Squeezing the most out of eigenvalue solvers on high-performance computers

AbstractThis paper describes modifications to many of the standard algorithms used in computing eigenvalues and eigenvectors of matrices. These modifications can dramatically increase the performance of the underlying software on high-performance computers without resorting to assembler language, without significantly influencing the floating-point operation count, and without affecting the roundoff-error properties of the algorithms. The techniques are applied to a wide variety of algorithms and are beneficial in various architectural settings

Orthogonal sets of data windows constructed from trigonometric polynomials

Suboptimal, easily computable substitutes for the discrete prolate-spheroidal windows used by Thomson for spectral estimation are given. Trigonometric coefficients and energy leakages of the window polynomials are tabulated

Reduced order feedback control equations for linear time and frequency domain analysis

An algorithm was developed which can be used to obtain the equations. In a more general context, the algorithm computes a real nonsingular similarity transformation matrix which reduces a real nonsymmetric matrix to block diagonal form, each block of which is a real quasi upper triangular matrix. The algorithm works with both defective and derogatory matrices and when and if it fails, the resultant output can be used as a guide for the reformulation of the mathematical equations that lead up to the ill conditioned matrix which could not be block diagonalized

ПОНИЖЕНИЕ ПОРЯДКА СЛОЖНЫХ МОДЕЛЕЙ С ПОМОЩЬЮ ИНСТРУМЕНТОВ ROBUST CONTROL TOOLBOX

The city of Berezniki, Perm region is located on an underworked mine area. For several years, the city has been experiencing active subsidence of the soil, which provoke the destruction of buil­dings. Therefore, for several years now, the city's buildings and structures have been monitored, which makes it possible to analyze the degree of subsidence. Models of a sufficiently high order are used for an accurate analysis of the situation and forecasting. The article is about a possibility of modeling the deformation of buildings associated with soil subsidence, as a result of mine workings in the city of Berezniki. The purpose of the study is to consider the capabilities of the ‘Robust Control Toolbox’ for reducing the order of complexity of models. An example of an eight-story building included in the collection of reference examples for reducing models of linear dynamic systems is used. Materials and methods. Typical steps for solving the problem of model reduction are presented, commands and tools used to solve this problem are described. The parameters of the model in the state space are determined, which has 48 states, which are displacements or rates of change. The singular values of Hankel are used to select states that can be neglected. The model is reduced using an adaptive error boundary. Reduction using the multiplicative error bound is considered. Comparison of the results of reduction of the model by all described methods is carried out, the choice of the best method of reduction of the model is substantiated. Results. An analysis of the approximation error was performed for all the methods. The maximum relative error has been calculated. An example of calculating the order of the model for a given error value of 5% is given. The order of the result model is 34 states with the error less is then 1%, which is less than the original model. As a result, the AFCs of the original and reduced models, as well as the transient processes of the models, were constructed. The plots in the frequency domain of the models practically coincide, which indicates an adequate description of the system. Conclusions. As a result, it was shown that it is possible to reduce the size of the model by 14 orders of magnitude, goal achieved.Город Березники Пермского края расположен на подработанной шахтной территории. Уже несколько лет в городе наблюдаются активные проседания почвы, которые провоцируют разрушения зданий. Поэтому уже несколько лет ведется мониторинг зданий и сооружений города, позволяющий анализировать степень оседания. Для точного анализа ситуации и прогнозирования используются модели достаточно высокого порядка. Работа посвящена возможности моделирования деформации зданий, связанных с проседанием почвы, в результате горных выработок в городе Березники. Целью исследования является рассмотрение возможностей пакета Robust Control Toolbox для понижения порядка сложности моделей на примере восьмиэтажного здания, входящего в сборник эталонных примеров для редукции моделей линейных динамических систем. Материалы и методы. Представлены типичные шаги для решения задачи редукции модели, описаны команды и инструменты, применяемые для решения этой задачи. Определены параметры модели в пространстве состояний, которая насчитывает 48 состояний, являющихся смещениями или скоростями изменения. С помощью сингулярных значений Ганкеля выбраны состояния, которыми можно пренебречь. Выполнено редуцирование модели с использованием адаптивной границы ошибки. Рассмотрено редуцирование с использованием границы мультипликативной ошибки. Выполнено сравнение результатов редуцирования модели всеми описанными способами, обоснован выбор наилучшего способа редуцирования модели. Результаты. Для всех методов выполнен анализ ошибки аппроксимации. Рассчитана максимальная относительная ошибка. Приведен пример расчета порядка модели для заданной величины ошибки в 5 %. Для такой ошибки порядок модели составил 34 состояния, что меньше исходной модели. Для модели с 34 состояниями величина ошибки составляет менее 1 %. В результате построены АФЧХ исходной и редуцированной модели, а также переходные процессы моделей. Графики в частотной области моделей практически совпадают, что говорит об адекватном описании системы. Заключение. В результате было показано, что возможно снизить размер модели на 14 порядков, цель достигнута

