Advanced nonlinear preconditioning methods for solving problems with high heterogeneities in geosciences

In numerical simulation, the discretization of physical models made up of partial differential equations often gives rise to non-linear algebraic systems. The resolution of these systems is generally not easy, due to the large number of unknowns and especially their non-linear character. Traditionally, this is done using Newton's method, which linearizes the problem around the current iterate. As a result, Newton's method often exhibits convergence problems, notably when the initial point is far from the true solution or when the system contains strong stiffnesses.
While the methods for solving linear systems and their preconditioning are relatively well established, the search for more robust and faster nonlinear techniques is still ongoing. For the last twenty years, several authors have been suggesting that preconditioning methods for linear systems, such as domain decomposition, should be transposed to nonlinear problems. It turns out that in addition to the gain in computation time, this approach also allows one to overcome the local character of Newton's method. In view of recent publications indicating the high potential of this approach for certain models of flow and transport in porous media, it is essential to scrutinize the relevance of these methods on more realistic geoscience models such as those of IFPEN.
In this work, we have chosen to study the RASPEN (Restricted Additive Schwarz with Exact Newton) method and the non-linear Schur techniques known for their simplicity and efficiency. We first focus on the single level versions in order to better understand the localization of the computations in the critical regions near the interfaces and the automatic construction of these interfaces. Then, we focus on the search for a better parallel scalability by investigating the two-level versions of these methods. The implementation aspects on modern computing architectures will also be addressed.

Keywords: nonlinear algebraic systems, Newton’s method, preconditioning, domain decomposition, Schur complement, parallel computing, flows in porous media

• Academic supervisor    BRENNER Konstantin (PhD), INRIA Sophia-Antipolis Méditerranée
• Doctoral School    ED SFA 364, http://www.ecoles-doctorales-aerospatiales.fr/fr/ecoles/37.htm
• IFPEN supervisor    TRAN Quang Huy (HDR), département Mathématiques Appliquées, quang-huy.tran@ifpen.fr, ORCID : 0000-0001-7771-3154
• PhD location    IFPEN (Rueil-Malmaison) and Laboratoire Jean Dieudonné (Nice)
• Duration and start date    3 years, starting in November 2023
• Employer    IFPEN (Rueil-Malmaison)
• Academic requirements    University Master degree in Numerical Analysis, Scientific Computing
• Language requirements    Fluency in English, willingness to learn French
• Other requirements    Matlab, Scilab, Python, C++