## 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.

### MSC:

 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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