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A robust and scalable multi-level domain decomposition preconditioner for multi-core architecture with large number of cores

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2020年第0期373卷 112614-000页

作者： Gratien, Jean-Marc Comp Sci Dept 1 & 4 Bois Preau F-92500 Rueil Malmaison France

With the evolution of High Performance Computing, multi-core and many-core systems are a common feature of new hardware architectures. The required programming efforts induced by the introduction of these architectures are challenging due to the increasing number of cores. Parallel programming models based on the data flow model and the task programming paradigm intend to fix this issue. Iterative linear solvers are a key part of petroleum reservoir simulation as they can represent up to 80% of the total computing time. In these algorithms, the standard preconditioning methods for large, sparse and unstructured matrices such as Incomplete LU Factorization (ILU) or algebraic Multigrid (AMG) fail to scale on shared-memory architectures with large number of cores. Multi-level domain decomposition (DDML) preconditioners recently introduced seem to be both numerically robust and scalable on emerging architectures because of their parallel nature. This paper proposes a parallel implementation of these preconditioners using the task programming paradigm with a data flow model. This approach is validated on linear systems extracted from realistic petroleum reservoir simulations. This shows that, given an appropriate coarse operator in such preconditioners, the method has good convergence rates while our implementation ensures interesting scalability on multi-core architectures. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Domain decomposition methods linear algebra Parallel computing Runtime systems Multi-core architecture Reservoir simulation

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Multi-parametric iterative algorithms for discrete periodic Lyapunov matrix equations

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IET CONTROL theory AND applications 2020年第2期14卷 334-342页

作者： Wu, Ai-Guo Zhang, Wen-Xue Zhang, Ying Harbin Inst Technol Shenzhen Sch Mech Engn & Automat Shenzhen 518055 Peoples R China Harbin Inst Technol Dept Control Sci & Engn Harbin 150001 Peoples R China

Two multi-parametric iterative algorithms are developed to solve the forward discrete periodic Lyapunov matrix equation associated with discrete-time linear periodic systems. An important feature of one of the proposed algorithms is that the information in the current and the last steps is used to update the iterative sequence. Necessary and sufficient conditions for these two algorithms to be convergent are provided in terms of the roots of a set of polynomial equations. Based on these conditions, a two-dimensional section method is established to search suboptimal tuning parameters for these two algorithms. In addition, convergence properties are also analysed for some special cases of the obtained algorithms. Finally, numerical examples are provided to illustrate the effectiveness of the proposed algorithms. It can be found that the presented algorithms have better convergence performance than some existing iterative algorithms by choosing proper tuning parameters.

关键词： time-varying systems matrix algebra Lyapunov matrix equations iterative methods discrete time systems polynomials periodic control linear systems Lyapunov methods multiparametric iterative algorithms discrete periodic Lyapunov matrix equations forward discrete periodic Lyapunov matrix equation discrete-time linear periodic systems iterative sequence necessary conditions sufficient conditions polynomial equations existing iterative algorithms

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Accelerating Parallel First-Principles Excited-State Calculation by Low-Rank Approximation with K-Means Clustering 22

Accelerating Parallel First-Principles Excited-State Calcula...

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Proceedings of the 51st International conference on Parallel Processing

作者： Qingcai Jiang Jielan Li Junshi Chen Xinming Qin Lingyun Wan Jinlong Yang Jie Liu Wei Hu Hong An University of Science and Technology of China China University of Science and Technology of China China and Lawrence Berkeley National Laboratory USA

