Residual Arnoldi method, theory, package and experiments

This thesis is concerned with the solution of large-scale eigenvalue problems. Although there are good algorithms for solving small dense eigenvalue problems, the large-scale eigenproblem has many open issues. The major difficulty faced by existing algorithms is the tradeoff of precision and time, especially when one is looking for interior or clustered eigenvalues. In this thesis, we present a new method called the residual Arnoldi method. This method has the desirable property that certain intermediate results can be computed in low precision without effecting the final accuracy of the solution. Thus we can reduce the computational cost without sacrificing accuracy. This thesis is divided into three parts. In the first, we develop the theoretical background of the residual Arnoldi method. In the second part, we describe RAPACK, a numerical package implementing the residual Arnoldi method. In the last part, numerical experiments illustrate the use of the package and show the practicality of the method

Experiences with Implementing Parallel Discrete-event Simulation on GPU

Modern graphics processing units (GPUs) offer much more computational power than recent CPUs by providing a vast number of simple, data-parallel, multithreaded cores. In this study, we focus on the use of a GPU to perform parallel discrete-event simulation. Our approach is to use a modified service time distribution function to allow more independent events to be processed in parallel. The implementation issues and alternative strategies will be discussed in detail. We describe and compare our experience and results in using Thrust and CUB, two open-source parallel algorithms libraries which resemble the C++ Standard Template Library, to build our tool. The experimental results show that our implementation can be two orders of magnitude faster than the sequential simulation for large-scale simulation models

Play as You Like: Timbre-enhanced Multi-modal Music Style Transfer

Style transfer of polyphonic music recordings is a challenging task when considering the modeling of diverse, imaginative, and reasonable music pieces in the style different from their original one. To achieve this, learning stable multi-modal representations for both domain-variant (i.e., style) and domain-invariant (i.e., content) information of music in an unsupervised manner is critical. In this paper, we propose an unsupervised music style transfer method without the need for parallel data. Besides, to characterize the multi-modal distribution of music pieces, we employ the Multi-modal Unsupervised Image-to-Image Translation (MUNIT) framework in the proposed system. This allows one to generate diverse outputs from the learned latent distributions representing contents and styles. Moreover, to better capture the granularity of sound, such as the perceptual dimensions of timbre and the nuance in instrument-specific performance, cognitively plausible features including mel-frequency cepstral coefficients (MFCC), spectral difference, and spectral envelope, are combined with the widely-used mel-spectrogram into a timber-enhanced multi-channel input representation. The Relativistic average Generative Adversarial Networks (RaGAN) is also utilized to achieve fast convergence and high stability. We conduct experiments on bilateral style transfer tasks among three different genres, namely piano solo, guitar solo, and string quartet. Results demonstrate the advantages of the proposed method in music style transfer with improved sound quality and in allowing users to manipulate the output

EIGENTEST: A Test Matrix Generator for Large-Scale Eigenproblems

Eigentest is a package that produces real test matrices with known eigensystems. A test matrix, called an eigenmat, is generated in a factored form, in which the user can specify the eigenvalues and has some control over the condition of the eigenvalues and eigenvectors. An eigenmat A of order n requires only O(n) storage for its representation. Auxiliary programs permit the computation of (A - sI)*b, (A - sI)'*b, inv(A - sI)*b, and inv(A - sI)*b in O(n) operations. A special routine computes specified eigenvectors of an eigenmat and the condition of its eigenvalue. Thus eigenmats are suitable for testing algorithms based on Krylov sequences, as well as others based on matrix-vector products. This paper introduces the eigenmat and describes implementations in Fortran~77, Fortran~95, C, and Matlab

On the Preconditioner of Conjugate Gradient Method a Power Grid Simulation Perspective

Preconditioned Conjugate Gradient (PCG) method has been demonstrated to be effective in solving large-scale linear systems for sparse and symmetric positive definite matrices. One critical problem in PCG is to design a good preconditioner, which can significantly reduce the runtime while keeping memory usage efficient. Universal preconditioners are simple and easy to construct, but their effectiveness is highly problem dependent. on the other hand, domain-specific preconditioners that explore the underlying physical meaning of the matrices usually work better but are difficult to design. in this paper, we study the problem in the context of power grid simulation, and develop a novel preconditioner based on the power grid structure through simple circuit simulations. Experimental results show 43% reduction in the number of iterations and 23% speedup over existing universal preconditioners. © 2011 IEEE

Analysis of the Residual Arnoldi Method

The Arnoldi method generates a nested squences of orthonormal bases $U_{1},U_{2}, \ldots$ by orthonormalizing $Au_{k}$ against $U_{k}$. Frequently these bases contain increasingly accurate approximations of eigenparis from the periphery of the spectrum of $A$. However, the convergence of these approximations stagnates if $u_{k}$ is contaminated by error. It has been observed that if one chooses a Rayleigh--Ritz approximation $(\mu_{k}, z_{k})$ to a chosen target eigenpair $(\lambda, x)$ and orthonormalizes the residual $Az_{k - }\mu_{k} z_{k}$, the approximations to $x$ (but not the other eigenvectors) continue to converge, even when the residual is contaminated by error. The same is true of the shift-invert variant of Arnoldi's method. In this paper we give a mathematical analysis of these new methods