# CASI Project

### Design and analysis of mathematical methods for high performance scientific computing

### I. Foreword

The Toulouse site brings together a large and varied community around digital simulation, particularly in relation to aeronautics and earth observation. The complexity of the phenomena simulated in these applications gives rise to problems characterized by their large size, which often makes them out of reach of conventional methods commonly accessible. The aim of the project team is to propose a framework of collaboration between IRIT and IMT researchers, allowing their skills to be pooled, related to mathematical analysis, numerical methods, optimal exploitation of computing architectures to solve boundary problems.

The themes of the team will be articulated around four topics

- Algorithms for Large Structured Linear / Singular Problems,
- Data assimilation and uncertainty quantification,
- Multi-domain and multi-scale analysis to deal with large-scale non-linearity,
- Tools and paradigms for parallel programming.

### II. Main axes of the project team

### 1. Algorithms for Large Structured Linear Problems

Methods of solving numerical simulation problems often lead to systems of equations, linear or not. When the unknowns of the system live in functional spaces, it is a matter of discretization. We are interested in situations where the final resolution step is a linear system of large size, out of reach, for questions of precision or use of computer resources, techniques available in the literature. To tackle these difficulties, we will combine a mathematical approach based on functional analysis tools, mathematical modeling (asymptotic analysis, numerical analysis) and a computer-based approach based on algorithmic analysis and implementation on a parallel computer.

Topics will include:

a)** Specific problems:** for example, singular problems compatible (as encountered in various inverse problems including imaging) or the study of the asymptotic behavior of solutions of partial differential equations.

b) **The creation and analysis of algorithms.** Will be particularly concerned those resulting from the coupling between direct methods (supported in particular on graph theory and stability analysis) and Krylov type projection methods (in connection with the theory of approximation). Special interest may also be given to problems with multiple second members, and indefinite problems.

__ IRIT members:__ Patrick Amestoy, Ronan Guivarch, Pierre Jolivet, Daniel Ruiz

__ IMT members:__ Robin Bouclier, Franck Boyer, Frederic Couderc, Fabrice Deluzet, Michel Fournié.

### 2. Data assimilation and uncertainty quantification

Data assimilation brings together a set of techniques for predicting the state of a system from system dynamics equations and observations. This discipline, particularly adapted to large-scale environmental systems, has historically developed in a variational framework, based on Bayesian estimation, optimization and control. The need to deal with larger problems and to more finely model model errors, while providing predictability indicators, has stimulated the development of stochastic filtering techniques, alone or in addition to more conventional variational techniques. . This disciplinary field of hybrid assimilation (between variational and set-up) is in full expansion and will benefit from the combined expertise of the two institutes in control, variational methods, set-ups and statistical techniques, with the aim notably of creating and studying algorithms of the EnVar type.

Another area is also at the crossroads of the skills of the IMT and IRIT is the domain of the quantification of uncertainties in the physical and digital models, in which the treatment of space of dimensions, even modest constitutes a real challenge . The aim will be to create the conditions for collaboration between specialists in statistical sampling and experts in high performance computing.

** IRIT members:** Olivier Cots, Joseph Gergaud, Serge Gratton, Ehouarn Simon

** IMT members:** Luca Amodei, Jérôme Fehrenbach, Jérôme Monnier, Pascal Noble and Jean-Pierre Raymond.

### 3. Multi-domain and multi-scale non-linear analysis

Methods for decomposing domains into space and time paralleling methods are, especially with multigrid techniques, among the main approaches for increasing parallelism in numerical methods. They have become unavoidable for dealing with large problems from complex physical problems. These methods are widely developed to solve linear problems, but continue to pose a number of challenges in nonlinear frameworks. For example, the choice of the rough grid in subdomain methods, the adaptation of grids in algebraic multigrid methods or the consideration of non-linearity in these algorithms. It will be a question here of pooling algorithmic skills of institutes for non-linear problems, in numerical methods to solve systems of partial differential equations to produce new globally convergent algorithms in functional spaces.

__ IRIT members:__ Serge Gratton, Pierre Jolivet, Xavier Vasseur (ISAE)

__ IMT members:__ Christophe Besse, Robin Bouclier, Fabrice Deluzet, Francis Filbet, Pascal Noble, Marie-Hélène Vignal.

### 4. Tools and paradigms for parallel programming

At the heart of all the aforementioned collaborative themes is the concern for the search for numerical methods capable of taking advantage of current, parallel and sometimes heterogeneous architectures to deal with border problems. Developing efficient algorithms (computation time, memory, energy consumption) requires a constant effort on the side of programming tools, which must allow to solve, sometimes in real time fundamental questions, around the complexity in operations, in memory and the load balancing of the calculation. These theoretical questions will be at the heart of a collaboration between the two institutes and the meso-center Calmip.

__ IRIT members:__ Patrick Amestoy, Alfredo Buttari, Pierre Jolivet.

__ IMT members:__ Frédéric Couderc, Michel Fournié, Jacek Narski.

__ Calmip:__ Nicolas Renon