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 -

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.