Thời gian: 13g30-15g, Thứ Sáu, ngày 07 tháng 10 năm 2022

Địa điểm: Phòng I44

Title: Nonlinear preconditioning techniques and their applications in
scientific computing

Speaker: Feng-Nan Hwang (Department of Mathematics, National Central
University, Taiwan)

Abstract

An inexact Newton-type (IN) method is one of the most popular methods
for solving large, sparse, algebraic nonlinear systems of equations
due to their easy implementation and fast local convergence if a good
initial guess is available. However, the method suffers from slow
convergence or convergence failure for the unbalanced nonlinear
systems even in conjunction with globalization techniques, such as
line search backtracking, grid-sequencing, and parameter continuation.
Unbalanced nonlinear systems arise in many computational science and
engineering applications, for example, from the discretization of PDEs
whose solutions involve shocks, boundary and internal layers, or
singularity. Alternatively, nonlinear preconditioning techniques have
been shown numerically promising to enhance the robustness and improve
the efficiency of nonlinear iterative methods. In this talk, we begin
by reviewing linear preconditioning and then discuss its
generalization to nonlinear cases. We introduce some designs of the
nonlinear preconditioners based on nonlinear domain decomposition
methods, nonlinear elimination methods, and calculus of variations,
followed by their applications to incompressible and compressible
fluid flow simulation, flow control, and trajectory optimization
problems. We also provide a geometric interpretation of nonlinear
preconditioners through some simple problems. Finally, some
performance studies on PCs and PC clusters for benchmark problems and
real-world applications are presented to demonstrate the effectiveness
and efficiency of the preconditioned Newton methods compared to some
alternative methods.

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