ISBN: (纸本)9781450397339

First-principles time-dependent density functional theory (TDDFT) is a powerful tool to accurately describe the excited-state properties of molecules and solids in condensed matter physics, computational chemistry and materials science. However, a perceived drawback in TDDFT calculations is its ultrahigh computational cost and large memory usage especially for plane-wave basis set, confining its applications to large systems containing thousands of atoms. Here, we present a massively parallel implementation of linear-response TDDFT (LR-TDDFT) and reduce the complexity to by combining K-Means clustering based low-rank approximation with iterative eigensolve algorithm. Furthermore, we carefully design the parallel data and task distribution schemes to accommodate with the physical nature in different steps of the computation, also, several optimization methods are employed to effectively handle the matrix operations and data communications of constructing and diagonalizing the LR-TDDFT Hamiltonian. In particular, our method can significantly reduce the cost of computation and memory by nearly 2 orders of magnitude compared to conventional LR-TDDFT calculations. numerical results demonstrate that our implementation can gain an overall speedup of 10x and efficiently scale up to 12,288 CPU cores for large systems up to 4,096 atoms within dozens of seconds.

关键词： linear-response Quantum mechanic calculation Low-rank approximation Iterative eigensolver Time-dependent density functional theory Parallel implementation

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Some algebraic Aspects of Boolean Valued Analysis 1

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International Scientific conference on Modern methods, Problems and applications of Operator theory and Harmonic Analysis, OTHA 2020

作者： Kusraev, Anatoly G. Southern Mathematical Institute of the Vladikavkaz Scientific Center of the RAS Vladikavkaz Russia K. L. Khetagurov North-Ossetian State University Vladikavkaz Russia

ISBN: (数字)9783030774936

ISBN: (纸本)9783030774929

The article deals with a Boolean valued approach to some algebraic problems arising from functional analysis. The main results are as follows. (1) A universally complete vector lattice without locally one-dimensional bands contains an infinite direct sum of order dense sublattices each of which is a band preserving linear isomorphic (but not lattice isomorphic) copy of the whole lattice. (2) Every separated injective module over a semiprime rationally complete commutative ring admits a direct sum decomposition with homogeneous summands. (3) A semiprime rationally complete commutative ring properly embedded in a ring with projections K is a homogeneity ring of an additive mapping between appropriate K-modules. © 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

关键词： algebra

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A One-Pass Distributed and Private Sketch for Kernel Sums with applications to Machine Learning at Scale 21

A One-Pass Distributed and Private Sketch for Kernel Sums wi...

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ACM SIGSAC conference on Computer and Communications Security (ACM CCS)

作者： Coleman, Benjamin Shrivastava, Anshumali Rice Univ Houston TX 77005 USA

ISBN: (纸本)9781450384544

Differential privacy is a compelling privacy definition that explains the privacy-utility tradeoff via formal, provable guarantees. In machine learning, we often wish to release a function over a dataset while preserving differential privacy. Although there are general algorithms to solve this problem for any function, such methods can require hours to days to run on moderately sized datasets. As a result, most private algorithms address task-dependent functions for specific applications. In this work, we propose a general purpose private sketch, or small summary of the dataset, that supports machine learning tasks such as regression, classification, density estimation, and more. Our sketch is ideal for large-scale distributed settings because it is simple to implement, mergeable, and can be created with a one-pass streaming algorithm. At the heart of our proposal is the reduction of many machine learning objectives to kernel sums. Our sketch estimates these sums using randomized contingency tables that are indexed with locality-sensitive hashing. Existing alternatives for kernel sum estimation scale poorly, often exponentially slower with an increase in dimensions. In contrast, our sketch can quickly run on large high-dimensional datasets, such as the 65 million node Friendster graph, in a single pass that takes less than 20 minutes, which is otherwise infeasible with any known alternative. Exhaustive experiments show that the privacy-utility tradeoff of our method is competitive with existing algorithms, but at an order-of-magnitude smaller computational cost. We expect that our sketch will be practically useful for differential privacy in distributed, large-scale machine learning settings.

关键词： differential privacy scaling sketching machine learning locality-sensitive hashing

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Nonlinear Least Squares Problems Using Approximate Greatest Descent Method

Nonlinear Least Squares Problems Using Approximate Greatest ...