Adaptive heterogeneous parallelism for semi-empirical lattice dynamics in computational materials science.

With the variability in performance of the multitude of parallel environments available today, the conceptual overhead created by the need to anticipate runtime information to make design-time decisions has become overwhelming. Performance-critical applications and libraries carry implicit assumptions based on incidental metrics that are not portable to emerging computational platforms or even alternative contemporary architectures. Furthermore, the significance of runtime concerns such as makespan, energy efficiency and fault tolerance depends on the situational context. This thesis presents a case study in the application of both Mattsons prescriptive pattern-oriented approach and the more principled structured parallelism formalism to the computational simulation of inelastic neutron scattering spectra on hybrid CPU/GPU platforms. The original ad hoc implementation as well as new patternbased and structured implementations are evaluated for relative performance and scalability. Two new structural abstractions are introduced to facilitate adaptation by lazy optimisation and runtime feedback. A deferred-choice abstraction represents a unified space of alternative structural program variants, allowing static adaptation through model-specific exhaustive calibration with regards to the extrafunctional concerns of runtime, average instantaneous power and total energy usage. Instrumented queues serve as mechanism for structural composition and provide a representation of extrafunctional state that allows realisation of a market-based decentralised coordination heuristic for competitive resource allocation and the Lyapunov drift algorithm for cooperative scheduling

Image 100 procedures manual development: Applications system library definition and Image 100 software definition

An outline for an Image 100 procedures manual for Earth Resources Program image analysis was developed which sets forth guidelines that provide a basis for the preparation and updating of an Image 100 Procedures Manual. The scope of the outline was limited to definition of general features of a procedures manual together with special features of an interactive system. Computer programs were identified which should be implemented as part of an applications oriented library for the system

Solving Multiple Classes of Problems in Parallel with MATLAB*P

MATLAB [7] is one of the most widely used mathematical computing environments in technical computing. It is an interactive environment that provides high performance computational routines and an easy-to-use, C-like scripting language. Mathworks, the company that develops MATLAB, currently does not provide a version of MATLAB that can utilize parallel computing [9]. This has led to academic and commercial efforts outside Mathworks to build a parallel MATLAB, using a variety of approaches. MATLAB*P is a parallel MATLAB that focus on enhancing productivity by providing an easy to use parallel computing tool. Using syntaxes identical to regular MATLAB, it can be used to solve large scale algebraic problems as well as multiple small problems in parallel. This paper describes how the innovative combination of ’*p mode’ and ’MultiMATLAB/MultiOctave mode’ in MATLAB*P can be used to solve a large range of real world problems.Singapore-MIT Alliance (SMA

Asymmetries in symmetric quantum walks on two-dimensional networks

We study numerically the behavior of continuous-time quantum walks over networks which are topologically equivalent to square lattices. On short time scales, when placing the initial excitation at a corner of the network, we observe a fast, directed transport through the network to the opposite corner. This transport is not ballistic in nature, but rather produced by quantum mechanical interference. In the long time limit, certain walks show an asymmetric limiting probability distribution; this feature depends on the starting site and, remarkably, on the precise size of the network. The limiting probability distributions show patterns which are correlated with the initial condition. This might have consequences for the application of continuous time quantum walk algorithms.Comment: 9 pages, 12 figures, revtex