ILU and Machine Learning Based Preconditioning For The Discretized Incompressible Navier-Stokes Equations.

Abstract

Motivated by the numerical solution of the incompressible Navier-Stokes equations, this thesis studies numerical properties of threshold incomplete LU factorizations for nonsymmetric saddlepoint matrices. We consider preconditioned iterative Krylov-subspace methods, such as GMRES, to solve large and sparse linear algebraic systems that result from a Galerkin nite element (FE) discretizations of the linearized Navier-Stokes equations. The corresponding preconditioners are used to accelerate the convergence of the GMRES method. Stabilized and unstabilized nite element methods are used for the Navier-Stokes problem leading to systems of algebraic equations of a saddle point type, which has a 2 2-block structure. Numerical experiments for model problems of a driven cavity flow, and flow over a backward-facing step illustrate the performance of one-parameter and two-parameter ILU factorizations as preconditioners. We also introduce a Machine Learning (ML) based approach for building ILU factorizations for preconditioning. For this purpose, we use the tools well-developed in the scope of image segmentation. Image Segmentation is the process of partitioning an image into separate and distinct regions.The process has some similarities to building patterns for ILU-type preconditioner. In our interpretation, the segmented regions represent a non-zero pattern for L and U factors. We applied a convolutional neural network with the benchmark U-net architecture to predict non-zero patterns for ILU-type factorizations and further use the resulting preconditioners to solve the discrete linearized Navier-Stokes system.