this end, four main objectives are set: (1) select context-aware techniques that are compatible with fluid modelling, and generalization strategies available for them; (2) define the formulations by which they can be used to assist high-fidelity CFD solvers, and implement the tooling to do so; (3) explore an array of uses they can have for reduced order models, and define quality metrics with respect to preexisting methods; (4) assemble data on a new unseen case and on a multitude of existing cases to map out the generalization capacity of all the methods explored in the project.

**Principal investigators:** Luciano Drozda, Taraneh Sayadi (Professor, Cnam), Paola Cinnella (Professor, Sorbonne Université), Patrick Gallinari (Professor, Sorbonne Université).

### 7.3.5 IRT Saint Exupéry - R-Evol project

**Title:** Revolutionary - Evolutionary processes

**Duration:** 2020-2025

**Algo contact:** P. Mycek

**Funding:** PIA (Programme d'investissements d'avenir), industrial project partners.

**Partners:** IRT Saint Exupéry, Cerfacs, Cenaero, Capgemini, Liebherr, Expleo, INSA Toulouse.

**Summary:** The R-Evol (Evolutionary – Revolutionary processes) project aimed at building a bridge between systems engineering and multidisciplinary design analysis and optimization (MDAO) domains, thus contributing to the digital continuity, a big challenge for both transport and space industries. In addition, R-Evol aimed at enhancing the efficiency, robustness and reliability of the multidisciplinary design optimization (MDO) processes, extending MDO capabilities to multi-mission, and propagating uncertainties throughout the processes, thereby opening the door to a routine industrial use of MDO. Cerfacs was involved in a work package dealing with uncertainty propagation, quantification, and management. A follow-up project, R-Evol 2, started in late 2025.

### 7.3.6 LEFE MANU: ADA

**Title:** Advanced Algorithms for Data Assimilation

**Duration:** 2023-2025

**Contact:** Selime Gürol

**Funding:** CNRS/INSU, Lefe programme
**Summary:** The primary goal of the project is to incorporate and design stochastic randomized algorithms within deterministic variational data assimilation, with the aim of reducing the computational cost while maintaining the same level of accuracy. The first part of the project focused on using randomized algorithms to estimate eigen-spectrum of the linear system matrix and use these approximations within the LMP to accelerate the iterative solver, mainly the conjugate gradients method. The second part of the project focused on the use of randomized algorithms with a non-standard Gaussian distribution-zero mean and a symmetric positive definite covariance matrix-to accelerate convergence within the Ensemble of DAs (EDAs).

### 7.3.7 LEFE MANU: DOGODA

**Title:** Developments in Observation- and background-error covariance modelling for Global Ocean Data Assimilation

**Duration:** 2023-2025

**Contact:** Anthony Weaver

**Funding:** CNRS/INSU Lefe programme

**Summary:** A project on developments in observation- and background-error covariance modelling for global ocean data assimilation. The project addressed three research topics. The first topic concerned methods for accounting for correlated observation error in satellite altimeter data, with both wide-swath (e.g., SWOT) and traditional nadir instruments being considered. The second and third topics dealt with aspects of background-error covariance modelling. One topic involved revisiting the choice of velocity control variables in NEMOVAR with the aim of better exploiting ocean velocity information. The second topic concerned developments to the diffusion-based correlation model to improve the representation of anisotropic background-error correlations.

### 7.3.8 SANTANA

**Title:** SimulAtion Numérique mulTidisciplinaire Avancée pour les Nouveaux Aéronefs

**Duration:** 2024-2027

**Algo contact:** Carola Kruse

**Funding:** DGAC

**Partners:** Airbus, Cerfacs, DLR, ONERA.

**Summary:** Santana is a project funded by the DGAC that focuses on the development and enhancement of the CODA solver, which is mainly used for the computation of aerodynamic effects. Within this CFD software, the Newton-Krylov solver has been identified as the most computationally expensive component. The role of Cerfacs/Concace is to optimize the linear solution step in the Newton-Krylov solver, with the goal of achieving substantial reductions in computation time.

