On the Use of Incomplete LU Decomposition as a Preconditioning Technique for Density Fitting in Electronic Structure Computations

Incomplete factorization preconditioners combined with Krylov subspace accelerators are currently among the most effective methods for iteratively solving large systems of linear equations. In this paper we consider the use of a dual threshold incomplete LU factorization (ILUT) preconditioner for the iterative solution of the linear equation systems encountered when performing electronic structure calculations that involve density fitting. Two questions are addressed, how the overall performance of the ILUT method varies as a function of the accuracy of the preconditioning matrix, and whether it is possible to make approximations to the original matrix on which the LU decomposition is based and still obtain a good preconditioner. With respect to overall performance both computational and memory storage requirements are considered, while in terms of approximations both those based on numerical and physical arguments are considered. The results indicate that under the right circumstances the ILUT method is superior to fully direct approaches such as singular value decomposition.

[1]  Christof Hättig,et al.  CC2 excitation energy calculations on large molecules using the resolution of the identity approximation , 2000 .

[2]  Alistair P. Rendell,et al.  COUPLED-CLUSTER THEORY EMPLOYING APPROXIMATE INTEGRALS : AN APPROACH TO AVOID THE INPUT/OUTPUT AND STORAGE BOTTLENECKS , 1994 .

[3]  S. Ten-no,et al.  Density fitting for the decomposition of three-electron integrals in explicitly correlated electronic structure theory , 2003 .

[4]  J. Connolly,et al.  On first‐row diatomic molecules and local density models , 1979 .

[5]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[6]  Martin W. Feyereisen,et al.  Use of approximate integrals in ab initio theory. An application in MP2 energy calculations , 1993 .

[7]  Valeria Simoncini,et al.  Recent computational developments in Krylov subspace methods for linear systems , 2007, Numer. Linear Algebra Appl..

[8]  Frederick R. Manby,et al.  Density fitting in second-order linear-r12 Møller–Plesset perturbation theory , 2003 .

[9]  J. L. Whitten,et al.  Coulombic potential energy integrals and approximations , 1973 .

[10]  W. Klopper.  A hybrid scheme for the resolution-of-the-identity approximation in second-order Møller-Plesset linear-r(12) perturbation theory. , 2004, The Journal of chemical physics.

[11]  Christof Hättig,et al.  Optimization of auxiliary basis sets for RI-MP2 and RI-CC2 calculations: Core–valence and quintuple-ζ basis sets for H to Ar and QZVPP basis sets for Li to Kr , 2005 .

[12]  Jun Zhang,et al.  BILUM: Block Versions of Multielimination and Multilevel ILU Preconditioner for General Sparse Linear Systems , 1999, SIAM J. Sci. Comput..

[13]  Frederick R. Manby,et al.  Fast linear scaling second-order Møller-Plesset perturbation theory (MP2) using local and density fitting approximations , 2003 .

[14]  M. Benzi.  Preconditioning techniques for large linear systems: a survey , 2002 .

[15]  Holger Patzelt,et al.  RI-MP2: optimized auxiliary basis sets and demonstration of efficiency , 1998 .

[16]  Robert J. Harrison,et al.  An implementation of RI–SCF on parallel computers , 1997 .

[17]  Youcef Saad,et al.  A Basic Tool Kit for Sparse Matrix Computations , 1990 .

[18]  Rick A. Kendall,et al.  The impact of the resolution of the identity approximate integral method on modern ab initio algorithm development , 1997 .

[19]  David A. H. Jacobs,et al.  The State of the Art in Numerical Analysis. , 1978 .

[20]  Yousef Saad,et al.  ILUT: A dual threshold incomplete LU factorization , 1994, Numer. Linear Algebra Appl..

[21]  D. Bernholdt,et al.  Large-scale correlated electronic structure calculations: the RI-MP2 method on parallel computers , 1996 .

[22]  S. Ten-no,et al.  Three-center expansion of electron repulsion integrals with linear combination of atomic electron distributions , 1995 .

[23]  Frederick R. Manby,et al.  Linear scaling local coupled cluster theory with density fitting. Part I: 4-external integrals , 2003 .

[24]  Florian Weigend,et al.  A fully direct RI-HF algorithm: Implementation, optimised auxiliary basis sets, demonstration of accuracy and efficiency , 2002 .

[25]  Frederick R. Manby,et al.  Fast Hartree–Fock theory using local density fitting approximations , 2004 .

[26]  Seiichiro Ten-no,et al.  Multiconfiguration self‐consistent field procedure employing linear combination of atomic‐electron distributions , 1996 .

[27]  Rui Yang,et al.  Automatically generated Coulomb fitting basis sets: design and accuracy for systems containing H to Kr. , 2007, The Journal of chemical physics.

[28]  F. Weigend,et al.  Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations , 2002 .

[29]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .