TY - JOUR
T1 - A High-Quality Preconditioning Technique for Multi-Length-Scale Symmetric Positive Definite Linear Systems
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 469
EP - 484
PY - 2009
DA - 2009/02
SN - 2
DO - http://doi.org/10.4208/nmtma.2009.m9008s
UR - https://global-sci.org/intro/article_detail/nmtma/6036.html
KW - Preconditioning, multi-length-scale, incomplete Cholesky factorization, quantum Monte
Carlo simulation.
AB - <p style="text-align: justify;">We study preconditioning techniques used in conjunction with the conjugate
gradient method for solving multi-length-scale symmetric positive definite linear systems originating from the quantum Monte Carlo simulation of electron interaction of
correlated materials. Existing preconditioning techniques are not designed to be adaptive to varying numerical properties of the multi-length-scale systems. In this paper,
we propose a hybrid incomplete Cholesky (HIC) preconditioner and demonstrate its
adaptivity to the multi-length-scale systems. In addition, we propose an extension of
the compressed sparse column with row access (CSCR) sparse matrix storage format to
efficiently accommodate the data access pattern to compute the HIC preconditioner. We
show that for moderately correlated materials, the HIC preconditioner achieves the optimal linear scaling of the simulation. The development of a linear-scaling preconditioner
for strongly correlated materials remains an open topic.</p>