## 8 Dissemination

### 8.1 Promoting scientific activities

#### 8.1.1 Scientific events: organisation

**General chair, scientific chair**

* Workshop AISSAI 2026 in Toulouse on AI for HPC: Luciano Drozda
### 8.1.2 Scientific events: selection

#### Member of conference programme committees

* Preconditioning techniques for scientific and industrial applications 2026: Selime Gürol.

* PDSEC: Luc Giraud.

* Luc Giraud is member of the Gene Golub SIAM Summer School. The thirteen Gene Golub SIAM Summer School was entitled "Frontiers in multi-dimensional pattern formation", Concordia University - Montreal, Quebec, CA - August 4th to 15th, 2025.

### 8.1.3 Journal

**Reviewer - reviewing activities** SIAM SISC; SIAM SIMAX; Combustion and Flame; International Journal for Numerical Methods in Engineering; Quarterly Journal of the Royal Meteorological Society.

### 8.1.4 Invited talks

* A journey through some numerical linear algebra algorithms with variable accuracy storage; Luc Giraud, Emmanuel Agullo, Olivier Coulaud, Martina Iannacito, Mohammad Issa, Gilles Marait, Miroslav Rozloznik CAS-ANLA 2025 - The Chinese Academy of Sciences Workshop on Approximate computing in Numerical Linear Algebra, Apr 2025, Beijing, China.

### 8.1.5 Scientific expertise

* Luc Giraud is:

* Member of the board on Modelization, Simulation and data analysis of the Competitiveness Cluster for Aeronautics, Space and Embedded Systems.

* Member of the scientific council of the ONERA Lab LMA2S (Laboratoire de Mathématiques Appliquées à l'Aéronautique et au Spatial).

* Member of the scientific council of GDR Calcul,

* Member of the expert committee for the thematic committee on algorithmics, mathematics, and quantum Technologies at GENCI.

* Selime Gürol is a member of the Scientific Advisory Committee (SAC) of ECMWF.

* Carola Kruse is referee for Icelandic Research Fund proposals.

* Anthony Weaver is:

* ECMWF Fellow.

* Member of the Data Assimilation Task Team (DA-TT) of the international programme Ocean-Predict.

## 8.2 Teaching - Supervision - Juries - Educational and pedagogical outreach

* Post graduate level/Master:

- L. Drozda: An introduction to Artificial Intelligence, 2h, ISAE-SUPAERO + Cerfacs (session in Data Assimilation training course), Toulouse.

- L. Giraud: Introduction to intensive computing and related programming tools 20h, INSA Toulouse; Advanced numerical linear algebra 10h, ENSEEIHT Toulouse.
- S. Gürol: ISAE Inverse Problems and Introduction to Data Assimilation (30 hours) + Data Assimilation training course at Cerfacs (1 week, organization of the training course and core lecturer).

- C. Kruse: Iterative methods in linear algebra, 28h, ENSEEIHT Toulouse.

- P. Mycek: Multifidelity methods 14h, ModIA (cursus en alternance, INSA/N7), Toulouse.

- A. Weaver: Core lecturer for the 5-day Cerfacs DA training course.

### 8.2.1 Supervision

* PhD: Olivier Goux; Prise en compte des corrélations d'erreur d'observation en assimilation vari-ationnelle de données océéniques: application aux données altimétriques; defended 20 February 2025, Université de Toulouse, A. Weaver, S. Gürol, O. Guillet (Météo-France).

* PhD: Laura Risley; Assimilation of future ocean current measurements using satellites; defended 16 September 2025, University of Reading, A. Lawless (U. Reading), M. Martin (Met Office), A. Weaver.

* PhD in progress: El Mehdi Ettaouchi; Nonlinear domain decomposition methods: analysis and industrial applications; started March 2023, L. Giraud, C. Kruse, N. Tardieu (EDF).

* PhD in progress: Théo Briquet; machine learning techniques for rank prediction of H-matrices; started October 2023, L. Giraud, P. Mycek, G. Sylvand (Airbus CR & T).

* PhD in progress: Oussama Mouhtal, Randomized algorithms for data assimilation; started on November 2023, Y. Diouane (Polytechnique Montréal), S. Gürol, and D. Orban (Polytechnique Montréal).

