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REML estimation with intrinsic Matern dependence in the spatial linear mixed model

ELECTRONIC JOURNAL OF STATISTICS 2016年第2期10卷 2856-2893页

作者： Dutta, Somak Mondal, Debashis Iowa State Univ Ames IA 50011 USA Oregon State Univ Corvallis OR 97331 USA

We present a new matrix-free residual maximum likelihood (REML) analysis for irregularly spaced spatial data, where observations usually represent average values over very small regions that are interpreted as points. The REML analysis is obtained after embedding the sampling locations in a fine scale rectangular lattice, treating unobserved sites as missing data. The spatial random fields considered here are based on fractional Laplacian differencing on the lattice and they are unique in approximating continuum intrinsic Matern dependence. Here, using the h-likelihood method, we derive REML estimating equations that allow for singular precision matrices, estimation of covariate effects, prediction of unobserved spatial effects and REML estimation of precision parameters as a solution to an explicit gamma non-linear model. Furthermore, we devise a sophisticated computational algorithm that enables us to achieve scalable matrix-free statistical computations. In particular, these matrix-free computations include the use of (1) the two-dimensional discrete cosine transformation that arises in the spectral decomposition of the precision matrix of our spatial random fields and that allows fast matrix-free matrix-vector multiplication, (2) a matrix-free pre-conditioned lanczos algorithm that solves non-sparse matrix equations with linear constraints, (3) a matrix-free Hutchinson's trace estimator that stochastically approximates the trace of a matrix, (4) a robust trust region method that always finds a local maximum of the non-concave residual log-likelihood function and (5) some preliminary computations of the log REML likelihood function based on Taylor series approximation. Using computer experiments, we provide further understanding on not just the number and values but also the basins of attraction of the local and global maxima of the REML function. This understanding significantly simplifies the problem of finding global maxima. We further demonstrate through comp

关键词： Arsenic contamination discrete cosine transform fractional differencing Hutchinson's trace approximations incomplete Cholesky lanczos algorithm long range dependence power variogram matrix-free computations trust region method

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A Distance-Based Spectral Clustering Approach with Applications to Network Community Detection 23

A Distance-Based Spectral Clustering Approach with Applicati...

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23rd ISPE Inc. International Conference on Transdisciplinary Engineering

作者： Shen, Gang Ye, Dongmei Huazhong Univ Sci & Technol Sch Software Engn Wuhan 430074 Peoples R China

ISBN: (纸本)9781614997030;9781614997023

Object clustering is a fundamental task in many data analysis and pattern understanding applications by providing insights into detecting the underlying structures of a large collection of samples. In this paper, we present our work on a novel spectral clustering algorithm that partitions a collection of objects using the spectrum of adistance matrix. If the nodes in a metric space can be associated with a well defined distance, the distance matrix is almost negative definite, implying that the eigenvector for the smallest eigenvalues of this matrix can be used as an approximation of the solution to a quadratic form partition problem. It is proved that this smallest eigenvalue is equivalent to the second largest singular value. Therefore lanczos iterative algorithm can be applied to findingthe eigenvalues efficiently. We adapted this algorithm to the distributed network community detection problem using a decentralized multi-agent framework, and tested the effectiveness of the proposed approach with simulations.

关键词： Spectral clustering network community detection multi-agent systems almost negative definite matrices lanczos algorithm

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Solving topological lattice models on coprocessors

Solving topological lattice models on coprocessors

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作者： Topi Siro Aalto University

学位级别：博士

Understanding how the various electronic properties of matter emerge from the motion and interaction of electrons has been an important goal of physics since the early 1900s. One important tool has been the study of quantum lattice models, which can be considered as simpliﬁed depictions of solids. Provided that the system is small enough, it is possible to solve the low energy spectrum accurately with numerical methods. Like most problems in modern physics, studying lattice models requires extensive numerical computation. Traditionally, computer programs have been written to run on the central processing unit, but in recent years, various new parallel computing coprocessors have been introduced. Graphics processing units, which were originally added to render images on the computer screen, can now also be used for general purpose computation. Another new platform is the Xeon Phi coprocessor, speciﬁcally designed to accelerate parallel programs. Both of these coprocessors are parallel systems, where there are hundreds or thousands of computational threads running concurrently. This poses challenges in designing and implementing algorithms that beneﬁt from the parallelism. In this Thesis, we implement the exact diagonalization method on graphics processors and the Xeon Phi. We apply it on topological lattice models, which have been under intense study recently. They feature topological phases that cannot be explained with Landau's symmetry- breaking theory, but instead require studying the topological properties of the ground state. One key quantity in identifying the phases is the Chern number that is related to the transverse conductance in quantum Hall phases. In the so called checkerboard model, We show that the topological ground state can withstand strong local impurities. With increasing impurity density, we observe transitions to a metallic state and an insulating state. In another model, the Haldane-Hubbard model, we study the phase diagram with changing on-s

