Multirate behavior of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) is characterized by widely separated time constants in different components of the solution or different additive terms of the right-hand side. Here, classical multirate schemes are dedicated solvers, which apply (e.g.) micro and macro steps to resolve fast and slow changes in a transient simulation accordingly. The use of extrapolation and interpolation procedures is a genuine way for coupling the different parts, which are defined on different time grids.
This paper contains for the first time, to the best knowledge of the authors, a complete convergence theory for inter/extrapolation-based multirate schemes for both ODEs and DAEs of index one, which are based on the fully-decoupled approach, the slowest-first and the fastest-first approach. The convergence theory is based on linking these schemes to multirate dynamic iteration schemes, i.e., dynamic iteration schemes without further iterations. This link defines naturally stability conditions for the DAE case.
This work considers multirate generalized-structure additively partitioned Runge-Kutta (MrGARK) methods for solving stiff systems of ordinary differential equations (ODEs) with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate cou…
We use matrix stability analysis for a singlerate and multirate predictor-corrector scheme (PC) used to solve the incompressible Navier-Stokes equations (INSE) in overlapping grids. By simplifying the stability analysis with the unsteady heat equation in 1D, we demonstrate that, as expected, the stability of the PC scheme increases with increase in the resolution and overlap of subdomains. For singlerate timestepping, we also find that the high-order PC scheme is stable when the number of corrector iterations ($Q$) is odd. This difference in the stability of odd- and even-$Q$ is novel and has …
Aitken extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduced that enables successive decomposition of high Eigen values of the iteration matrix to enable convergence. While extrapolation of a convergent fixed point iteration using a geometric series sum is a known form of Aitken acceleration, it is shown in this paper that the same formula can be used to estimate the solution of sets of linear equations from diverging Gauss Seidel iterations. In both conve…
In this paper, we are concerned with the global pressure formulation of immiscible incompressible two-phase flow between different rock types. We develop for this problem two robust schemes based on domain decomposition (DD) methods and operator-splitting techniques. The first scheme follows a sequential procedure in which the (global) pressure, the saturation-advection and the saturation-diffusion problems are fully decoupled. In this scheme, each problem is treated individually using various DD approaches and specialized numerical methods. The coupling between the different problems is expli…
Schwarz Modulus Based Matrix Splittings with Minimal Polynomial Extrapolation Acceleration for linear complementarity problems arising from American option pricing
Accelerating AI Performance using Anderson Extrapolation on GPUs
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Applied Physics Program, Physical Sciences and Engineering Division（物理科学与工程分部应用物理项目）；Extreme Computing Research Center, Computer, Electrical and Mathematical Sciences and Engineering Division（计算机、电气与数学科学与工程分部极端计算研究中心）
This article introduces new acceleration methods for fixed-point iterations. Extrapolations are computed using two or three mappings alternately and a new type of step length is proposed with good properties for nonlinear applications. The methods require no problem-specific adaptation and are especially efficient in high-dimensional contexts. Their computation uses few objective function evaluations, no matrix inversion and little extra memory. A convergence analysis is followed by eight applications including gradient descent acceleration for constrained and unconstrained optimization. Perfo…
An extrapolation cascadic multigrid method combined with a fourth order compact scheme for 3D poisson equation
School of Mathematics and Statistics, Central South University（中南大学数学与统计学院）；School of Aerospace Engineering and Applied Mechanics, Tongji University（同济大学航空航天与力学学院）
Optimization acceleration techniques such as momentum play a key role in state-of-the-art machine learning algorithms. Recently, generic vector sequence extrapolation techniques, such as regularized nonlinear acceleration (RNA) of Scieur et al., were proposed and shown to accelerate fixed point iterations. In contrast to RNA which computes extrapolation coefficients by (approximately) setting the gradient of the objective function to zero at the extrapolated point, we propose a more direct approach, which we call direct nonlinear acceleration (DNA). In DNA, we aim to minimize (an approximation…