* PhD in progress: Sai Aakash Dasari; Scalable multigrid methods for tokamak geometries; started October 2024, C. Kruse, P. Mycek.

* PhD in progress: Richard Faucheron, Hybrid data assimilation and deep learning system for Earth monitoring; started on October 2024, S. Gürol and S. Valero-Valbuena (CESBIO).

* PhD in progress: Alexandre Malhene; Abstraction of subspace methods in numerical linear algebra; started October 2024, E. Agullo (Inria), L. Giraud.

* PhD in progress: Jordan Ramassamy-Moutoussamy, Learning dynamical system through differentiable data assimilation and machine learning; started on November 2024, S. Gratton (AN-ITI/IRIT), S. Gürol and S. Zhang (IRIT).

* PhD in progress: Aurélien Theret, Near-wall flow reconstruction using Generative AI models; started on January 2025, L. Drozda and N. Odier (Cerfacs).

* PhD in progress: Ariadna Perello-Achfari, Emulation of data assimilation in a global spherical framework; started on September 2025, L. Berre (Météo-France) V. Chabot (Météo-France) and S. Gürol.

* PhD in progress: Younes Essafouri, Development of explainability methods for AI-based regional weather forecasting models; started on October 2025, L. Drozda, L. Raynaud (Météo-France), and L. Risser (Institut de Mathématiques de Toulouse).

* Post-doc in progress: Ieva Daužickaitė, Randomized algorithms for ensemble of variational data assimilations; started on November 2024, L. Berre (Météo-France), M. Destouches (Météo-France), S. Gürol and A. Weaver.

* Post-doc in progress: Rachid El Montassir, AI-based statistical downscaling of climate variables; started on January 2025, L. Drozda, O. Pannekoucke (Météo-France / Cerfacs), L. Terray (Cerfacs).
* Post-doc in progress: Olivier Goux, Development of methods for modelling observation-error correlations; started in March 2025, A. Weaver.

* Post-doc in progress: Théo Boivin, Advanced parallel linear solvers for the CFD simulation software CODA; started in April 2025, L. Giraud, C. Kruse, and P. Mycek.

* Post-doc (finised Feb 2025): Hadrien Godé, Extreme Scale Multigrid through Artificial Intelligence, C. Kruse.

## 8.2.2 Thesis committees

### 8.2.3 Juries

PhD defense

* El Hachimi Anas, “Tensor-Based Computational Methods: Algorithms, Theory, and Applications”; Spécialité: Mathématiques Appliquées, Université du Littoral Côte d'Opale and Université Mohammed VI Polytechnique, Maroc; referees: Nicolas Gillis, Université de Mons - Luc Giraud, Inria, Stefano Serra Capizzano - Université Insubria; members: Lahcen Maniar - Université Cadi Ayyad Marrakech, Hassane Sadok (president) - Université du Littoral Côte d'Opale, Françoise Tisseur - Université de Manchester, Khalide Jbilou - Université du Littoral Côte d'Opale, Ahmed Ratnani - Université Mohammed 6 Polytechnique; Jul. 22, 2025.

* Mouhssine Abdellatif, “Vector extrapolation methods with applications to geometric multigrid and nonlinear least-squares problems”; Spécialité: Mathématiques Appliquées, University of the Littoral Opal Coast and Mohammed VI Polytechnic University; referees: Kees Vuik - Delft University of Technology, Martin Gander - University of Geneva, Luc Giraud - Inria; members: Lahcen Maniar (president) - Cadi Ayyad University, Carole Rosier - University of the Littoral Opal Coast, Ahmed Ratnani - Mohammed VI Polytechnic University, Hassane Sadok - University of the Littoral Opal Coast; Dec. 29, 2025.

HDR defense

* Nicole Spilanne, “High Performance Krylov Subspace Solvers with Preconditioning and Deflation”; Habilitation à Diriger des Recherches Institut Polytechnique de Paris CMAP (CNRS, École polytechnique); referees: Grégoire Allaire, Professor at École polytechnique - Martin Gander, Professor at Université de Genève - Yousef Saad, Professor at University of Minesota; members: Stéphanie Chaillat, CNRS Senior Researcher at EPFL - Marc Embree, Professor at Virginia Tech - Virginie Erlacher, Professor at École des Ponts - Luc Giraud, Senior Researcher at Inria (President) - Laura Grigori, Professor at EPFL - Axel Klawonn, Professor at University of Cologne; Oct. 24, 2025.

