Eigenvalue Problems for Exponential-Type Kernels
Eigenvalue Problems for Exponential-Type Kernels
复制标题
指数型核的特征值问题
DOI：
10.1515/cmam-2018-0186
复制
发表时间：
2019
影响因子：
1.3
通讯作者：
P. Vassilevski
中科院分区：
文献类型：
--
作者：
Difeng Cai;P. Vassilevski
Abstract We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate | λ k - λ k , h | ≤ C k h 2 {\lvert\lambda_{k}-\lambda_{k,h}\rvert\leq C_{k}h^{2}} in the case of lowest order approximation for both Galerkin and Nyström methods, where h is the mesh size, λ k {\lambda_{k}} and λ k , h {\lambda_{k,h}} are the exact and approximate kth largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that | λ k , h ( G ) - λ k , h ( N ) | ≤ C h 2 {|\lambda^{(G)}_{k,h}-\lambda^{(N)}_{k,h}|\leq Ch^{2}} , where λ k , h ( G ) {\lambda^{(G)}_{k,h}} and λ k , h ( N ) {\lambda^{(N)}_{k,h}} denote the kth largest eigenvalues computed by Galerkin and Nyström methods, respectively, and C is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments.
Abstract We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate | λ k - λ k , h | ≤ C k h 2 {\lvert\lambda_{k}-\lambda_{k,h}\rvert\leq C_{k}h^{2}} in the case of lowest order approximation for both Galerkin and Nyström methods, where h is the mesh size, λ k {\lambda_{k}} and λ k , h {\lambda_{k,h}} are the exact and approximate kth largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that | λ k , h ( G ) - λ k , h ( N ) | ≤ C h 2 {|\lambda^{(G)}_{k,h}-\lambda^{(N)}_{k,h}|\leq Ch^{2}} , where λ k , h ( G ) {\lambda^{(G)}_{k,h}} and λ k , h ( N ) {\lambda^{(N)}_{k,h}} denote the kth largest eigenvalues computed by Galerkin and Nyström methods, respectively, and C is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments.