• Journal of applied mathematics & informatics
• /
• 제25권1_2호
• /
• pp.435-443
• /
• 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
• /
• 제22권1_2호
• /
• pp.467-474
• /
• 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.

• 한국지능시스템학회:학술대회논문집
• /
• 한국퍼지및지능시스템학회 2001년도 춘계학술대회 학술발표 논문집
• /
• pp.69-72
• /
• 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.

• 한국지능시스템학회논문지
• /
• 제11권4호
• /
• pp.340-346
• /
• 2001

본 논문은 GBAM (general bidirectional associative memory) 모델을 위한 새로운 설계방법을 제시한다. GBAM 모델에 대한 이론적 고찰을 바탕으로, GBAM 기방 양방향 연상 메모리의 설계 문제가 GEVP (generalized eigenvalue problem)로 불리는 최적화 문제로 표현될 수 있음을 밝힌다. 설계 과정에서 등장하는 GEVP 문제들은 최근에 개발된 내부점 방법에 의하여 주어진 허용 오차 이내에서 효과적으로 풀릴 수 있으므로, 본 논문에서 확립된 설계 절차는 매우 실용적이다. 제안된 설계 절차에 대한 적용 가능성은 관련 연구에서 고려되었던 간단한 설계 예제를 통하여 예시된다.

• 한국지능시스템학회논문지
• /
• 제10권2호
• /
• pp.161-167
• /
• 2000

본 논문에서는 양방향 연상 기능을 효과적으로 수행할 수 있는 BAM(bidirectional associative memory)의 설계방법론을 제안한다. 먼저 BAM의 성질에 관한 이론적 고찰을 바탕으로 하여 주어진 패턴 쌍을 안정하게 그리고 높은 오차수정율(error correction ratio)을 가지고 저장할 수 있는 BAM을 찾는 문제를 제약조건이 있는 최적화 문제로 공식화한다 다음과정에서 이 최적화 문제를 GEVP(generalized eigenvalue problem)로 변환함으로써 최근에 개발된 내부점 방법(interior point method)을 통하여 해가 구해질 수 있도록 한다. 제안된 설계 방법론의 적용가능성은 예제를 통해 확인된다.

• 제어로봇시스템학회:학술대회논문집
• /
• 제어로봇시스템학회 1992년도 한국자동제어학술회의논문집(국제학술편); KOEX, Seoul; 19-21 Oct. 1992
• /
• pp.197-202
• /
• 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
• /
• 제어로봇시스템학회 1992년도 한국자동제어학술회의논문집(국제학술편); KOEX, Seoul; 19-21 Oct. 1992
• /
• pp.575-580
• /
• 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
• /
• 제11권1_2호
• /
• pp.305-316
• /
• 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.

• 한국정보과학회논문지:소프트웨어및응용
• /
• 제27권5호
• /
• pp.463-472
• /
• 2000

본 논문에서는 연상 메모리 기능을 수행하는 셀룰라 신경망(Cellular Neural Network)의 설계를 위한 새로운 방법론을 제안한다. 먼저, 셀룰라 신경망 모델의 기본적 특성들을 소개한 후, 최적 성능을 가지고 이진 원형 패턴들을 저장할 수 있는 셀룰라 신경망 모델의 설계 방법을 제약 조건이 가해진 최적화 문제로 공식화한다. 다음으로 이 문제의 제약 조건을 선형 행렬 부등식(Linear Matrix Inequalities)을 포함하는 부등식의 형태로 변환시킬 수 있음을 관찰한다. 마지막으로 셀룰라 신경망 최적 설계 문제를 내부점 방법(interior point method)에 의해 효율적으로 풀릴 수 있는 일반화된 고유값 문제(Genaralized EigenValue Problem)로 변환한다. 본 논문에서 제시하는 셀룰라 신경망 설계 방법론은 공간 변형 형판 셀룰라 신경망과 공간 불변 형판 셀룰라 신경망 설계에 모두 적용될 수 있다. 설계 예제를 통해 제안된 방법의 유효성을 검증한다.