PhD follow-up committee (Comité de Suivi Individuel des doctorants - CSI)

* Thomas Buey, “Taking into account horizontal correlations of satellite observation errors in the AROME data assimilation system”, Centre National de Recherches Météorologiques – UMR3589, Toulouse; Supervisors: Nadia Fourrié (CNRM), Oliver Guillet (CNRM), Olivier Audoin (CNRM); CSI members: Benjamin Ménétrier (Norwegian Meteorological Institute, Oslo), Loïk Berre (CNRM, Toulouse), Selime Gürol, Jean-Marie Lalande (CNRM, Lannion).

* Tom Caruso, "Solution de grands systèmes linéaires creux par des méthodes de décomposition de domaine à précision mixte", LJLL, Sorbonne Université; supervisors: Frédéric Nataf (CNRS, LJLL, Inria), Pierre Jolivet (CNRS, LIP6); CSI Members: Luc Giraud, Fabienne Jézéquel (Sorbonne Université, LIP6).

* Hélène Henon, “Multifidelity variational data assimilation for ocean forecasting”, Inria; supervisor: Arthur Vidard (Inria); CSI members: S. Gürol (external expert) and Prieur Clémentine (doctoral school referent).
* Karmijn Hoogveld, "Algorithmes parallèles pour le calcul approximé", IRIT, Université de Toulouse; supervisors: Alfredo Buttari (CNRS, IRIT), Théo Mary (CNRS, LIP6), CSI members: Cédric Févotte (CNRS, IRIT), Luc Giraud.

# 9 Scientific publications of the year

The complete list of publications of the ALGO team is available at [hal.science/ALGO](https://hal.science/ALGO).

## 9.1 Journal articles

[1] F. Böhm, D. Bauer, N. Kohl, C. Alappat, D. Thönnes, M. Mohr, H. Köstler, and U. Rüde. "Code Generation and Performance Engineering for Matrix-Free Finite Element Methods on Hybrid Tetrahedral Grids." In: *SIAM Journal on Scientific Computing* 47.1 (2025), B131–B159. DOI: 10.1137/24M1653756. URL: https://hal.science/hal-04963804 (cited on p. 17).

[2] J. Briant, P. Mycek, M. Destouches, O. Goux, S. Gratton, S. Gürol, E. Simon, and A. T. Weaver. "A filtered multilevel Monte Carlo method for estimating the expectation of cell-centered discretized random fields." In: *International Journal for Uncertainty Quantification* (2025). DOI: 10.1615/Int.J.UncertaintyQuantification.2025057754. URL: https://hal.science/hal-04337114 (cited on p. 13).

[3] M. Cances, L. Giraud, M. Bauerheim, L. Y. Gicquel, and F. Nicoud. "Robustness and reliability of state-space, frame-based modeling for thermoacoustics." In: *Journal of Computational Physics* 520 (2025), p. 113472. DOI: 10.1016/j.jcp.2024.113472. URL: https://hal.science/hal-04734953 (cited on p. 17).

[4] E. Carson and I. Daužickaitė. "Mixed Precision Sketching for Least-Squares Problems and its Application in GMRES-Based Iterative Refinement." In: *SIAM Journal on Matrix Analysis and Applications* (2025). DOI: 10.1137/24M1702246. URL: https://hal.science/hal-05247973 (cited on p. 14).

[5] E. Carson and I. Daužickaitė. "A Comparison of Mixed Precision Iterative Refinement Approaches for Least-Squares Problems." In: *SIAM Journal on Matrix Analysis and Applications* 46.2 (2025), pp. 1117–1144. DOI: 10.1137/24M1664927. URL: https://hal.science/hal-05043779 (cited on p. 14).

[6] M. Chrust, A. Weaver, P. Browne, H. Zuo, and M. A. Balmaseda. "Impact of ensemble-based hybrid background-error covariances in ECMWF's next-generation ocean reanalysis system." In: *Quarterly Journal of the Royal Meteorological Society* 151.767 (2025), Part B e4914. DOI: 10.1002/qj.4914. URL: https://hal.science/hal-04903750 (cited on p. 17).

[7] O. Coulaud, L. Giraud, and M. Iannacito. "A note on TT-GMRES for the solution of parametric linear systems." In: *Electronic Transactions on Numerical Analysis* 62 (2025), pp. 163–187. DOI: 10.1553/etna\_vol62s163. URL: https://inria.hal.science/hal-04986230 (cited on p. 10).

[8] O. Coulaud, L. Giraud, and M. Iannacito. "On some orthogonalization schemes in Tensor Train format." In: *BIT Numerical Mathematics* 65.45 (2025). DOI: 10.1007/s10543-025-01086-5. URL: https://inria.hal.science/hal-05343109 (cited on p. 10).

[9] M. Destouches, P. Mycek, S. Gürol, A. Weaver, S. Gratton, and E. Simon. "Multilevel Monte Carlo methods for ensemble variational data assimilation." In: *Nonlinear Processes in Geophysics* 32.2 (2025), pp. 167–187. DOI: 10.5194/npg-32-167-2025. URL: https://hal.science/hal-04869802 (cited on p. 14).

[10] A. Dumitrasc, C. Kruse, and U. Rüde. "Generalized Golub–Kahan Bidiagonalization for Non-symmetric Saddle-Point Systems." In: *SIAM Journal on Matrix Analysis and Applications* 46.1 (2025), pp. 370–392. DOI: 10.1137/23M160760X. URL: https://hal.science/hal-04963962 (cited on p. 11).
[11] R. Espoeys, L. Brevault, M. Balesdent, S. Ricci, and P. Mycek. "Multifidelity Bayesian Sequential Optimization and Reliability Assessment for Aerospace Systems Design." In: *Journal of Aerospace Information Systems* 663 (2025), pp. 1-24. DOI: 10.2514/1.I011614. URL: https://hal.science/hal-05244435.

[12] E. M. Ettaouchi, L. Giraud, C. Kruse, and N. Tardieu. "Convergence analysis of overlapping domain decomposition preconditioners for nonlinear problems." In: *SIAM Journal on Scientific Computing* 47.6 (2025), A3464-A3493. DOI: 10.1137/25M1735585. URL: https://hal.science/hal-05412939 (cited on p. 15).

[13] O. Goux, A. T. Weaver, S. Gürol, O. Guillet, and A. Piacentini. "On the impact of observation-error correlations in data assimilation, with application to along-track altimeter data." In: *Quarterly Journal of the Royal Meteorological Society* 151.772 (2025). DOI: 10.1002/qj.5026. URL: https://hal.science/hal-05137373 (cited on p. 16).

[14] P. Leleux, C. Schwarz, M. Kühn, C. Kruse, and U. Rüde. "Complexity analysis and scalability of a matrix-free extrapolated geometric multigrid solver for curvilinear coordinates representations from fusion plasma applications." In: *Journal of Parallel and Distributed Computing* 205 (2025), p. 105143. DOI: 10.1016/j.jpdc.2025.105143. URL: https://hal.science/hal-05164801 (cited on p. 13).

[15] D. Mignac, J. Waters, D. J. Lea, M. J. Martin, J. While, A. T. Weaver, A. Vidard, C. Guiavarc'H, D. Storkey, D. Ford, E. W. Blockley, J. Baker, K. Haines, M. R. Price, M. J. Bell, and R. Renshaw. "Improvements to the Met Office's global ocean-sea ice forecasting system including model and data assimilation changes." In: *Geoscientific Model Development* 18.11 (2025), pp. 3405-3425. DOI: 10.5194/gmd-18-3405-2025. URL: https://hal.science/hal-05113966 (cited on p. 16).

[16] R. E. Montassir, L. Drozda, O. Pannekoucke, L. Terray, H. Omrani, and L. Verny. "Training loss matters: exploring new fit-for-purpose loss combinations for statistical downscaling." In: (2025). DOI: 10.22541/essoar.175977614.45331660/v1. URL: http://dx.doi.org/10.22541/essoar.175977614.45331660/v1 (cited on p. 18).