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International conference on Computer and Drone applications (IConDA)

作者： Christina Nguk Ling Eu Hendra Gunawan Harno King Hann Lim College of Computing Informatics & Media Universiti Teknologi MARA Kota Samarahan Malaysia Faculty of Science & Technology Sanata Dharma University Yogyakarta Indonesia Dept. of Electrical & Comp. Engineering Curtin University Malaysia Miri Malaysias

ISBN: (纸本)9781665492362

Many numerical methods have been developed and modified to solve nonlinear least squares (NLS) problems as unconstrained optimization problems. One of the challenges regarding the existing numerical methods in solving NLS problems is expensive computation of the Hessian matrix at every iteration. Hence, most numerical methods employ a truncated Hessian, which, if the residuals are large, it may not be a good approximation of the Hessian matrix. In this paper, the approximate greatest descent (AGD) method is proposed to so lve NLS problems. The algorithm of the AGD method, which uses the full Hessian matrix, is constructed in a logical, rational, and geometrical way. numerical differentiation is employed to numerically calculate the derivatives of the function. The convergence analysis of the AGD method is following the Lyapunov function theorem whereby monotonic decreasing property and properly nested level sets of the objective function ensures its convergence. Coincidentally, the iterative equation of the AGD method resembles the Levenberg-Marquadrt (LM) method, even though both methods derived differently. numerical experiments have shown that both AGD and LM methods are able to produce stable trajectories. The AGD method, on the ° ther hand, is more efficient, reliable, and robust because convergence can be achieved with fewer iterations in less time.

关键词： Level set Reliability theory linear programming Mathematical models Robustness Trajectory Iterative methods

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Finite-time control of periodic systems with event-triggering mechanisms

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IET CONTROL theory AND applications 2020年第8期14卷 1012-1021页

作者： Yang, Liu Gao, Yabin Zhao, Yuxin Wu, Ligang Harbin Engn Univ Coll Automat Harbin 150001 Peoples R China Harbin Inst Technol Dept Control Sci & Engn Harbin 150001 Peoples R China

This study is concerned with the event-triggered finite-time control of periodic systems via a piecewise method. The continuous-time periodic system is formulated by finite-number linear time-invariant subsystems, of which the system parameters are determined by the piecewise approximation method. In consideration of the mitigation of network communication in networked control loops, some event-triggering mechanisms are designed in terms of the networks between the sensor (or controller) and the controller (or actuator). Three different event-triggered control schemes are designed for the networked periodic system. The finite-time stability of the resulting closed-loop system is analysed by using a Lyapunov function approach with a well-defined matrix-valued function. The controller gains depending on the piecewise parameters are thus designed with several matrix inequalities that can be solved by off-line procedures. Finally, a numerical example is provided to illustrate the effectiveness of the proposed design scheme.

关键词： continuous time systems Lyapunov methods matrix algebra periodic control closed loop systems stability linear matrix inequalities control system synthesis time-varying systems linear systems event-triggering mechanisms sensor or controller different event-triggered control schemes networked periodic system finite-time stability controller gains piecewise parameters periodic systems event-triggered finite-time control piecewise method continuous-time periodic system finite-number time-invariant subsystems system parameters piecewise approximation method network communication networked control loops

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Fast quantum algorithm for differential equations

arXiv

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arXiv 2023年

作者： Bagherimehrab, Mohsen Nakaji, Kouhei Wiebe, Nathan Aspuru-Guzik, Alán Chemical Physics Theory Group Department of Chemistry University of Toronto TorontoON Canada Department of Computer Science University of Toronto TorontoON Canada 1-1-1 Umezono Ibaraki Tsukuba305-8568 Japan Quantum Computing Center Keio University 3-14-1 Hiyoshi Kohoku-ku Kanagawa Yokohama223-8522 Japan Pacific Northwest National Laboratory RichlandWA United States Canadian Institute for Advanced Research TorontoON Canada Vector Institute for Artificial Intelligence TorontoON Canada Department of Chemical Engineering & Applied Chemistry University of Toronto TorontoON Canada Department of Materials Science & Engineering University of Toronto TorontoON Canada

