Block Jacobi/Gauss-Seidel preconditioning for GLT sequences, and GLH sequences
For researchers using GLT theory to analyze preconditioners for differential equation discretizations, this provides theoretical justification for the efficiency of block Jacobi/Gauss-Seidel preconditioning.
The paper proves that block Jacobi and block Gauss-Seidel preconditioners for GLT sequences preserve the GLT property with the same symbol, enabling prediction of their efficiency, and extends the Fasino-Tilli theorem to GLH sequences.
The theory of generalized locally Toeplitz (GLT) sequences is an apparatus for computing the spectral and singular value distribution of sequences of matrices that possess a (possibly hidden) Toeplitz-like structure. These sequences, which are known as GLT sequences, arise in several applications, including the discretization of differential equations. Associated with any GLT sequence is a special function called symbol. In this paper, we prove that, if $\{A_n\}_n$ is a GLT sequence with symbol $κ$ and $P_n$ is any block Jacobi or block Gauss-Seidel preconditioner for $A_n$ with a fixed number of blocks independent of $n$, then $\{P_n\}_n$ is a GLT sequence with symbol $κ$, just like $\{A_n\}_n$. This result allows us to predict a remarkable efficiency of block Jacobi/Gauss-Seidel preconditioning for GLT sequences, which is in fact illustrated through numerical experiments. It also allows us to extend the Fasino-Tilli theorem on the zero distribution of Hankel matrix sequences generated by $L^1$ functions to a larger class of matrix sequences called generalized locally Hankel (GLH) sequences.