Advanced Methods For Solving Nonlinear Eigenvalue Problems Of Generalized Phase Optimization

Author : Yunshen Zhou
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.6/5 (624 download)

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Download or read book Solution of Nonlinear Eigenvalue Problems Arising from Constrained Rayleigh Quotient Optimization and Resonant Modes Computation of Accelerator Cavity written by Yunshen Zhou and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation consists of two parts: constrained Rayleigh quotient optimization problems and its application in image segmentation, and Pade approximation linearization algorithm to solve nonlinear eigenvalue problems arising from resonant modes computation of accelerator cavity. In the first part, we consider the following constrained Rayleigh quotient optimization problem [formula omitted], where A is an n*n real symmetric matrix and C is an n*m real matrix. Usually, m“n. The problem is also known as the constrained eigenvalue problem in the literature because it becomes an eigenvalue problem if the linear constraint C[superscript T]x = b is removed. We start by equivalently transforming CRQopt into an optimization problem, called LGopt, of minimizing the Lagrangian multiplier of CRQopt, and then an eigenvalue problem, called QEPmin, of finding the smallest eigenvalue of a quadratic eigenvalue problem. Although such equivalences has been discussed in the literature, it appears to be the first time that these equivalences are rigorously justified. Then we propose to numerically solve LGopt and QEPmin by the Krylov subspace projection method via the Lanczos process. The basic idea, as the Lanczos method for the symmetric eigenvalue problem, is to first reduce LGopt and QEPmin by projecting them onto Krylov subspaces to yield problems of the same types but of much smaller sizes, and then solve the reduced problems by some direct methods, which is either a secular equation solver (in the case of LGopt) or an eigensolver (in the case of QEPmin). The resulting algorithm is called the Lanczos algorithm. We perform convergence analysis for the proposed method and obtain error bounds. The sharpness of the error bound is demonstrated by artificial examples, although in applications the method often converges much faster than the bounds suggest. Finally, we apply the Lanczos algorithm to semi-supervised learning in the context of constrained clustering. In the second part, we propose a method to solve nonlinear eigenvalue problems with low-rank nonlinear terms. We first apply rational approximants to the nonlinear functions and transform the nonlinear eigenvalue problem to a rational eigenvalue problem. Then we transform the rational eigenvalue problem to a linear eigenvalue problem by trimmed linearization. For solving the linear eigenvalue problem, we provide a method to compute the matrix-vector multiplication for the shift-invert Arnoldi method, which is suitable for any shift. Moreover, the method of matrix-vector multiplication is designed to make the arithmetic as real as possible. We show the effectiveness of our Pade approximate linearization (PAL) method by comparing it with the fully rational Krylov method for nonlinear eigenvalue problems (NLEIGS) and compact rational Krylov (CORK) methods to solve nonlinear eigenvalue problems arising from finite element electromagnetic simulations in accelerator modeling. For these problems, we provide a method to choose the expansion point of rational approximants to reduce the residual for the eigenpairs by up to three digits. Numerical examples show that our PAL algorithm runs 48% to 87% faster than NLEIGS and CORK algorithm with comparable results.