• 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.

• 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.

• 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.

• 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.

• 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.

• 제어로봇시스템학회:학술대회논문집
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• 1992.10b
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• pp.197-202
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• 1992

In this paper, we consider the synthesis of mixed H$_{2}$/H$_{\infty}$ controllers such that the closed-loop poles are located in a specified region in the complex plane. Using solution to a generalized Riccati equation and a change of variable technique, it is shown that this synthesis problem can be reduced to a convex optimization problem over a bounded subset of matrices. This convex programming can be further reduced to Generalized Eigenvalue Minimization Problem where Interior Point method has been recently developed to efficiently solve this problem..

• 제어로봇시스템학회:학술대회논문집
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• 1992.10b
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• pp.575-580
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• 1992

Mixed H$_{2}$/H$_{\infty}$ robust control synthesis is considered for finite dimensional linear time-invariant systems under the presence of diagonal structured uncertainties. Such uncertainties arise for instance when there is real perturbation in the nominal model of the state space system or when modeling multiple (unstructured) uncertainty at different locations in the feedback loop. This synthesis problem is reduced to convex optimization problem over a bounded subset of matrices as well as diagonal matrix having certain structure. For computational purpose, this convex optimization problem is further reduced into Generalized Eigenvalue Minimization Problem where a powerful algorithm based on interior point method has been recently developed..

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.305-316
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• 2003

Recently, an iterative algorithm for finding the interior eigenvalues of a definite matrix by CG-type method has been proposed. This method compares to the inverse power method. The given matrices A, and B are assumed to be large and sparse, and SPD( Symmetric Positive Definite) The CG scheme for the optimization of the Rayleigh quotient has been proven a very attractive and promising technique for large sparse eigenproblems for smallest eigenvalue. Also, it is very amenable to parallel computations, like the CG method for the linear systems. A proper choice of the preconditioner significantly improves the convergence of the CG scheme. But for parallel computations we need to find an efficient parallel preconditioner. Our candidates we ILU(0) in the wave-front order, ILU(0) in the multi-coloring order, Point-SSOR(Symmetric Successive Overrelaxation), and Multi-Color Block SSOR preconditioner. Wavefront order is a simple way to increase parallelism in the natural order, and Multi-coloring realizes a parallelism of order(N), where N is the order of the matrix. Another choice is the Multi-Color Block SSOR(Symmetric Successive OverRelaxation) preconditioning. Block SSOR is a symmetric preconditioner which is expected to minimize the interprocessor communication due to the blocking. We implemented the results on the CRAY-T3E with 128 nodes. The MPI (Message Passing Interface) library was adopted for the interprocessor communications. The test problem was drawn from the discretizations of partial differential equations by finite difference methods. The results show that for small number of processors Multi-Color ILU(0) has the best performance, while for large number of processors Multi-Color Block SSOR performs the best.

• Journal of KIISE:Software and Applications
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• v.27 no.5
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• pp.463-472
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• 2000

In this paper, we consider the problem of realizing associative memories via cellular neural network(CNN). After introducing qualitative properties of the CNN model, we formulate the synthesis of CNN that can store given binary vectors with optimal performance as a constrained optimization problem. Next, we observe that this problem's constraints can be transformed into simple inequalities involving linear matrix inequalities(LMIs). Finally, we reformulate the synthesis problem as a generalized eigenvalue problem(GEVP), which can be efficiently solved by recently developed interior point methods. Proposed method can be applied to both space varying template CNNs and space-invariant template CNNs. The validity of the proposed approach is illustrated by design examples.