Browsing by Author "Lin, L"

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    A Mathematical Theory of Optimal Milestoning (with a Detour via Exact Milestoning)
    (2017-04-23) Lin, L; Lu, J; Vanden-Eijnden, E
    Milestoning is a computational procedure that reduces the dynamics of complex systems to memoryless jumps between intermediates, or milestones, and only retains some information about the probability of these jumps and the time lags between them. Here we analyze a variant of this procedure, termed optimal milestoning, which relies on a specific choice of milestones to capture exactly some kinetic features of the original dynamical system. In particular, we prove that optimal milestoning permits the exact calculation of the mean first passage times (MFPT) between any two milestones. In so doing, we also analyze another variant of the method, called exact milestoning, which also permits the exact calculation of certain MFPTs, but at the price of retaining more information about the original system's dynamics. Finally, we discuss importance sampling strategies based on optimal and exact milestoning that can be used to bypass the simulation of the original system when estimating the statistical quantities used in these methods.
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    Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation
    (Journal of Computational Physics, 2012-02-20) Lin, L; Lu, J; Ying, L; E, W
    Kohn-Sham density functional theory is one of the most widely used electronic structure theories. In the pseudopotential framework, uniform discretization of the Kohn-Sham Hamiltonian generally results in a large number of basis functions per atom in order to resolve the rapid oscillations of the Kohn-Sham orbitals around the nuclei. Previous attempts to reduce the number of basis functions per atom include the usage of atomic orbitals and similar objects, but the atomic orbitals generally require fine tuning in order to reach high accuracy. We present a novel discretization scheme that adaptively and systematically builds the rapid oscillations of the Kohn-Sham orbitals around the nuclei as well as environmental effects into the basis functions. The resulting basis functions are localized in the real space, and are discontinuous in the global domain. The continuous Kohn-Sham orbitals and the electron density are evaluated from the discontinuous basis functions using the discontinuous Galerkin (DG) framework. Our method is implemented in parallel and the current implementation is able to handle systems with at least thousands of atoms. Numerical examples indicate that our method can reach very high accuracy (less than 1. meV) with a very small number (4-40) of basis functions per atom. © 2011 Elsevier Inc.
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    Analysis of time reversible born-oppenheimer molecular dynamics
    (Entropy, 2014-01-01) Lin, L; Lu, J; Shao, S
    We analyze the time reversible Born-Oppenheimer molecular dynamics (TRBOMD) scheme, which preserves the time reversibility of the Born-Oppenheimer molecular dynamics even with non-convergent self-consistent field iteration. In the linear response regime, we derive the stability condition, as well as the accuracy of TRBOMD for computing physical properties, such as the phonon frequency obtained from the molecular dynamics simulation. We connect and compare TRBOMD with Car-Parrinello molecular dynamics in terms of accuracy and stability. We further discuss the accuracy of TRBOMD beyond the linear response regime for non-equilibrium dynamics of nuclei. Our results are demonstrated through numerical experiments using a simplified one-dimensional model for Kohn-Sham density functional theory. ©2013 by the author; licensee MDPI, Basel, Switzerland.
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    Bayesian analysis of multi-type recurrent events and dependent termination with nonparametric covariate functions
    (Statistical Methods in Medical Research, 2015) Lin, L; Luo, S; Chen, BE; Davis, BR
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    Bayesian regression model for recurrent event data with event-varying covariate effects and event effect
    (Journal of Applied Statistics, 2017) Lin, L; Luo, S; Davis, BR
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    Decay estimates of discretized Green’s functions for Schrödinger type operators
    (Science China Mathematics, 2016-08-01) Lin, L; Lu, J
    © 2016, Science China Press and Springer-Verlag Berlin Heidelberg.For a sparse non-singular matrix A, generally A−1 is a dense matrix. However, for a class of matrices, A−1 can be a matrix with off-diagonal decay properties, i.e., |Aij−1| decays fast to 0 with respect to the increase of a properly defined distance between i and j. Here we consider the off-diagonal decay properties of discretized Green’s functions for Schrödinger type operators. We provide decay estimates for discretized Green’s functions obtained from the finite difference discretization, and from a variant of the pseudo-spectral discretization. The asymptotic decay rate in our estimate is independent of the domain size and of the discretization parameter. We verify the decay estimate with numerical results for one-dimensional Schrödinger type operators.
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    ELSI: A Unified Software Interface for Kohn-Sham Electronic Structure Solvers
    (2017-07-01) Yu, VWZ; Corsetti, F; García, A; Huhn, WP; Jacquelin, M; Jia, W; Lange, B; Lin, L; Lu, J; Mi, W; Seifitokaldani, A; Vázquez-Mayagoitia, Á; Yang, C; Yang, H; Blum, V
    Solving the electronic structure from a generalized or standard eigenproblem is often the bottleneck in large scale calculations based on Kohn-Sham density-functional theory. This problem must be addressed by essentially all current electronic structure codes, based on similar matrix expressions, and by high-performance computation. We here present a unified software interface, ELSI, to access different strategies that address the Kohn-Sham eigenvalue problem. Currently supported algorithms include the dense generalized eigensolver library ELPA, the orbital minimization method implemented in libOMM, and the pole expansion and selected inversion (PEXSI) approach with lower computational complexity for semilocal density functionals. The ELSI interface aims to simplify the implementation and optimal use of the different strategies, by offering (a) a unified software framework designed for the electronic structure solvers in Kohn-Sham density-functional theory; (b) reasonable default parameters for a chosen solver; (c) automatic conversion between input and internal working matrix formats, and in the future (d) recommendation of the optimal solver depending on the specific problem. Comparative benchmarks are shown for system sizes up to 11,520 atoms (172,800 basis functions) on distributed memory supercomputing architectures.
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    Fast construction of hierarchical matrix representation from matrix-vector multiplication
    (Journal of Computational Physics, 2011-05-10) Lin, L; Lu, J; Ying, L
    We develop a hierarchical matrix construction algorithm using matrix-vector multiplications, based on the randomized singular value decomposition of low-rank matrices. The algorithm uses O(logn) applications of the matrix on structured random test vectors and O(nlogn) extra computational cost, where n is the dimension of the unknown matrix. Numerical examples on constructing Green's functions for elliptic operators in two dimensions show efficiency and accuracy of the proposed algorithm. © 2011 Elsevier Inc.
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    Multipole representation of the Fermi operator with application to the electronic structure analysis of metallic systems
    (Physical Review B - Condensed Matter and Materials Physics, 2009-03-03) Lin, L; Lu, J; Car, R; Weinan, E
    We propose a multipole representation of the Fermi-Dirac function and the Fermi operator and use this representation to develop algorithms for electronic structure analysis of metallic systems. The algorithm is quite simple and efficient. Its computational cost scales logarithmically with βΔ where β is the inverse temperature and Δ is the width of the spectrum of the discretized Hamiltonian matrix. © 2009 The American Physical Society.
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    Neural-PDE: a RNN based neural network for solving time dependent PDEs
    (COMMUNICATIONS IN INFORMATION AND SYSTEMS, 2022) Hu, Y; Zhao, T; Xu, S; Lin, L; Xu, Z