• Journal of the Computational Structural Engineering Institute of Korea
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• v.20 no.6
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• pp.743-749
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• 2007

Algebraic substructuring (AS) is a state-of-the-art method in eigenvalue computations, especially for large size problems, but, originally, it was designed to calculate only the smallest eigenvalues. In this paper, an updated version of AS is proposed to calculate the interior eigenvalues over a specified range by using a shift value, which is referred to as the shifted AS. Numerical experiments demonstrate that the proposed method has better efficiency to compute numerous interior eigenvalues for the finite element models of structural problems than a Lanczos-type method.

• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.435-443
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• 2007

Computing the interior spectrum of large sparse generalized eigenvalue problems $Ax\;=\;{\lambda}Bx$, where A and b are large sparse and SPD(Symmetric Positive Definite), is often required in areas such as structural mechanics and quantum chemistry, to name a few. Recently, CG-type methods have been found useful and hence, very amenable to parallel computation for very large problems. Also, as in the case of linear systems proper choice of preconditioning is known to accelerate the rate of convergence. After the smallest eigenpair is found we use the orthogonal deflation technique to find the next m-1 eigenvalues, which is also suitable for parallelization. This offers advantages over Jacobi-Davidson methods with partial shifts, which requires re-computation of preconditioner matrx with new shifts. We consider as preconditioners Incomplete LU(ILU)(0) in two variants, ever-relaxation(SOR), and Point-symmetric SOR(SSOR). We set m to be 5. We conducted our experiments on matrices from discretizations of partial differential equations by finite difference method. The generated matrices has dimensions up to 4 million and total number of processors are 32. MPI(Message Passing Interface) library was used for interprocessor communications. Our results show that in general the Multi-Color ILU(0) gives the best performance.

• Bulletin of the Korean Mathematical Society
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• v.32 no.1
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• pp.19-23
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• 1995

We will estimate the lower bound of the first nonzero Neumann and Dirichlet eigenvalue of Laplacian equation on compact Riemannian manifold M with boundary. In case that the boundary of M has positive second fundamental form elements, Ly-Yau[3] gave the lower bound of the first nonzero neumann eigenvalue $\eta_1$. In case that the second fundamental form elements of $\partial$M is bounded below by negative constant, Roger Chen[4] investigated the lower bound of $\eta_1$. In [1], [2], we obtained the lower bound of the first nonzero Neumann eigenvalue is estimated under the condtion that the second fundamental form elements of boundary is bounded below by zero. Moreover, I realize that "the interior rolling $\varepsilon$ - ball condition" is not necessary when the first Dirichlet eigenvalue was estimated in [1].ed in [1].

• Journal of the Korean Mathematical Society
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• v.56 no.4
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• pp.857-867
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• 2019

Suppose that G = (V, E) is a connected locally finite graph with the vertex set V and the edge set E. Let ${\Omega}{\subset}V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph G $$\{-{\Delta}_{pu}={\lambda}K(x){\mid}u{\mid}^{p-2}u+f(x,u),\;x{\in}{\Omega}^{\circ},\\u=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}^{\circ}$ and ${\partial}{\Omega}$ denote the interior and the boundary of ${\Omega}$, respectively, ${\Delta}_p$ is the discrete p-Laplacian, K(x) is a given function which may change sign, ${\lambda}$ is the eigenvalue parameter and f(x, u) has exponential growth. We prove the existence and monotonicity of the principal eigenvalue of the corresponding eigenvalue problem. Furthermore, we also obtain the existence of a positive solution by using variational methods.

• Journal of the Korean Institute of Intelligent Systems
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• v.10 no.2
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• pp.161-167
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• 2000

In this paper, we propose a design method for BAMs(bidirectiona1 associative memories) which can perform the function of bidirectional association efficiently. Based on the theoretical investigation about the properties of BAMs, we first formulate the problem of finding a BAM that can store the given pattern pairs as stable states with high error correction ratio in the form of a constrained optimization problem. Next, we transform the constrained optimization problem into a GEVP(genera1ized eigenvalue problem), which can be solved by recently developed interior point methods. The applicability of the proposed method is illustrated via design examples.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.467-474
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• 2006

We present and study the stability and convergence of a deflation-preconditioned conjugate gradient(PCG) scheme for the interior generalized eigenvalue problem $Ax = {\lambda}Bx$, where A and B are large sparse symmetric positive definite matrices. Numerical experiments are also presented to support our theoretical results.

• Communications of the Korean Mathematical Society
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• v.16 no.3
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• pp.437-443
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• 2001

In this paper the inverse problems for the Dirac Operator are studied. A set of values of eigenfunctions in some internal point and spectrum are taken as a data. Uniqueness theorems are obtained. The approach that was used in the investigation of inverse problems for interior spectral data of the Sturm-Liouville operator is employed.

• The KIPS Transactions:PartA
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• v.10A no.2
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• pp.137-140
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• 2003

Recently, CG (Conjugate Gradient) scheme for the optimization of the Rayleigh quotient has been proven a very attractive and promising technique for interior eigenvalues for the following eigenvalue problem, Ax＝λx (1) The given matrix A is assummed to be large and sparse, and symmetric. Also, the method is very amenable to parallel computations. A proper choice of the preconditioner significantly improves the convergence of the CG scheme. We compare the parallel preconditioners for the computation of the interior eigenvalues of a symmetric matrix by CG-type method. The considered preconditioners are Point-SSOR, ILU (0) in the multi-coloring order, and Multi-Color Block SSOR (Symmetric Succesive OverRelaxation). We conducted our experiments on the CRAY­T3E with 128 nodes. The MPI (Message Passing Interface) library was adopted for the interprocessor communications. The test matrices are up to $512{\times}512$ in dimensions and were created from the discretizations of the elliptic PDE. All things considered the MC-BSSOR seems to be most robust preconditioner.

• Journal of the Korean Institute of Intelligent Systems
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• v.11 no.4
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• pp.340-346
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• 2001

This paper proposes a new design method for the GBAM: (general bidirectional associative memory) model. Based on theoretical investigations on the GBAM: model, it is shown that the design of the GBAM:-based bidirectional associative memeories can be formulated as optimization problems called GEVPs (generalized eigenvalue problems). Since the GEVPs arising in the procedure can be efficiently solved within a given tolerance by the recently developed interior point methods, the design procedure established in this paper is very useful in practice. The applicability of the proposed design procedure is demonstrated by simple design examples considered in related studies.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2001.05a
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• pp.69-72
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• 2001

In this paper, we propose a new design method for bidirectional associative memories model with high error correction ratio. We extend the conventional GBAM model using bias terms and formulate a design procedure in the form of a constrained optimization problem. The constrained optimization problem is then transformed into a GEVP(generalized eigenvalue problem), which can be efficiently solved by recently developed interior point methods. The effectiveness of the proposed approach is illustrated by a example.