关键词： lattice model GPU CUDA Xeon Phi lanczos algorithm

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Solution of time-dependent PDE through rapid estimation of block Gaussian quadrature nodes

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LINEAR ALGEBRA AND ITS APPLICATIONS 2015年 468卷 233-259页

作者： Palchak, Elisabeth M. Cibotarica, Alexandru Lambers, James V. Pearl River Community Coll Hattiesburg MS 39401 USA Univ So Mississippi Dept Math Hattiesburg MS 39406 USA

The stiffness of systems of ODEs that arise from spatial discretization of PDEs causes difficulties for both explicit and implicit time-stepping methods. Krylov Subspace Spectral (KSS) methods present a balance between the efficiency of explicit methods and the stability of implicit methods by computing each Fourier coefficient from an individualized approximation of the solution operator of the PDE. While KSS methods are explicit methods that exhibit a high order of accuracy and stability similar to that of implicit methods, their efficiency needs to be improved. A previous asymptotic study of block lanczos iteration yielded estimates of extremal block Gaussian quadrature nodes for each Fourier component and led to an improvement in efficiency. In this paper, a more detailed asymptotic study is performed in order to rapidly estimate all nodes, thus drastically reducing computational expense without sacrificing accuracy. Numerical results demonstrate that the new node estimation scheme does in fact accomplish these aims. (C) 2014 Elsevier Inc. All rights reserved.

关键词： lanczos algorithm Spectral methods Gaussian quadrature Heat equation Wave equation

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Exact numerical methods for a many-body Wannier-Stark system

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COMPUTER PHYSICS COMMUNICATIONS 2015年 186卷 19-30页

作者： Parra-Murillo, Carlos A. Madronero, Javier Wimberger, Sandro Heidelberg Univ Inst Theoret Phys D-69120 Heidelberg Germany Univ Valle Dept Fis Cali Colombia Univ Parma Dipartimento Fis & Sci Terra I-43124 Parma Italy Ist Nazl Fis Nucl Grp Collegato Parma Sez Milano Bicocca Milan Italy

We present exact methods for the numerical integration of the Wannier-Stark system in a many-body scenario including two Bloch bands. Our ab initio approaches allow for the treatment of a few-body problem with bosonic statistics and strong interparticle interaction. The numerical implementation is based on the lanczos algorithm for the diagonalization of large, but sparse symmetric Floquet matrices. We analyze the scheme efficiency in terms of the computational time, which is shown to scale polynomially with the size of the system. The numerically computed eigensystem is applied to the analysis of the Floquet Hamiltonian describing our problem. We show that this allows, for instance, for the efficient detection and characterization of avoided crossings and their statistical analysis. We finally compare the efficiency of our lanczos diagonalization for computing the temporal evolution of our many-body system with an explicit fourth order Runge-Kutta integration. Both implementations heavily exploit efficient matrix vector multiplication schemes. Our results should permit an extrapolation of the applicability of exact methods to increasing sizes of generic many-body quantum problems with bosonic statistics. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Floquet problem Bose-Hubbard model Wannier-Stark system lanczos algorithm Quantum chaos Ultracold atoms Optical lattices

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Krylov subspace methods for the Dirac equation

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COMPUTER PHYSICS COMMUNICATIONS 2015年 188卷 189-197页

作者： Beerwerth, Randolf Bauke, Heiko Max Planck Inst Kernphys D-69117 Heidelberg Germany

The lanczos algorithm is evaluated for solving the time-independent as well as the time-dependent Dirac equation with arbitrary electromagnetic fields. We demonstrate that the lanczos algorithm can yield very precise eigenenergies and allows very precise time propagation of relativistic wave packets. The unboundedness of the Dirac Hamiltonian does not hinder the applicability of the lanczos algorithm. As the lanczos algorithm requires only matrix vector products and inner products, which both can be efficiently parallelized, it is an ideal method for large-scale calculations. The excellent parallelization capabilities are demonstrated by a parallel implementation of the Dirac lanczos propagator utilizing the Message Passing Interface standard. Program summary Program title: Dirac_Laczos Catalogue identifier: AEUY_v1_0 Program summary URL: http://***/summaries/AEUY_v1_*** Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://***/licence/*** No. of lines in distributed program, including test data, etc.: 526428 No. of bytes in distributed program, including test data, etc.: 2 181 729 Distribution format: *** Programming language: C++11. Computer: Multi-core systems or cluster computers. Operating system: Any. Has the code been vectorized or parallelized?: Parallelized using MPI. RAM: Typically 10 megabyte to 1 gigabyte depending on the chosen problem size Classification: 2.7. External routines: Boost [1], LAPACK [2] Nature of problem: Solving the time-dependent Dirac equation in two spatial dimensions Solution method: lanczos propagator Running time: Depending on the problem size and computer hardware typically several minutes to several days References: [1] Boost C++ Libraries, http://*** [2] LAPACK-Linear Algebra PACKage, http://***/lapack/ (C) 2014 Elsevier B.V. All rights reserved.

