Carol S. WOODWARD, Lawrence Livermore National Laboratory, USA:
A reconsideration of fixed point methods for nonlinear systems
Newton-Krylov methods have proven to be very effective for solution of large-scale, nonlinear systems of equations resulting from discretizations of PDEs. However, increasing complexities and newer models are giving rise to nonlinear systems with characteristics that challenge this commonly used method. In particular, for many problems, Jacobian information may not be available or it may be too costly to compute. Moreover, linear system solves required to update the linear model within each Newton iteration may be too costly on newer machine architectures.
Fixed point iteration methods have not been as commonly used for PDE systems due to their slow convergence rate. However, these methods do not require Jacobian information nor do they require a linear system solve. In addition, recent work has employed Anderson acceleration as a way to speed up fixed point iterations.
In this presentation, we will discuss reasons for success of Newton’s method as well as its weaknesses. Fixed point and Anderson acceleration will be presented along with a summary of known convergence results for this accelerated method. Results will show benefits from this method for a number of applications. In addition, the impacts of these methods will be discussed for large-scale problems on next generation architectures.
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC. LLNL-ABS-663073.
Friday, December 4, 10:30 a.m. to 11:30 a.m., Turing lecture hall, building 1, Paris-Rocquencourt center. Coffee from 10:15 a.m.