Partial differential equations (PDEs) are ubiquitous in science and engineering. Prior quantum algorithms for solving the system of linear algebraic equations obtained from discretizing a PDE have a computational complexity that scales at least linearly with the condition number κ of the matrices involved in the computation. For many practical applications, κ scales polynomially with the size N of the matrices, rendering a polynomial-in-N complexity for these algorithms. Here we present a quantum algorithm with a complexity that is polylogarithmic in N but is independent of κ for a large class of PDEs. Our algorithm generates a quantum state that enables extracting features of the solution. Central to our methodology is using a wavelet basis as an auxiliary system of coordinates in which the condition number of associated matrices is independent of N by a simple diagonal preconditioner. We present numerical simulations showing the effect of the wavelet preconditioner for several differential equations. Our work could provide a practical way to boost the performance of quantum-simulation algorithms where standard methods are used for discretization. © 2023, CC BY.

关键词： Matrix algebra

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Computational Science in the 17th Century. numerical Solution of algebraic Equations: Digit–by–Digit Computation 5th

Computational Science in the 17th Century. Numerical Solutio...

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5th International conference on numerical Analysis and Optimization: theory, methods, applications and Technology Transfer, NAOV 2020

作者： Steihaug, Trond Department of Informatics University of Bergen Box 7803 Bergen5020 Norway

ISBN: (纸本)9783030720391

In this paper we give a complete overview of test–problems by Viète from 1600, Harriot from 1631 and Oughtred from 1647. The original material is not easily accessible due to archaic language and lack of conciseness. Viéte’s method was gradually elucidated by the subsequent writers Harriot and Oughtred using symbols and being more concise. However, the method is presented in tables and from the layout of the tables it is difficult to find the general principle. Many authors have therefore described Viète’s process inaccurately and in this paper we give a precise description of the divisor used in the process which has been verified on all the test–problems. The process of Viète is an iterative method computing one digit of the root in each iteration and has a linear rate of convergence and we argue that the digit–by–digit process lost its attractiveness with the publications in 1685 and 1690 of the Newton-Raphson method which doubles the number of digits for each iteration. © 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

关键词： Newton-Raphson method

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4th International conference on Analysis and Applied Mathematics, ICAAM 2018

4th International Conference on Analysis and Applied Mathema...

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4th International conference on Analysis and Applied Mathematics, ICAAM 2018

ISBN: (纸本)9783030692919

The proceedings contain 21 papers. The special focus in this conference is on Analysis and Applied Mathematics. The topics include: On the Stability of Schrödinger Type Involutory Differential Equations;preface;attractors for Weak Solutions of a Regularized Model of Viscoelastic Mediums Motion with Memory in Non-autonomous Case;on the Ternary Semigroups of Continuous Mappings;a Unified numerical Method for Solving System of Nonlinear Wave Equations;Comparison of the Rate of Induced Intrinsic Pathway of Apoptosis on COLO-320 and COLO-741;a Note on Representation Variety of Abelian Groups and Reidemeister Torsion;a Space-Dependent Source Identification Problem for Hyperbolic-Parabolic Equations;on the Stability of the Time Delay Telegraph Equation with Neumann Condition;a Note on a Hyperbolic-Parabolic Problem with Involution;numerical Solution of a Parabolic Source Identification Problem with Involution and Neumann Condition;the Structure of Fractional Spaces Generated by a Two-Dimensional Difference Neutron Transport Operator and Its applications;uncertainty Type Principles for Radial Derivatives;investigation of Finite-Difference Analogue of the Integral Geometry Problem with a Weight Function;boundary Conditions of Volume Hyperbolic Potential in a Domain with Curvilinear Boundary;basic theory of Impulsive Quaternion-Valued linear Systems;six Point Implicit methods for the Approximation of the Derivatives of the Solution of First Type Boundary Value Problem for Heat Equation;identification Elliptic Problem with Dirichlet and Integral Conditions;on Solvability of Some Boundary Value Problems with Involution for the Biharmonic Equation.

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