Eigenvalue Problems for Exponential-Type Kernels

Journal Article · Sat Apr 06 00:00:00 EDT 2019 · Computational Methods in Applied Mathematics

DOI:https://doi.org/10.1515/cmam-2018-0186· OSTI ID:1669237

Cai, Difeng ^[1]; Vassilevski, Panayot S. ^[2]

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1. Purdue Univ., West Lafayette, IN (United States). Dept. of Mathematics
2. Portland State Univ., OR (United States). Dept. of Mathematics and Statistics; Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing

Here, We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate $\left|{\lambda }_k-{\lambda }_{k,h}\right|\le C_k{h}^2$ in the case of lowest order approximation for both Galerkin and Nystrom methods, where h is the mesh size, ${\lambda }_k$ and ${\lambda }_{k,h}$ are the exact and approximate kth largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that $\left|{\lambda }_{k,h}^{\left(G\right)}-{\lambda }_{k,h}^{\left(N\right)}\right|\le C{h}^2$ , where ${\lambda }_{k,h}^{\left(G\right)}$ and ${\lambda }_{k,h}^{\left(N\right)}$ denote the kth largest eigenvalues computed by Galerkin and Nystrom methods, respectively, and C is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments.

• Research Organization: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: AC52-07NA27344
• OSTI ID: 1669237
• Report Number(s): LLNL-JRNL--755085; 942021
• Journal Information: Computational Methods in Applied Mathematics, Journal Name: Computational Methods in Applied Mathematics Journal Issue: 1 Vol. 20; ISSN 1609-4840
• Publisher: de Gruyter
• Country of Publication: United States
• Language: English

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