关键词： Dirac equation Krylov subspace methods lanczos algorithm

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Two new methods for computing vibrational energy levels

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CANADIAN JOURNAL OF CHEMISTRY 2015年第6期93卷 589-593页

作者： Carrington, Tucker, Jr. Queens Univ Dept Chem Kingston ON K7L 3N6 Canada

I review two new ideas for coping with the size of large product basis sets and large product grids when one computes vibrational energy levels. The first is based on a tensor reduction scheme. It exploits advantages of a sum-of-products potential. The key idea is to use a basis each of whose function is a sum of optimized products and to compress the number of terms in each basis function. When the potential does not have the sum-of-products form, it is usually necessary to use quadrature. The second idea uses a nondirect product grid that has structure and is therefore compatible with efficient matrix-vector products.

关键词： theory rovibrational spectroscopy iterative methods lanczos algorithm

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Decentralized Eigenvalue algorithms for Distributed Signal Detection in Wireless Networks

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2015年第2期63卷 427-440页

作者： Penna, Federico Stanczak, Slawomir Samsung Mobile Solut Lab MSL San Diego CA 92121 USA Heinrich Hertz Inst Nachrichtentech Berlin GmbH Fraunhofer Inst Telecommun D-10587 Berlin Germany

In this paper, we derive and analyze two algorithms-referred to as decentralized power method (DPM) and decentralized lanczos algorithm (DLA)-for distributed computation of one (the largest) or multiple eigenvalues of a sample covariance matrix over a wireless network. The proposed algorithms, based on sequential average consensus steps for computations of matrix-vector products and inner vector products, are first shown to be equivalent to their centralized counterparts in the case of exact distributed consensus. Then, closed-form expressions of the error introduced by nonideal consensus are derived for both algorithms. The error of the DPM is shown to vanish asymptotically under given conditions on the sequence of consensus errors. Finally, we consider applications to spectrum sensing in cognitive radio networks, and we show that virtually all eigenvalue-based tests proposed in the literature can be implemented in a distributed setting using either the DPM or the DLA. Simulation results are presented that validate the effectiveness of the proposed algorithms in conditions of practical interest (large-scale networks, small number of samples, and limited number of iterations).

关键词： Average consensus eigenvalue-based signal detection lanczos algorithm power method

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An h-likelihood method for spatial mixed linear models based on intrinsic auto-regressions

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JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY 2015年第3期77卷 699-726页

作者： Dutta, Somak Mondal, Debashis Univ Chicago Chicago IL 60637 USA Oregon State Univ Corvallis OR 97331 USA

We consider sparse spatial mixed linear models, particularly those described by Besag and Higdon, and develop an h-likelihood method for their statistical inference. The method proposed allows for singular precision matrices, as it produces estimates that coincide with those from the residual maximum likelihood based on appropriate differencing of the data and has a novel approach to estimating precision parameters by a gamma linear model. Furthermore, we generalize the h-likelihood method to include continuum spatial variations by making explicit use of scaling limit connections between Gaussian intrinsic Markov random fields on regular arrays and the de Wijs process. Keeping various applications of spatial mixed linear models in mind, we devise a novel sparse conjugate gradient algorithm that allows us to achieve fast matrix-free statistical computations. We provide two applications. The first is an extensive analysis of an agricultural variety trial that brings forward various new aspects of nearest neighbour adjustment such as effects on statistical analyses to changes of scale and use of implicit continuum spatial formulation. The second application concerns an analysis of a large cotton field which gives a focus to matrix-free computations. The paper closes with some further considerations, such as applications to irregularly spaced data, use of the parametric bootstrap and some generalizations to the Gaussian Matern mixed effect models.

关键词： Hutchinson's trace approximations Incomplete Cholesky decomposition lanczos algorithm Matrix-free computations Nearest neighbour adjustment Residual maximum likelihood Variety trials de Wijs process

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Parallelization of the lanczos algorithm on Multi-core Platforms

Parallelization of the Lanczos Algorithm on Multi-core Platf...

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11th International Conference on Distributed Computing and Networking

作者： Bhattacherjee, Souvik Das, Abhijit Indian Inst Technol Dept Comp Sci & Engn Kharagpur 721302 W Bengal India

ISBN: (纸本)9783642113215

In this paper, we report our parallel implementations of the lanczos sparse linear system solving algorithm over large prime fields, on a multi-core platform. We employ several load-balancing methods suited to these platforms. We have carried out process-level and thread-level parallel implementations under two different arithmetic libraries, and the best speedup obtained is 6.57 on eight cores. To the best of our knowledge;no implementation of the lanczos algorithm on a multi-core platform is ever reported in the literature. Moreover, we seem to have achieved significantly larger speedup compared to all previously reported implementations of this algorithm.

关键词： Sparse linear system lanczos algorithm modular arithmetic prime field multi-core machine parallelization load balancing

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