# Browsing by Author "Ying, L"

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Item Open Access 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, WKohn-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.Item Open Access Compression of the electron repulsion integral tensor in tensor hypercontraction format with cubic scaling cost(Journal of Computational Physics, 2015-12-01) Lu, J; Ying, L© 2015 Elsevier Inc.Electron repulsion integral tensor has ubiquitous applications in electronic structure computations. In this work, we propose an algorithm which compresses the electron repulsion tensor into the tensor hypercontraction format with O(nN2logN) computational cost, where N is the number of orbital functions and n is the number of spatial grid points that the discretization of each orbital function has. The algorithm is based on a novel strategy of density fitting using a selection of a subset of spatial grid points to approximate the pair products of orbital functions on the whole domain.Item Open Access Crystal image analysis using 2D synchrosqueezed transforms(Multiscale Modeling and Simulation, 2015-01-01) Yang, H; Lu, J; Ying, L© 2015 Society for Industrial and Applied Mathematics.We propose efficient algorithms based on a band-limited version of 2D synchrosqueezed transforms to extract mesoscopic and microscopic information from atomic crystal images. The methods analyze atomic crystal images as an assemblage of nonoverlapping segments of 2D general intrinsic mode type functions, which are superpositions of nonlinear wave-like components. In particular, crystal defects are interpreted as the irregularity of local energy; crystal rotations are described as the angle deviation of local wave vectors from their references; the gradient of a crystal elastic deformation can be obtained by a linear system generated by local wave vectors. Several numerical examples of synthetic and real crystal images are provided to illustrate the efficiency, robustness, and reliability of our methods.Item Open Access Efficient construction of tensor ring representations from sampling(2017-11-30) Khoo, Y; Lu, J; Ying, LIn this note we propose an efficient method to compress a high dimensional function into a tensor ring format, based on alternating least-squares (ALS). Since the function has size exponential in $d$ where $d$ is the number of dimensions, we propose efficient sampling scheme to obtain $O(d)$ important samples in order to learn the tensor ring. Furthermore, we devise an initialization method for ALS that allows fast convergence in practice. Numerical examples show that to approximate a function with similar accuracy, the tensor ring format provided by the proposed method has less parameters than tensor-train format and also better respects the structure of the original function.Item Open Access Fast construction of hierarchical matrix representation from matrix-vector multiplication(Journal of Computational Physics, 2011-05-10) Lin, L; Lu, J; Ying, LWe 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.Item Open Access Multidimensional Butterfly Factorization(Applied and Computational Harmonic Analysis) Li, Y; Yang, H; Ying, LItem Open Access Oscillatory data analysis and fast algorithms for integral operators(2015-06-14) Yang, HItem Open Access Solving parametric PDE problems with artificial neural networks(2017-11-30) Khoo, Y; Lu, J; Ying, LThe curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of physical quantities derived from PDE can be captured by a few features on the space of random coefficients. Based on such observation, we propose using neural-network, a technique gaining prominence in machine learning tasks, to parameterize the physical quantity of interest as a function of random input coefficients. The simplicity and accuracy of the approach are demonstrated through notable examples of PDEs in engineering and physics.Item Open Access Sparsifying preconditioner for soliton calculations(Journal of Computational Physics, 2016-06-15) Lu, J; Ying, L© 2016 Elsevier Inc.We develop a robust and efficient method for soliton calculations for nonlinear Schrödinger equations. The method is based on the recently developed sparsifying preconditioner combined with Newton's iterative method. The performance of the method is demonstrated by numerical examples of gap solitons in the context of nonlinear optics.