Hanoi Institute of Mathematics

Hanoi, Vietnam

Hanoi Institute of Mathematics

Hanoi, Vietnam
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Lee G.M.,Pukyong National University | Sach P.H.,Hanoi Institute of Mathematics
Journal of Global Optimization | Year: 2010

In this paper, we give sufficient conditions for the upper semicontinuity property of the solution mapping of a parametric generalized vector quasiequilibrium problem with mixed relations and moving cones. The main result is proven under the assumption that moving cones have local openness/local closedness properties and set-valued maps are cone-semicontinuous in a sense weaker than the usual sense of semicontinuity. The nonemptiness and the compactness of the solution set are also investigated. © Springer Science+Business Media, LLC. 2009.


Hao D.N.,Hanoi Institute of Mathematics | Van Duc N.,Vinh University
Inverse Problems | Year: 2011

Let H be a Hilbert space with the norm ∥ ∥ and A(t) (0 ≤ t ≤ T ) be positive self-adjoint unbounded operators from D(A(t)) ⊂ H to H. In the paper, we establish stability estimates of Hölder type and propose a regularizationmethod for the ill-posed-backward parabolic equation with time-dependent coefficients {ut + A(t)u = 0, 0 < t < T{ ∥u(T ) - f ∥ ≤ ε, f ε H, ε > 0. Our stability estimates improve the related results by Krein (1957 Dokl. Akad. Nauk SSSR 114 1162-5), and Agmon and Nirenberg (1963 Commun. Pure Appl. Math. 16 121-239). Our regularization method with a priori and a posteriori parameter choice yields error estimates of Hölder type. This is the only result when a regularization method for backward parabolic equations with time-dependent coefficients provides a convergence rate. © 2011 IOP Publishing Ltd.


Ho D.N.,Hanoi Institute of Mathematics | Thanh P.X.,Hanoi University of Science and Technology | Lesnic D.,University of Leeds | Johansson B.T.,University of Birmingham
International Journal of Computer Mathematics | Year: 2012

In this paper, we investigate a variational method for a multi-dimensional inverse heat conduction problem in Lipschitz domains. We regularize the problem by using the boundary element method coupled with the conjugate gradient method. We prove the convergence of this scheme with and without Tikhonov regularization. Numerical examples are given to show the efficiency of the scheme. © 2012 Taylor & Francis.


Sach P.H.,Hanoi Institute of Mathematics | Minh N.B.,Hanoi University
Journal of Global Optimization | Year: 2013

This paper gives sufficient conditions for the continuity of the solution mappings of parametric non-weak vector Ky Fan inequality problems with moving cones. The main results of the paper are new and are obtained under an assumption different from the known density hypothesis. They are written in terms of nonlinear scalarization functions associated to the data of the problems under consideration. Verifiable conditions are given, and examples are provided. © 2012 Springer Science+Business Media New York.


Doan T.S.,TU Dresden | Doan T.S.,Hanoi Institute of Mathematics | Kalauch A.,TU Dresden | Siegmund S.,TU Dresden | Wirth F.R.,University of Würzburg
Systems and Control Letters | Year: 2010

We deal with dynamic equations on time scales, where we characterize the positivity of a system. Uniform exponential stability of a system is determined by the spectrum of its matrix. We investigate the corresponding stability radii with respect to structured perturbations and show that, for positive systems, the complex and the real stability radius coincide. © 2010 Elsevier B.V. All rights reserved.


Thanh N.T.,Austrian Academy of Sciences | Sahli H.,Vrije Universiteit Brussel | Hao D.N.,Hanoi Institute of Mathematics
Inverse Problems in Science and Engineering | Year: 2011

The application of infrared (IR) thermography to the detection and characterization of buried landmines (more generally, buried objects) is introduced. The problem is aimed at detecting the presence of objects buried under the ground and characterizing them by estimating their thermal and geometrical properties using IR measurements on the soil surface. Mathematically, this problem can be stated as an inverse problem for reconstructing a piecewise constant coefficient of a three-dimensional heat equation in a parallelepiped from only one measurement taken at one plane of its boundary (the air-soil interface). Due to the lack of spatial information in the observed data, this problem is extremely ill-posed. In order to reduce its ill-posedness, keeping in mind the application of detecting buried landmines we make use of some simplification steps and propose a twostep method for solving it. The performance of the proposed algorithm is illustrated with numerical examples. © 2011 Taylor & Francis.


