arXiv subjects

Lexing Ying

Publications and source records attributed to Lexing Ying.

At least 181 records · Page 10Linked to original sources

arXiv2014

Sparsifying Preconditioner for the Lippmann-Schwinger Equation

The Lippmann-Schwinger equation is an integral equation formulation for acoustic and electromagnetic scattering from an inhomogeneous media and quantum scattering from a localized potential. We present the sparsifying preconditioner for accelerating the iterative solution of the Lippmann-Schwinger equation. This new preconditioner transforms the discretized Lippmann-Schwinger equation into sparse form and leverages the efficient sparse linear algebra algorithms for computing an approximate inverse. This preconditioner is efficient and easy to implement. When combined with standard iterative methods, it results in almost frequency-independent iteration counts. We provide 2D and 3D numerical results to demonstrate the effectiveness of this new preconditioner.

math.NA↗

arXiv2014

Sparsifying preconditioner for pseudospectral approximations of indefinite systems on periodic structures

This paper introduces the sparsifying preconditioner for the pseudospectral approximation of highly indefinite systems on periodic structures, which include the frequency-domain response problems of the Helmholtz equation and the Schrödinger equation as examples. This approach transforms the dense system of the pseudospectral discretization approximately into an sparse system via an equivalent integral reformulation and a specially-designed sparsifying operator. The resulting sparse system is then solved efficiently with sparse linear algebra algorithms and serves as a reasonably accurate preconditioner. When combined with standard iterative methods, this new preconditioner results in small iteration counts. Numerical results are provided for the Helmholtz equation and the Schrödinger in both 2D and 3D to demonstrate the effectiveness of this new preconditioner.

math.NA↗

arXiv2014

Pole expansion for solving a type of parametrized linear systems in electronic structure calculations

We present a new method for solving parametrized linear systems. Under certain assumptions on the parametrization, solutions to the linear systems for all parameters can be accurately approximated by linear combinations of solutions to linear systems for a small set of fixed parameters. Combined with either direct solvers or preconditioned iterative solvers for each linear system with a fixed parameter, the method is particularly suitable for situations when solutions to a large number of distinct parameters or a large number of right hand sides are required. The method is also simple to parallelize. We demonstrate the applicability of the method to the calculation of the response functions in electronic structure theory. We demonstrate the numerical performance of the method using a benzene molecule and a DNA molecule.

math.NA↗

arXiv2013

A parallel butterfly algorithm

The butterfly algorithm is a fast algorithm which approximately evaluates a discrete analogue of the integral transform \int K(x,y) g(y) dy at large numbers of target points when the kernel, K(x,y), is approximately low-rank when restricted to subdomains satisfying a certain simple geometric condition. In d dimensions with O(N^d) quasi-uniformly distributed source and target points, when each appropriate submatrix of K is approximately rank-r, the running time of the algorithm is at most O(r^2 N^d log N). A parallelization of the butterfly algorithm is introduced which, assuming a message latency of αand per-process inverse bandwidth of β, executes in at most O(r^2 N^d/p log N + βr N^d/p + α)log p) time using p processes. This parallel algorithm was then instantiated in the form of the open-source DistButterfly library for the special case where K(x,y)=exp(i Φ(x,y)), where Φ(x,y) is a black-box, sufficiently smooth, real-valued phase function. Experiments on Blue Gene/Q demonstrate impressive strong-scaling results for important classes of phase functions. Using quasi-uniform sources, hyperbolic Radon transforms and an analogue of a 3D generalized Radon transform were respectively observed to strong-scale from 1-node/16-cores up to 1024-nodes/16,384-cores with greater than 90% and 82% efficiency, respectively.

math.NA↗

arXiv2013

A parallel directional Fast Multipole Method

This paper introduces a parallel directional fast multipole method (FMM) for solving N-body problems with highly oscillatory kernels, with a focus on the Helmholtz kernel in three dimensions. This class of oscillatory kernels requires a more restrictive low-rank criterion than that of the low-frequency regime, and thus effective parallelizations must adapt to the modified data dependencies. We propose a simple partition at a fixed level of the octree and show that, if the partitions are properly balanced between p processes, the overall runtime is essentially O(N log N/p+ p). By the structure of the low-rank criterion, we are able to avoid communication at the top of the octree. We demonstrate the effectiveness of our parallelization on several challenging models.

math.NA↗

arXiv2013

Synchrosqueezed Curvelet Transform for 2D Mode Decomposition

This paper introduces the synchrosqueezed curvelet transform as an optimal tool for 2D mode decomposition of wavefronts or banded wave-like components. The synchrosqueezed curvelet transform consists of a generalized curvelet transform with application dependent geometric scaling parameters, and a synchrosqueezing technique for a sharpened phase space representation. In the case of a superposition of banded wave-like components with well-separated wave-vectors, it is proved that the synchrosqueezed curvelet transform is capable of recognizing each component and precisely estimating local wave-vectors. A discrete analogue of the continuous transform and several clustering models for decomposition are proposed in detail. Some numerical examples with synthetic and real data are provided to demonstrate the above properties of the proposed transform.

math.NA↗

arXiv2013

A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations

A parallelization of a sweeping preconditioner for 3D Helmholtz equations without large cavities is introduced and benchmarked for several challenging velocity models. The setup and application costs of the sequential preconditioner are shown to be O(γ^2 N^{4/3}) and O(γ N log N), where γ(ω) denotes the modestly frequency-dependent number of grid points per Perfectly Matched Layer. Several computational and memory improvements are introduced relative to using black-box sparse-direct solvers for the auxiliary problems, and competitive runtimes and iteration counts are reported for high-frequency problems distributed over thousands of cores. Two open-source packages are released along with this paper: "Parallel Sweeping Preconditioner (PSP)" and the underlying distributed multifrontal solver, "Clique".

math.NA↗

arXiv2012

Element orbitals for Kohn-Sham density functional theory

We present a method to discretize the Kohn-Sham Hamiltonian matrix in the pseudopotential framework by a small set of basis functions automatically contracted from a uniform basis set such as planewaves. Each basis function is localized around an element, which is a small part of the global domain containing multiple atoms. We demonstrate that the resulting basis set achieves meV accuracy for 3D densely packed systems with a small number of basis functions per atom. The procedure is applicable to insulating and metallic systems.

physics.comp-ph↗

arXiv2011

Optimized local basis set for Kohn-Sham density functional theory

We develop a technique for generating a set of optimized local basis functions to solve models in the Kohn-Sham density functional theory for both insulating and metallic systems. The optimized local basis functions are obtained by solving a minimization problem in an admissible set determined by a large number of primitive basis functions. Using the optimized local basis set, the electron energy and the atomic force can be calculated accurately with a small number of basis functions. The Pulay force is systematically controlled and is not required to be calculated, which makes the optimized local basis set an ideal tool for ab initio molecular dynamics and structure optimization. We also propose a preconditioned Newton-GMRES method to obtain the optimized local basis functions in practice. The optimized local basis set is able to achieve high accuracy with a small number of basis functions per atom when applied to a one dimensional model problem.

physics.comp-ph↗

arXiv2011

Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation

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 1meV) with a very small number ($4\sim 40$) of basis functions per atom.

math.NA↗

arXiv2010

Fast construction of hierarchical matrix representation from matrix-vector multiplication

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 $\mathcal{O}(\log n)$ applications of the matrix on structured random test vectors and $\mathcal{O}(n \log n)$ 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.

math.NA↗

arXiv2010

Sweeping Preconditioner for the Helmholtz Equation: Hierarchical Matrix Representation

The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable coefficient Helmholtz equation including very high frequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea of this approach is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The GMRES solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the three dimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach.

math.NA↗

arXiv2010

Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers

This paper introduces a new sweeping preconditioner for the iterative solution of the variable coefficient Helmholtz equation in two and three dimensions. The algorithms follow the general structure of constructing an approximate $LDL^t$ factorization by eliminating the unknowns layer by layer starting from an absorbing layer or boundary condition. The central idea of this paper is to approximate the Schur complement matrices of the factorization using moving perfectly matched layers (PMLs) introduced in the interior of the domain. Applying each Schur complement matrix is equivalent to solving a quasi-1D problem with a banded LU factorization in the 2D case and to solving a quasi-2D problem with a multifrontal method in the 3D case. The resulting preconditioner has linear application cost and the preconditioned iterative solver converges in a number of iterations that is essentially indefinite of the number of unknowns or the frequency. Numerical results are presented in both two and three dimensions to demonstrate the efficiency of this new preconditioner.

math.NA↗

arXiv2009

Pole-based approximation of Fermi-Dirac function

Two approaches for the efficient rational approximation of the Fermi-Dirac function are discussed: one uses the contour integral representation and conformal mapping and the other is based on a version of the multipole representation of the Fermi-Dirac function that uses only simple poles. Both representations have logarithmic computational complexity. They are of great interest for electronic structure calculations.

