A multigrid method to solve large scale Sylvester equations. (English) Zbl 1154.65024

The authors consider the Sylvester matrix equation \(AX-XB+ C=0\), where \(A\) and \(-B\) are stiffness matrices from the discretization of a linear elliptic partial differential operator, the matrix \(C\) is of low rank given in factorized form. They develop a multigrid method for computing a low rank approximation to the solution. They use the usual Jacobian smoother and standard prolongation and restriction operators but extend the basic multigrid cycle by a projection step that ensures that the rank of each iterate is bounded. The algorithm is of linear complexity instead of quadratic complexity.


65F30 Other matrix algorithms (MSC2010)
65F05 Direct numerical methods for linear systems and matrix inversion
65F50 Computational methods for sparse matrices
15A24 Matrix equations and identities
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