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TWO-GRID AND MULTIPLE-GRID ARNOLDI FOR EIGENVALUES
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DOI：10.1137/16M1062260.png)
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Abstract
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A new approach is given for computing eigenvalues and eigenvectors of large matrices. Multiple grids are combined with the Arnoldi method in order to solve difficult problems. First, a two-grid method computes eigenvectors on a coarse grid and improves them on the fine grid. On the fine grid, an Arnoldi-type method is used that, unlike standard Arnoldi methods, can accept initial approximate eigenvectors. This two-grid approach can vastly improve Krylov computations. It also succeeds for problems that regular multigrid cannot handle. Analysis is given to explain why fine grid convergence rates can sometimes unexpectedly match that of regular Arnoldi on the fine grid in spite of having many initial guess vectors. Near-Krylov theory is developed for this. Finally, this new method is generalized for more grid levels to produce a multiple-grid Arnoldi method.
Keywords:
eigenvalues
Arnoldi
multigrid
Krylov
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