TY - JOUR
T1 - A block α-circulant based preconditioned MINRES method for wave equations
AU - Lin, Xue lei
AU - Hon, Sean
N1 - The work of Sean Hon was supported in part by the Hong Kong RGC under grant 22300921, a start-up grant from the Croucher Foundation, and a Tier 2 Start-up Grant from Hong Kong Baptist University. The work of Xue-lei Lin was supported by research grants: 2021M702281 from China Postdoctoral Science Foundation, 12301480 from NSFC, HA45001143 from Harbin Institute of Technology, Shenzhen, HA11409084 from Shenzhen.
Publisher Copyright:
© 2024 IMACS.
PY - 2024/11/6
Y1 - 2024/11/6
N2 - In this work, we propose an absolute value block α-circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Motivated by the absolute value block circulant preconditioner proposed in McDonald et al. (2018) [40], we propose an absolute value version of the block α-circulant preconditioner. Since the original block α-circulant preconditioner is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Our proposed preconditioner is the first Hermitian positive definite variant of the block α-circulant preconditioner for the concerned wave equations, which fills the gap between block α-circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for a properly chosen α the MINRES solver with the proposed preconditioner achieves a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance the concerned problem. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved.
AB - In this work, we propose an absolute value block α-circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Motivated by the absolute value block circulant preconditioner proposed in McDonald et al. (2018) [40], we propose an absolute value version of the block α-circulant preconditioner. Since the original block α-circulant preconditioner is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Our proposed preconditioner is the first Hermitian positive definite variant of the block α-circulant preconditioner for the concerned wave equations, which fills the gap between block α-circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for a properly chosen α the MINRES solver with the proposed preconditioner achieves a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance the concerned problem. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved.
KW - Block Toeplitz matrix
KW - Convergence of MINRES solver
KW - Wave equations
KW - Absolute value block 𝛼-circulant
KW - preconditioners
UR - http://www.scopus.com/inward/record.url?scp=85208681727&partnerID=8YFLogxK
U2 - 10.1016/j.apnum.2024.10.020
DO - 10.1016/j.apnum.2024.10.020
M3 - Journal article
AN - SCOPUS:85208681727
SN - 0168-9274
VL - 209
SP - 1
EP - 17
JO - Applied Numerical Mathematics
JF - Applied Numerical Mathematics
ER -