math.NA↗

arXiv2008

A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators

This paper is concerned with the fast computation of Fourier integral operators of the general form $\int_{\R^d} e^{2\piıΦ(x,k)} f(k) d k$, where $k$ is a frequency variable, $Φ(x,k)$ is a phase function obeying a standard homogeneity condition, and $f$ is a given input. This is of interest for such fundamental computations are connected with the problem of finding numerical solutions to wave equations, and also frequently arise in many applications including reflection seismology, curvilinear tomography and others. In two dimensions, when the input and output are sampled on $N \times N$ Cartesian grids, a direct evaluation requires $O(N^4)$ operations, which is often times prohibitively expensive. This paper introduces a novel algorithm running in $O(N^2 \log N)$ time, i. e. with near-optimal computational complexity, and whose overall structure follows that of the butterfly algorithm [Michielssen and Boag, IEEE Trans Antennas Propagat 44 (1996), 1086-1093]. Underlying this algorithm is a mathematical insight concerning the restriction of the kernel $e^{2\piıΦ(x,k)}$ to subsets of the time and frequency domains. Whenever these subsets obey a simple geometric condition, the restricted kernel has approximately low-rank; we propose constructing such low-rank approximations using a special interpolation scheme, which prefactors the oscillatory component, interpolates the remaining nonoscillatory part and, lastly, remodulates the outcome. A byproduct of this scheme is that the whole algorithm is highly efficient in terms of memory requirement. Numerical results demonstrate the performance and illustrate the empirical properties of this algorithm.

math.NA↗

arXiv2008

Discrete Symbol Calculus

This paper deals with efficient numerical representation and manipulation of differential and integral operators as symbols in phase-space, i.e., functions of space $x$ and frequency $ξ$. The symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equations allow to write fast-converging, non-asymptotic expansions in adequate systems of rational Chebyshev functions or hierarchical splines. The classical results of closedness of such symbol classes under multiplication, inversion and taking the square root translate into practical iterative algorithms for realizing these operations directly in the proposed expansions. Because symbol-based numerical methods handle operators and not functions, their complexity depends on the desired resolution $N$ very weakly, typically only through $\log N$ factors. We present three applications to computational problems related to wave propagation: 1) preconditioning the Helmholtz equation, 2) decomposing wavefields into one-way components and 3) depth-stepping in reflection seismology.

math.NA↗

arXiv2008

Scattering in flatland: Efficient representations via wave atoms

This paper presents a numerical compression strategy for the boundary integral equation of acoustic scattering in two dimensions. These equations have oscillatory kernels that we represent in a basis of wave atoms, and compress by thresholding the small coefficients to zero. This phenomenon was perhaps first observed in 1993 by Bradie, Coifman, and Grossman, in the context of local Fourier bases \cite{BCG}. Their results have since then been extended in various ways. The purpose of this paper is to bridge a theoretical gap and prove that a well-chosen fixed expansion, the nonstandard wave atom form, provides a compression of the acoustic single and double layer potentials with wave number $k$ as $O(k)$-by-$O(k)$ matrices with $O(k^{1+1/\infty})$ nonnegligible entries, with a constant that depends on the relative $\ell_2$ accuracy $\eps$ in an acceptable way. The argument assumes smooth, separated, and not necessarily convex scatterers in two dimensions. The essential features of wave atoms that enable to write this result as a theorem is a sharp time-frequency localization that wavelet packets do not obey, and a parabolic scaling wavelength $\sim$ (essential diameter)${}^2$. Numerical experiments support the estimate and show that this wave atom representation may be of interest for applications where the same scattering problem needs to be solved for many boundary conditions, for example, the computation of radar cross sections.

math.NA↗

arXiv2008

Fast Directional Computation for the High Frequency Helmholtz Kernel in Two Dimensions

This paper introduces a directional multiscale algorithm for the two dimensional $N$-body problem of the Helmholtz kernel with applications to high frequency scattering. The algorithm follows the approach in [Engquist and Ying, SIAM Journal on Scientific Computing, 29 (4), 2007] where the three dimensional case was studied. The main observation is that, for two regions that follow a directional parabolic geometric configuration, the interaction between the points in these two regions through the Helmholtz kernel is approximately low rank. We propose an improved randomized procedure for generating the low rank representations. Based on these representations, we organize the computation of the far field interaction in a multidirectional and multiscale way to achieve maximum efficiency. The proposed algorithm is accurate and has the optimal $O(N\log N)$ complexity for problems from two dimensional scattering applications. We present numerical results for several test examples to illustrate the algorithm and its application to two dimensional high frequency scattering problems.

math.NA↗