[17] A. Tremodeux, A. Benoit, E. Agullo, T. Herault, L. Giraud, and Y. Robert. "Fault-tolerant numerical iterative algorithms at scale." In: *International Journal of High Performance Computing Applications* (2025). URL: https://inria.hal.science/hal-05234063 (cited on p. 13).

[18] A. Zekri, O. Coulaud, L. Giraud, and J.-R. Poirier. "A Tensor Train solver for the Magnetic Moment Method." In: *IEEE Transactions on Magnetics* (2025), pp. 1-1. DOI: 10.1109/TMAG.2025.3618391. URL: https://hal.science/hal-05311300 (cited on p. 11).

## 9.2 Conferences

[19] I. Dauzickaite, S. Gürol, M. Destouches, L. Berre, M. Chrust, and A. T. Weaver. "Accelerating an ensemble of data assimilations." In: *Journées Scientifiques LEFE/MANU 2025*. Plouzané, France, 2025. URL: https://hal.science/hal-05503772.

[20] I. Dauzickaite, S. Gürol, M. Destouches, L. Berre, M. Chrust, and A. T. Weaver. "Accelerating an ensemble of variational data assimilations with randomized preconditioning." In: *Workshop on Approximate Computing in Numerical Linear Algebra*. Paris, France, 2025. URL: https://hal.science/hal-05503764.

[21] L. Giraud, E. Agullo, O. Coulaud, M. Iannacito, M. Issa, G. Marait, and M. Rozloznik. "A journey through some numerical linear algebra algorithms with variable accuracy storage." In: *CAS-ANLA 2025 - The Chinese Academy of Sciences Workshop on Approximate computing in Numerical Linear Algebra*. Beijing, China, 2025. URL: https://hal.science/hal-05049142.
[22] H. Godé, C. Kruse, R. Angersbach, H. Köstler, M. Bauerheim, and U. Rüde. "Comparison of Multigrid and Machine Learning-Based Poisson Solvers." In: *Parallel Processing and Applied Mathematics 15th International Conference, PPAM 2024*. Ed. by R. Wyrzykowski, J. Dongarra, E. Deelman, and K. Karczewski. Vol. Lecture Notes in Computer Science. Lecture Notes in Computer Science LNCS-15581. Ostrava, Czech Republic: Springer Nature Switzerland, 2024, pp. 174–189. DOI: 10.1007/978-3-031-85703-4\_12. URL: https://hal.science/hal-05164864.

[23] A.-K. M. E. Maarouf, L. Giraud, A. Guermouche, and T. Guignon. "Sparse Matrix Ordering for Fine Grain Parallel Triangular Solve Using SIMD." In: *Parallel Processing and Applied Mathematics (PPAM 2024)*. Vol. LNCS. Lecture Notes in Computer Science 15579. Czestochowa University of Technology and Technical University of Ostrava. Ostrava, Czech Republic: Springer Nature Switzerland, 2024, pp. 51–64. DOI: 10.1007/978-3-031-85697-6\_4. URL: https://hal.science/hal-05185404.

[24] A. Malhéné, E. Agullo, L. Giraud, and M. Rozloznik. "Practical stopping criteria for GMRES." In: *Sparse Days meeting 2025*. Toulouse, France, 2025. URL: https://inria.hal.science/hal-05200599.

[25] S. Nakov, E. Agullo, A. Buttari, L. Giraud, and G. Marait. "Scalable Domain Decomposition Methods for Large Sparse Linear Systems on Modern Architectures." In: *COMPAS 2025 - Conference francophone d'informatique en Parallélisme, Architecture et Système*. Bordeaux, France, 2025. URL: https://inria.hal.science/hal-05248772.

[26] S. Nakov, E. Agullo, A. Buttari, L. Giraud, and G. Marait. "Scalable Domain Decomposition Methods for Large Sparse Linear Systems on Modern Architectures." In: *Sparse Days 2025*. Toulouse, France, 2025. URL: https://inria.hal.science/hal-05248757.

