472 research outputs found
A Survey on Intelligent Iterative Methods for Solving Sparse Linear Algebraic Equations
Efficiently solving sparse linear algebraic equations is an important
research topic of numerical simulation. Commonly used approaches include direct
methods and iterative methods. Compared with the direct methods, the iterative
methods have lower computational complexity and memory consumption, and are
thus often used to solve large-scale sparse linear equations. However, there
are numerous iterative methods, parameters and components needed to be
carefully chosen, and an inappropriate combination may eventually lead to an
inefficient solution process in practice. With the development of deep
learning, intelligent iterative methods become popular in these years, which
can intelligently make a sufficiently good combination, optimize the parameters
and components in accordance with the properties of the input matrix. This
survey then reviews these intelligent iterative methods. To be clearer, we
shall divide our discussion into three aspects: a method aspect, a component
aspect and a parameter aspect. Moreover, we summarize the existing work and
propose potential research directions that may deserve a deep investigation
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
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