Deflation and projection methods applied to symmetric positive semi-definite systems

Elisabeth Ludwig, R Nabben, Jok Tang

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


Linear systems with a singular symmetric positive semi-definite matrix appear frequently in practice. This usually does not lead to difficulties for CG methods as long as these systems are consistent. However, the construction of a preconditioner, especially the construction of two-level and multilevel methods, becomes more complicated, since singular coarse grid matrices or Galerkin matrices may occur. Here we continue the work started in [21,22] where deflation is used for some special singular coefficient matrices. Here we show that deflation and other projection-type preconditioners can be applied to arbitrary singular problems without any difficulties. In each of these methods, a two-level preconditioner is involved where coarse-grid systems based on a singular Galerkin matrix should be solved. We prove that each projection operator consisting of a singular Galerkin matrix can be written as an operator with a nonsingular Galerkin matrix. Therefore many results that hold for nonsingular Galerkin matrices are also valid for problems with singular Galerkin matrices.

Original languageEnglish
Pages (from-to)253-273
Number of pages21
JournalLinear Algebra and Its Applications
Publication statusPublished - 15 Jan 2016


  • Coarse-grid corrections
  • Conjugate gradients
  • Deflation
  • Domain decomposition
  • Multigrid
  • Preconditioning
  • Projection methods
  • SPSD matrices
  • Two-grid schemes


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