On symmetric positive definite preconditioners for multiple saddle-point systems

We consider symmetric positive definite preconditioners for multiple saddle-point systems of block tridiagonal form, which can be applied within the MINRES algorithm. We describe such a preconditioner for which the preconditioned matrix has only two distint eigenvalues, 1 and -1, when the preconditioner is applied exactly. We discuss the relative merits of such an approach compared to a more widely studied block diagonal preconditioner, specify the computational work associated with applying the new preconditioner inexactly, and survey a number of theoretical results for the block diagonal case. Numerical results validate our theoretical findings.

Automated summary

This paper considers symmetric positive definite preconditioners for multiple saddle-point systems of block tridiagonal form, which can be applied within the MINRES algorithm. The authors describe a preconditioner that yields a preconditioned matrix with only two distinct eigenvalues, 1 and -1, when applied exactly. They discuss the advantages of this approach compared to a more widely studied block diagonal preconditioner, analyze the computational work associated with applying the new preconditioner inexactly, and survey various theoretical results for the block diagonal case. Numerical results confirm the authors' theoretical findings.