title: Truncated Conjugate Gradient: An Optimal Strategy for the Analytical Evaluation of the Many-Body Polarization Energy and Forces in Molecular Simulations.
abstract: We introduce a new class of methods, denoted as Truncated Conjugate Gradient(TCG), to solve the many-body polarization energy and its associated forces in molecular simulations (i.e. molecular dynamics (MD) and Monte Carlo). The method consists in a fixed number of Conjugate Gradient (CG) iterations. TCG approaches provide a scalable solution to the polarization problem at a user-chosen cost and a corresponding optimal accuracy. The optimality of the CG-method guarantees that the number of the required matrix-vector products are reduced to a minimum compared to other iterative methods. This family of methods is non-empirical, fully adaptive, and provides analytical gradients, avoiding therefore any energy drift in MD as compared to popular iterative solvers. Besides speed, one great advantage of this class of approximate methods is that their accuracy is systematically improvable. Indeed, as the CG-method is a Krylov subspace method, the associated error is monotonically reduced at each iteration. On top of that, two improvements can be proposed at virtually no cost: (i) the use of preconditioners can be employed, which leads to the Truncated Preconditioned Conjugate Gradient (TPCG); (ii) since the residual of the final step of the CG-method is available, one additional Picard fixed point iteration ("peek"), equivalent to one step of Jacobi Over Relaxation (JOR) with relaxation parameter ω, can be made at almost no cost. This method is denoted by TCG-n(ω). Black-box adaptive methods to find good choices of ω are provided and discussed. Results show that TPCG-3(ω) is converged to high accuracy (a few kcal/mol) for various types of systems including proteins and highly charged systems at the fixed cost of four matrix-vector products: three CG iterations plus the initial CG descent direction. Alternatively, T(P)CG-2(ω) provides robust results at a reduced cost (three matrix-vector products) and offers new perspectives for long polarizable MD as a production algorithm. The T(P)CG-1(ω) level provides less accurate solutions for inhomogeneous systems, but its applicability to well-conditioned problems such as water is remarkable, with only two matrix-vector product evaluations.
authors.affiliation: Laboratoire de Chimie Théorique, UPMC Université Paris 06, UMR 7617 , F-75005, Paris, France.
authors.forename: Félix
authors.initials: F
authors.lastname: Aviat
authors.affiliation: Inria Paris, F-75589 Paris Cedex 12, Université Paris-Est, CERMICS (ENPC) , Marne-la-Vallée, F-77455, France.
authors.forename: Antoine
authors.initials: A
authors.lastname: Levitt
authors.affiliation: MATHCCES, Department of Mathematics, RWTH Aachen University , Schinkelstraße 2, D-52062 Aachen, Germany.|Computational Biomedicine, Institute for Advanced Simulation IAS-5 and Institute of Neuroscience and Medicine INM-9, Forschungszentrum Jülich , Jülich, 52425, Germany.
authors.forename: Benjamin
authors.initials: B
authors.lastname: Stamm
authors.affiliation: Laboratoire Jacques-Louis Lions, UPMC Université Paris 06, UMR 7598 , F-75005, Paris, France.|Institut Universitaire de France , Paris Cedex 05, 75231, France.|Division of Applied Maths, Brown University , Providence, Rhode Island 02912, United States.
authors.forename: Yvon
authors.initials: Y
authors.lastname: Maday
authors.affiliation: Department of Biomedical Engineering, The University of Texas at Austin , Austin, Texas 78712, United States.
authors.forename: Pengyu
authors.initials: P
authors.lastname: Ren
authors.affiliation: Department of Chemistry, Washington University in Saint Louis , Campus Box 1134, One Brookings Drive, Saint Louis, Missouri 63130, United States.
authors.forename: Jay W
authors.initials: JW
authors.lastname: Ponder
authors.affiliation: Laboratoire de Chimie Théorique, UPMC Université Paris 06, UMR 7617 , F-75005, Paris, France.|Institut du Calcul et de la Simulation, UPMC Université Paris 06 , F-75005, Paris, France.
authors.forename: Louis
authors.initials: L
authors.lastname: Lagardère
authors.affiliation: Laboratoire de Chimie Théorique, UPMC Université Paris 06, UMR 7617 , F-75005, Paris, France.|Institut Universitaire de France , Paris Cedex 05, 75231, France.
authors.forename: Jean-Philip
authors.identifier: 0000-0001-6615-9426
authors.initials: JP
authors.lastname: Piquemal
mesh_terms: D000465:Algorithms; D056004:Molecular Dynamics Simulation; D009010:Monte Carlo Method; D013816:Thermodynamics