Truncated Conjugate Gradient (TCG): an optimal strategy for the analytical evaluation of the many-body polarization energy and forces in molecular simulations
Résumé
We introduce a new class of methods, denoted as Truncated Conjugate Gradient (TCG) methods, to solve the many-body polarization energy and its associated forces in molecular simulations encountered in molecular dynamics (MD) and Monte-Carlo techniques. The method consists of 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 and complexity. 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 fi xed point iteration ("peek"), equivalent to one step of Jacobi Over Relaxation (JOR) with relaxation parameter omega, can be made at almost no cost. This method is denoted by TCG-n(omega). Black box adaptive methods to find omega are provided and discussed. Results show that TPCG-3(omega) is converged to high accuracy for various types of systems including proteins and highly charged systems at the fixed cost of 4 matrix-vector products: (3 CG iterations+the initial CG descent direction) whereas T(P)CG-2(omega) provides robust results at a reduced cost (3 matrix-vector products) and off ers new perspectives for long polarizable MD as a production algorithm. The T(P)CG-1(omega) 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.
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