Hao D.N.,Hanoi Institute of Mathematics | Quyen T.N.T.,The University of Da nang
Inverse Problems | Year: 2011

We investigate the convergence rates for total variation regularization of the problem of identifying (i) the coefficient q in the Neumann problem for the elliptic equation -div(q∇u) = f in Ω, q∂u/∂n = g on Ω and (ii) the coefficient a in the Neumann problem for the elliptic equation , when u is imprecisely given by -Δu + au = f in Ω, q∂u/∂n = g on ∂Ω, when u is imprecisely given by zδ in Ω ∈ ℝ d, d ≥ 1. We regularize these problems by correspondingly minimizing the convex functionals 1/2 ∫ Ωq|∇(U(q)-z δ)| 2dx + ρ ∫ Ω|∇a| and 1/2 ∫ Ωq|∇(U(a)- z δ)| 2dx + 1/2 ∫ ω a(U(a))-z δ) 2dx + ρ ∫ Ω|∇a| over the admissible sets, where U(q) (U(a)) is the solution of the first (second) Neumann boundary value problem; ρ > 0 is the regularization parameter. Taking the solutions of these optimization problems as the regularized solutions to the corresponding identification problems, we obtain the convergence rates of them to a total variation-minimizing solution in the sense of the Bregman distance under relatively simple source conditions without the smallness requirement on the source functions. © 2011 IOP Publishing Ltd.


Hao D.N.,Hanoi Institute of Mathematics | Oanh N.T.N.,Thai Nguyen University
Inverse Problems in Science and Engineering | Year: 2016

We study the problem of determining the initial condition in parabolic equations with time-dependent coefficients from integral observations which can be regarded as generalizations of point-wise interior observations. Our approach is new in the sense that for determining the initial condition we do not assume that the data available in the whole space domain at the final moment or in a subset of the space domain during a certain time interval, but some integral observations during a time interval. We propose a variational method in combination with Tikhonov regularization for solving the problem and then discretize it by finite difference splitting methods. The discretized minimization problem is solved by the conjugate gradient method and tested on computer to show its efficiency. Also as a by-product of the variational method, we propose a numerical scheme for estimating the degree of ill-posedness of the problem. © 2016 Informa UK Limited, trading as Taylor & Francis Group


Hao D.N.,Hanoi Institute of Mathematics | Quyen T.N.T.,The University of Da nang
Inverse Problems | Year: 2010

We investigate the convergence rates for Tikhonov regularization of the problem of identifying (1) the coefficient q ε L ∞ (Ω) in the Dirichlet problem -div(qδu) = f in Ω, u = 0 on δω, and (2) the coefficient a ε L ∞ (Ω) in the Dirichlet problem ?Δu + au = f in Ω, u = 0 on δΩ, when u is imprecisely given by zδ ε H10 (Ω), ∥u-zδ∥ H1(Ω) ≤ δ, Ω ⊂ ℝd, d ≥ 1.We regularize these problems by correspondingly minimizing the strictly convex functionals Mathematical Equation Represented, where U(q) (U(a)) is the solution of the first (second) Dirichlet problem, ρ > 0 is the regularization parameter and q* (or a*) is an a priori estimate of q(or ). We prove that these functionals attain a unique global minimizer on the admissible sets. Further, we give very simple source conditions without the smallness requirement on the source functions which provide the convergence rate O( √ δ) for the regularized solutions. © 2010 IOP Publishing Ltd.


Hao D.N.,University of Leeds | Chuong L.H.,Hanoi Institute of Mathematics | Lesnic D.,University of Leeds
Computers and Mathematics with Applications | Year: 2012

In this paper, we use smoothing splines to deal with numerical differentiation. Some heuristic methods for choosing regularization parameters are proposed, including the L-curve method and the de Boor method. Numerical experiments are performed to illustrate the efficiency of these methods in comparison with other procedures. © 2011 Elsevier Ltd. All rights reserved.

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