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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. Source

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 Wurzburg
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. Source

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. Source

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. Source

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. Source

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