[27] Y. Xiang and L. Giraud. "Convolution neural operator preconditioning for the solution of some heterogeneous PDES." In: *DTE - AICOMAS 2025 - Digital Twins in Engineering & Artificial Intelligence and Computational Methods in Applied Science*. Kenjiro Terada, John Dolbow, Ferdinando Auricchio, Harald van Brummelen, Dominique Baillargeat, Charbel Farhat, Georges Karniadakis, Stefan Kollmannsberger, Trond Kvamsdal, Vissarion Papadopoulos, Ruben Sevilla. Paris, France, 2025. URL: https://inria.hal.science/hal-04961884.

[28] A. E. M. Zekri, O. Coulaud, L. Giraud, and J.-R. Poirier. "Tensor Train-Based Approach for the Magnetic Moment Method." In: *Compumag 2025 - 25th International Conference on the Computation of Electromagnetic Fields*. Naples, Italy, 2025. URL: https://hal.science/hal-05291496.

## 9.3 Doctoral dissertations and habilitation theses

[29] O. Goux. "Accounting for correlated observation error in variational ocean data assimilation: application to altimeter data." Theses. ISAE-SUPAERO, 2025. URL: https://hal.science/tel-04990207.

## 9.4 Reports & preprints

[30] O. Coulaud, L. Giraud, M. Iannacito, and M. Issa. *Solving eigenvalue problems in high dimensions using contour integration and Tensor Train format*. Tech. rep. RR-9586. Inria, 2025, p. 19. URL: https://inria.hal.science/hal-05017425 (cited on p. 11).

[31] I. Dauzickaite, M. A. Freitag, S. Gürol, A. S. Lawless, J. A. Scott, A. Ramage, and J. M. Tabeart. *An Introduction to Solving the Least-Squares Problem in Variational Data Assimilation*. Tech. rep. CERFACS, 2025. URL: https://hal.science/hal-05116307 (cited on p. 14).

[32] E. M. Ettaouchi, L. Giraud, C. Kruse, and N. Tardieu. *Convergence analysis of overlapping domain decomposition preconditioners for nonlinear problems*. Tech. rep. RR-9575. INRIA, 2025. URL: https://hal.science/hal-04958735.
[33] L. Giraud, C. Kruse, P. Mycek, M. Shpakovych, and Y. Xiang. Neural network preconditioning: a case study for the solution of the parametric Helmholtz equation. Tech. rep. RR-9593. Inria Centre at the University of Bordeaux, France, 2025. URL: https://hal.science/hal-05157038 (cited on p. 12).

[34] L. Giraud and Y. Xiang. A note on the partial convergence management for the solution of symmetric linear systems with multiple right-hand sides. Tech. rep. RR-9574. Inria Centre at the University of Bordeaux, 2025. URL: https://inria.hal.science/hal-04922247 (cited on p. 11).

[35] C. Kruse, A. Clara Ordonez Egas, and N. Tardieu. A Scalable and Parameter-Robust Preconditioner for a Second Gradient of Dilation Regularization Applied to a Mechanics Problem. Tech. rep. Cerfacs ; EDF R&D, 2025. URL: https://hal.science/hal-05209343 (cited on p. 12).

[36] J. Litz, P. Leleux, C. Kruse, J. Gedicke, and M. J. Kühn. Memory-and compute-optimized geometric multigrid GMGPolar for curvilinear coordinate representations -Applications to fusion plasma. Tech. rep. Cerfacs - Toulouse, France, 2025. URL: https://hal.science/hal-05153605 (cited on p. 12).

[37] J. M. Tabeart, S. Gürol, J. W. Pearson, and A. T. Weaver. "Block Alpha-Circulant Preconditioners for All-at-Once Diffusion-Based Covariance Operators." working paper or preprint. 2025. DOI: 10.48550/arXiv.2506.03947. URL: https://hal.science/hal-05175107 (cited on p. 15).

[38] A. Tremodeux, E. Agullo, A. Benoit, L. Giraud, T. Herault, and Y. Robert. Fault-tolerant numerical iterative algorithms at scale. Tech. rep. RR-9567. Inria Lyon, 2025. URL: https://inria.hal.science/hal-04872041.