The Experts below are selected from a list of 297 Experts worldwide ranked by ideXlab platform

Bor Plestenjak - One of the best experts on this subject based on the ideXlab platform.

  • A Sylvester–Arnoldi type method for the Generalized Eigenvalue Problem with two‐by‐two operator determinants
    Numerical Linear Algebra with Applications, 2015
    Co-Authors: Karl Meerbergen, Bor Plestenjak
    Abstract:

    In various applications, for instance in the detection of a Hopf bifurcation or in solving separable boundary value Problems using the two-parameter Eigenvalue Problem, one has to solve a Generalized Eigenvalue Problem of the form (B1 ⊗A2 −A1 ⊗B2)z = μ(B1 ⊗ C2 − C1 ⊗B2)z, where matrices are 2 × 2 operator determinants. We present efficient methods that can be used to compute a small subset of the Eigenvalues. For full matrices of moderate size we propose either the standard implicitly restarted Arnoldi or Krylov–Schur iteration with shift-and-invert transformation, performed efficiently by solving a Sylvester equation. For large Problems, it is more efficient to use subspace iteration based on low-rank approximations of the solution of the Sylvester equation combined with a Krylov–Schur method for the projected Problems.

  • a sylvester arnoldi type method for the Generalized Eigenvalue Problem with two by two operator determinants
    TW Reports, 2014
    Co-Authors: Karl Meerbergen, Bor Plestenjak
    Abstract:

    In various applications, for instance in the detection of a Hopf bifurcation or in solving separable boundary value Problems using the two-parameter Eigenvalue Problem, one has to solve a Generalized Eigenvalue Problem of the form (B1 ⊗A2 −A1 ⊗B2)z = μ(B1 ⊗ C2 − C1 ⊗B2)z, where matrices are 2 × 2 operator determinants. We present efficient methods that can be used to compute a small subset of the Eigenvalues. For full matrices of moderate size we propose either the standard implicitly restarted Arnoldi or Krylov–Schur iteration with shift-and-invert transformation, performed efficiently by solving a Sylvester equation. For large Problems, it is more efficient to use subspace iteration based on low-rank approximations of the solution of the Sylvester equation combined with a Krylov–Schur method for the projected Problems.

Karl Meerbergen - One of the best experts on this subject based on the ideXlab platform.

  • A Sylvester–Arnoldi type method for the Generalized Eigenvalue Problem with two‐by‐two operator determinants
    Numerical Linear Algebra with Applications, 2015
    Co-Authors: Karl Meerbergen, Bor Plestenjak
    Abstract:

    In various applications, for instance in the detection of a Hopf bifurcation or in solving separable boundary value Problems using the two-parameter Eigenvalue Problem, one has to solve a Generalized Eigenvalue Problem of the form (B1 ⊗A2 −A1 ⊗B2)z = μ(B1 ⊗ C2 − C1 ⊗B2)z, where matrices are 2 × 2 operator determinants. We present efficient methods that can be used to compute a small subset of the Eigenvalues. For full matrices of moderate size we propose either the standard implicitly restarted Arnoldi or Krylov–Schur iteration with shift-and-invert transformation, performed efficiently by solving a Sylvester equation. For large Problems, it is more efficient to use subspace iteration based on low-rank approximations of the solution of the Sylvester equation combined with a Krylov–Schur method for the projected Problems.

  • a sylvester arnoldi type method for the Generalized Eigenvalue Problem with two by two operator determinants
    TW Reports, 2014
    Co-Authors: Karl Meerbergen, Bor Plestenjak
    Abstract:

    In various applications, for instance in the detection of a Hopf bifurcation or in solving separable boundary value Problems using the two-parameter Eigenvalue Problem, one has to solve a Generalized Eigenvalue Problem of the form (B1 ⊗A2 −A1 ⊗B2)z = μ(B1 ⊗ C2 − C1 ⊗B2)z, where matrices are 2 × 2 operator determinants. We present efficient methods that can be used to compute a small subset of the Eigenvalues. For full matrices of moderate size we propose either the standard implicitly restarted Arnoldi or Krylov–Schur iteration with shift-and-invert transformation, performed efficiently by solving a Sylvester equation. For large Problems, it is more efficient to use subspace iteration based on low-rank approximations of the solution of the Sylvester equation combined with a Krylov–Schur method for the projected Problems.

Ioannis Dassios - One of the best experts on this subject based on the ideXlab platform.

  • primal and dual Generalized Eigenvalue Problems for power systems small signal stability analysis
    IEEE Transactions on Power Systems, 2017
    Co-Authors: Federico Milano, Ioannis Dassios
    Abstract:

    The paper presents a comprehensive study of small-signal stability analysis of power systems based on matrix pencils and the Generalized Eigenvalue Problem. Both primal and dual formulations of the Generalized Eigenvalue Problem are considered and solved through a variety of state-of-the-art solvers. The paper also discusses the impact on the performance of the solvers of two formulations of the equations modelling the power systems, namely, the explicit and semi-implicit form of differential-algebraic equations. The case study illustrates the theoretical aspects and numerical features of these formulations and solvers through two real-world systems, namely, a 1,479-bus model of the all-island Irish system, and a 21,177-bus model of the ENTSO-E network.

Tong Zhang - One of the best experts on this subject based on the ideXlab platform.

  • sparse Generalized Eigenvalue Problem optimal statistical rates via truncated rayleigh flow
    Journal of The Royal Statistical Society Series B-statistical Methodology, 2018
    Co-Authors: Kean Ming Tan, Zhaoran Wang, Han Liu, Tong Zhang
    Abstract:

    The sparse Generalized Eigenvalue Problem (GEP) plays a pivotal role in a large family of high dimensional statistical models, including sparse Fisher's discriminant analysis, canonical correlation analysis and sufficient dimension reduction. The sparse GEP involves solving a non‐convex optimization Problem. Most existing methods and theory in the context of specific statistical models that are special cases of the sparse GEP require restrictive structural assumptions on the input matrices. We propose a two‐stage computational framework to solve the sparse GEP. At the first stage, we solve a convex relaxation of the sparse GEP. Taking the solution as an initial value, we then exploit a non‐convex optimization perspective and propose the truncated Rayleigh flow method (which we call ‘rifle’) to estimate the leading Generalized eigenvector. We show that rifle converges linearly to a solution with the optimal statistical rate of convergence. Theoretically, our method significantly improves on the existing literature by eliminating structural assumptions on the input matrices. To achieve this, our analysis involves two key ingredients: a new analysis of the gradient‐based method on non‐convex objective functions, and a fine‐grained characterization of the evolution of sparsity patterns along the solution path. Thorough numerical studies are provided to validate the theoretical results.

U. Helmke - One of the best experts on this subject based on the ideXlab platform.

  • ICASSP - A new algorithm for the Generalized Eigenvalue Problem
    1997 IEEE International Conference on Acoustics Speech and Signal Processing, 1
    Co-Authors: K. Huper, U. Helmke
    Abstract:

    The Problem of finding the Generalized Eigenvalues and eigenvectors of a pair of real symmetric matrices A and B, with B>0, can be viewed as a smooth optimization Problem on a smooth manifold. We present a cost function approach to the Generalized Eigenvalue Problem which is posed on the product of the n-sphere and Euclidian space R. The critical point set of this cost function is studied. An algorithm is presented based on constrained optimization. A proof of local quadratic convergence is given.