4 research outputs found
Smoothing analysis of two-color distributive relaxation for solving 2D Stokes flow by multigrid method
Smoothing properties of two-color distributive relaxation for solving a
two-dimensional (2D) Stokes flow by multigrid method are theoretically
investigated by using the local Fourier analysis (LFA) method. The governing
equation of the 2D Stokes flow in consideration is discretized with the
non-staggered grid and an added pressure stabilization term with stabilized
parameters to be determined is introduced into the discretization system in
order to enhance the smoothing effectiveness in the analysis. So, an important
problem caused by the added pressure stabilization term is how to determine a
suitable zone of parameters in the added term. To that end, theoretically, a
two-color distributive relaxation, developed on the two-color Jacobi point
relaxation, is established for the 2D Stokes flow. Firstly, a mathematical
constitution based on the Fourier modes with various frequency components is
constructed as a base of the two-color smoothing analysis, in which the related
Fourier representation is presented by the form of two-color Jacobi point
relaxation. Then, an optimal one-stage relaxation parameter and related
smoothing factor for the two-color distributive relaxation are applied to the
discretization system, and an analytical expression of the parameter zone on
the added pressure stabilization term is established by LFA. The obtained
analytical results show that numerical schemes for solving 2D Stokes flow by
multigrid method on the two-color distributive relaxation have a specific
convergence zone on the parameters of the added pressure stabilization term,
and the property of convergence is independent of mesh size, but depends on the
parameters of the pressure stabilization term
Learning Relaxation for Multigrid
During the last decade, Neural Networks (NNs) have proved to be extremely
effective tools in many fields of engineering, including autonomous vehicles,
medical diagnosis and search engines, and even in art creation. Indeed, NNs
often decisively outperform traditional algorithms. One area that is only
recently attracting significant interest is using NNs for designing numerical
solvers, particularly for discretized partial differential equations. Several
recent papers have considered employing NNs for developing multigrid methods,
which are a leading computational tool for solving discretized partial
differential equations and other sparse-matrix problems. We extend these new
ideas, focusing on so-called relaxation operators (also called smoothers),
which are an important component of the multigrid algorithm that has not yet
received much attention in this context. We explore an approach for using NNs
to learn relaxation parameters for an ensemble of diffusion operators with
random coefficients, for Jacobi type smoothers and for 4Color GaussSeidel
smoothers. The latter yield exceptionally efficient and easy to parallelize
Successive Over Relaxation (SOR) smoothers. Moreover, this work demonstrates
that learning relaxation parameters on relatively small grids using a two-grid
method and Gelfand's formula as a loss function can be implemented easily.
These methods efficiently produce nearly-optimal parameters, thereby
significantly improving the convergence rate of multigrid algorithms on large
grids.Comment: This research was carried out under the supervision of Prof. Irad
Yavneh and Prof. Ron Kimmel. XeLate