• 대한수학회논문집
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• 제11권4호
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• pp.1147-1157
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• 1996

Nonlinear ill-posed problems arise in many application including parameter estimation and inverse scattering. We introduce a least squares regularization method to solve nonlinear ill-posed problems with constraints robustly and efficiently. The regularization method uses Trust-Region approach to handle the constraints on variables. The Generalized Cross Validation is used to choose the regularization parameter in computational tests. Numerical results are given to exhibit faster convergence of the method over other methods.

• Journal of applied mathematics & informatics
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• 제29권3_4호
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• pp.909-920
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• 2011

A regularized Newton method for nonlinear ill-posed problems is considered. In each Newton step an implicit iterative method with an appropriate stopping rule is proposed and analyzed. Under certain assumptions on the nonlinear operator, the convergence of the algorithm is proved and the algorithm is stable if the discrepancy principle is used to terminate the outer iteration. Numerical experiment shows the effectiveness of the method.

• Journal of applied mathematics & informatics
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• 제7권1호
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• pp.29-40
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• 2000

We provide semilocal convergence theorems for Newton-like methods in Banach space using outer and generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Frechet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least square problems and ill-posed nonlinear operator equations.

• Journal of applied mathematics & informatics
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• 제27권3_4호
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• pp.621-638
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• 2009

The linear system Ax = b will have (i) no solution, (ii) only one non-trivial (trivial) solution, or (iii) infinity of solutions. Our focus will be on cases (ii) and (iii). The mathematical models of many real-world problems give rise to (a) ill-conditioned linear systems, (b) singular linear systems (A is singular with all its linearly independent rows are sufficiently linearly independent), or (c) ill-conditioned singular linear systems (A is singular with some or all of its strictly linearly independent rows are near-linearly dependent). This article highlights the scope and need of a randomized algorithm for ill-conditioned/singular systems when a reasonably narrow domain of a solution vector is specified. Further, it stresses that with the increasing computing power, the importance of randomized algorithms is also increasing. It also points out that, for many optimization linear/nonlinear problems, randomized algorithms are increasingly dominating the deterministic approaches and, for some problems such as the traveling salesman problem, randomized algorithms are the only alternatives.

• 대한수학회논문집
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• 제17권2호
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• pp.349-361
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• 2002

The ill-posed elliptic wave propagation problems can be transformed into well-posed initial value problems of the reflection and transmission operators characterizing the material structure of the given model by the combination of wave field splitting and invariant imbedding methods. In general, the derived scattering operator equations are of first-order in range, nonlinear, nonlocal, and stiff and oscillatory with a subtle fixed and movable singularity structure. The phase space and path integral analysis reveals that construction and reconstruction algorithms depend crucially on a detailed symbol analysis of the scattering operators. Some information about the singularity structure of the scattering operator symbols is presented and analyzed in the transversely homogeneous limit.

• 호남수학학술지
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• 제36권4호
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• pp.831-852
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• 2014

The obstacle shape reconstruction problem has been known to be difficult to solve since it is highly nonlinear and severely ill-posed. The use of local or locally supported basis functions for the problem has been addressed for many years. However, to the authors' knowledge, any research report on the proper usage of local or locally supported basis functions for the shape reconstruction has not been appeared in the literature due to many difficulties. The aim of this paper is to introduce the general concepts and methodologies for the proper choice and their implementation of locally supported basis functions through the two-dimensional Helmholtz equation. The implementations are based on the complex nonlinear parameter estimation (CNPE) formula and its robust algorithm developed recently by the authors. The basic concepts and ideas are simple. The derivation of the necessary properties needed for the shape reconstructions are elementary. However, the capturing abilities for the local geometry of the obstacle are superior to those by conventional methods, the trial and errors, due to the proper implementation and the CNPE algorithm. Several numerical experiments are performed to show the power of the proposed method. The fundamental ideas and methodologies described in this paper can be applied to many other shape reconstruction problems.

• 대한수학회보
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• 제29권1호
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• pp.125-135
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• 1992

In various viscus flow problems it has been the custom to replace the convective derivative by the ordinary partial derivative in problems for which the data are small. In this paper we consider the Benard Convection problem with small data and compare the solution of this problem (assumed to exist) with that of the linearized system resulting from dropping the nonlinear terms in the expression for the convective derivative. The objective of the present work is to derive an estimate for the error introduced in neglecting the convective inertia terms. In fact, we derive an explicit bound for the L$_{2}$ error. Indeed, if the initial data are O(.epsilon.) where .epsilon. << 1, and the Rayleigh number is sufficiently small, we show that this error is bounded by the product of a term of O(.epsilon.$^{2}$) times a decaying exponential in time. The results of the present paper then give a justification for linearizing the Benard Convection problem. We remark that although our results are derived for classical solutions, extensions to appropriately defined weak solutions are obvious. Throughout this paper we will make use of a comma to denote partial differentiation and adopt the summation convention of summing over repeated indices (in a term of an expression) from one to three. As reference to work of continuous dependence on modelling and initial data, we mention the papers of Payne and Sather [8], Ames [2] Adelson [1], Bennett [3], Payne et al. [9], and Song [11,12,13,14]. Also, a similar analysis of a micropolar fluid problem backward in time (an ill-posed problem) was given by Payne and Straughan [10] and Payne [7].

• 지구물리와물리탐사
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• 제7권3호
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• pp.207-212
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• 2004

자기지전류(MT) 자료의 3차원 역산에 대해 소개한다. MT 자료의 역산 문제는 기본적으로 악조건이므로 유일한 해가 존재하지 않는다. 이러한 비유일성을 줄이고 정확한 역산해를 구하기 위해서는 역산 시 사전정보를 추가하는 제약조건을 가해야 한다. 물리탐사 분야에서 비선형 역산에 사용되는 가장 일반적인 방법은 일련의 선형화된 역산문제를 푸는 Gauss-Newton법이다. 이 알고리듬은 수렴 시, 모델 공간에서 역산문제에 대한 목적함수를 최소화하는 최적해를 준다. 그러나 이러한 반복적 선형화기법은 3차원 MT 역산의 경우 Jacobian 행렬을 구하기 힘들기 때문에 그 유용성에 한계가 있다. 이러한 어려움은 CG법에 의해 완화할 수 있다. 선형 CG법은 Gauss-Newton 반복의 각 단계를 근사적으로 풀기 위해서 사용된다. 한편 비선형 CG법은 목적함수의 최소화에 직접적으로 적용된다. 이들 CG법은 Jacobian 행렬의 계산 및 대형 선형방정식의 해를 반복 당 세 번의 모델링으로 대치할 수 있어서 3차원 역산에 적합하다.