43 research outputs found
A Massively Parallel Algorithm for the Approximate Calculation of Inverse p-th Roots of Large Sparse Matrices
We present the submatrix method, a highly parallelizable method for the
approximate calculation of inverse p-th roots of large sparse symmetric
matrices which are required in different scientific applications. We follow the
idea of Approximate Computing, allowing imprecision in the final result in
order to be able to utilize the sparsity of the input matrix and to allow
massively parallel execution. For an n x n matrix, the proposed algorithm
allows to distribute the calculations over n nodes with only little
communication overhead. The approximate result matrix exhibits the same
sparsity pattern as the input matrix, allowing for efficient reuse of allocated
data structures.
We evaluate the algorithm with respect to the error that it introduces into
calculated results, as well as its performance and scalability. We demonstrate
that the error is relatively limited for well-conditioned matrices and that
results are still valuable for error-resilient applications like
preconditioning even for ill-conditioned matrices. We discuss the execution
time and scaling of the algorithm on a theoretical level and present a
distributed implementation of the algorithm using MPI and OpenMP. We
demonstrate the scalability of this implementation by running it on a
high-performance compute cluster comprised of 1024 CPU cores, showing a speedup
of 665x compared to single-threaded execution
A Fast Monte Carlo algorithm for evaluating matrix functions with application in complex networks
We propose a novel stochastic algorithm that randomly samples entire rows and
columns of the matrix as a way to approximate an arbitrary matrix function.
This contrasts with the "classical" Monte Carlo method which only works with
one entry at a time, resulting in a significant better convergence rate than
the "classical" approach. To assess the applicability of our method, we compute
the subgraph centrality and total communicability of several large networks. In
all benchmarks analyzed so far, the performance of our method was significantly
superior to the competition, being able to scale up to 64 CPU cores with a
remarkable efficiency.Comment: Submitted to the Journal of Scientific Computin
Dynamics of the N-link pendulum: A fractional perspective
This paper addresses the dynamics of an N-link planar pendulum in the perspective of fractional calculus. The proposed methodology leads to a novel point of view for signal propagation as a time-space wave within the system mechanical structure. Numerical simulations show the effectiveness of the approach both for signal visualisation and limit cycle detection.info:eu-repo/semantics/publishedVersio
Taylor's theorem for matrix functions with applications to condition number estimation
We derive an explicit formula for the remainder term of a Taylor polynomial of a matrix function. This formula generalizes a known result for the remainder of the Taylor polynomial for an analytic function of a complex scalar. We investigate some consequences of this result, which culminate in new upper bounds for the level-1 and level-2 condition numbers of a matrix function in terms of the pseudospectrum of the matrix. Numerical experiments show that, although the bounds can be pessimistic, they can be computed much faster than the standard methods. This makes the upper bounds ideal for a quick estimation of the condition number whilst a more accurate (and expensive) method can be used if further accuracy is required. They are also easily applicable to more complicated matrix functions for which no specialized condition number estimators are currently available
Rational Krylov methods for functions of matrices with applications to fractional partial differential equations
In this paper, we propose a new choice of poles to define reliable rational
Krylov methods. These methods are used for approximating function of positive
definite matrices. In particular, the fractional power and the fractional
resolvent are considered because of their importance in the numerical solution
of fractional partial differential equations. The results of the numerical
experiments we have carried out on some fractional models confirm that the
proposed approach is promising
Spectrum analysis of LTI continuous-time systems with constant delays: A literature overview of some recent results
In recent decades, increasingly intensive research attention has been given to dynamical systems containing delays and those affected by the after-effect phenomenon. Such research covers a wide range of human activities and the solutions of related engineering problems often require interdisciplinary cooperation. The knowledge of the spectrum of these so-called time-delay systems (TDSs) is very crucial for the analysis of their dynamical properties, especially stability, periodicity, and dumping effect. A great volume of mathematical methods and techniques to analyze the spectrum of the TDSs have been developed and further applied in the most recent times. Although a broad family of nonlinear, stochastic, sampled-data, time-variant or time-varying-delay systems has been considered, the study of the most fundamental continuous linear time-invariant (LTI) TDSs with fixed delays is still the dominant research direction with ever-increasing new results and novel applications. This paper is primarily aimed at a (systematic) literature overview of recent (mostly published between 2013 to 2017) advances regarding the spectrum analysis of the LTI-TDSs. Specifically, a total of 137 collected articles-which are most closely related to the research area-are eventually reviewed. There are two main objectives of this review paper: First, to provide the reader with a detailed literature survey on the selected recent results on the topic and Second, to suggest possible future research directions to be tackled by scientists and engineers in the field. © 2013 IEEE.MSMT-7778/2014, FEDER, European Regional Development Fund; LO1303, FEDER, European Regional Development Fund; CZ.1.05/2.1.00/19.0376, FEDER, European Regional Development FundEuropean Regional Development Fund through the Project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19.0376]; National Sustainability Program Project [LO1303 (MSMT-7778/2014)
Efficient computation of matrix power-vector products: application for space-fractional diffusion problems
A novel algorithm is proposed for computing matrix-vector products A^\alpha v, where A is a symmetric positive semidefinite sparse matrix and \alpha > 0. The method can be applied for the efficient implementation of the matrix transformation method to solve space-fractional diffusion problems. The performance of the new algorithm is studied in a comparison with the conventional MATLAB subroutines to compute matrix powers