Publications (52)42.59 Total impact
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ABSTRACT: A novel inverse problem which consists of the simultaneous determination of a source together with the temperature in the heat equation from integral observations is investigated. These integral observations are weighted averages of the temperature over the space domain and over the time interval. The heat source is sought in the form of a sum of two space and timedependent unknown components in order to ensure the uniqueness of a solution. The local existence and uniqueness of the solution in classical Hölder spaces are proved. The inverse problem is linear, but it is illposed because small errors in the input integral observations cause large errors in the output source. For a stable reconstruction a variational leastsquares method with or without penalization is employed. The gradient of the functional which is minimized is calculated explicitly and the conjugate gradient method is applied. Numerical results obtained for several benchmark test examples show accurate and stable numerical reconstructions of the heat source.Journal of Computational and Applied Mathematics 01/2014; 264:82–98. · 0.99 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We study the problem of identifying the spatially varying diffusion coefficient [Inline formula] in the boundary value problems for the elliptic equation [Inline formula] in [Inline formula], [Inline formula] on [Inline formula] and [Inline formula] on [Inline formula], [Inline formula], [Inline formula], when the solution [Inline formula] is imprecisely given by [Inline formula] with [Inline formula] and [Inline formula]. The finite element method is applied to a convex energy functional with Tikhonov regularization for solving this coefficient identification problem. We show the [Inline formula]convergence of finite element solutions to the unique minimum norm solution of the identification problem. Furthermore, convergence rates of the method are established under certain source conditions.Applicable Analysis 01/2014; 93(7). · 0.71 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The determination of the space or timedependent heat transfer coefficient which links the boundary temperature to the heat flux through a thirdkind Robin boundary condition in transient heat conduction is investigated. The reconstruction uses average surface temperature measurements. In both cases of the space or timedependent unknown heat transfer coefficient the inverse problems are nonlinear and ill posed. Leastsquares penalized variational formulations are proposed and new formulae for the gradients are derived. Numerical results obtained using the nonlinear conjugate gradient method combined with a boundary element direct solver are presented and discussed.Inverse Problems 09/2013; 29(9):095020. · 1.90 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: 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, (ii) the coefficient a in the Neumann problem for the elliptic equation −Δu+au=f in Ω, ∂u/∂n=g on ∂Ω, Ω⊂Rd, d⩾1, when u is imprecisely given by zδ∈H1(Ω), ‖u−zδ‖H1(Ω)⩽δ, δ>0. We regularize these problems by correspondingly minimizing the strictly convex functionals12∫Ωq∇(U(q)−zδ)2dx+ρ(12‖q‖L2(Ω)2+∫Ω∇q), and12∫Ω∇(U(a)−zδ)2dx+12∫Ωa(U(a)−zδ)2dx+ρ(12‖a‖L2(Ω)2+∫Ω∇a) over 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 the solution of the inverse problem in the sense of the Bregman distance and in the L2norm under relatively simple source conditions without the smallness requirement on the source functions.Journal of Mathematical Analysis and Applications  J MATH ANAL APPL. 04/2012;  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we use smoothing splines to deal with numerical differentiation. Some heuristic methods for choosing regularization parameters are proposed, including the LLcurve method and the de Boor method. Numerical experiments are performed to illustrate the efficiency of these methods in comparison with other procedures.Computers & Mathematics with Applications. 01/2012; 63:816826.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we investigate a variational method for a multidimensional 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.International Journal of Computer Mathematics  IJCM. 01/2012;  [Show abstract] [Hide abstract]
ABSTRACT: We investigate the convergence rates for Tikhonov regularization of the problem of simultaneously estimating the coefficients q and a in the Neumann problem for the elliptic equation ${{{\rm div}(q \nabla u) + au = f \;{\rm in}\; \Omega, q{\partial u}/{\partial n} = g}}$ on the boundary ${{\partial\Omega, \Omega \subset \mathbb{R}^d, d \geq 1}}$ , when u is imprecisely given by ${{{z^\delta} \in {H^1}(\Omega), \uz^\delta\_{H^1(\Omega)}\le\delta, \delta > 0}}$ . We regularize this problem by minimizing the strictly convex functional of (q, a) $$\begin{array}{lll}\int\limits_{\Omega}\left(q \nabla (U(q,a)z^{\delta})^2 + a(U(q,a)z^{\delta})^2\right)dx\\\quad+\rho\left(\qq^*\^2_{L^2(\Omega)} + \aa^*\^2_{L^2(\Omega)}\right)\end{array}$$ over the admissible set K, where ρ > 0 is the regularization parameter and (q*, a*) is an a priori estimate of the true pair (q, a) which is identified, and consider the unique solution of these minimization problem as the regularized one to that of the inverse problem. We obtain the convergence rate ${{{\mathcal {O}}(\sqrt{\delta})}}$ , as δ → 0 and ρ ~ δ, for the regularized solutions under the simple and weak source condition $${\rm there\;is\;a\;function}\;w^* \in V^*\;{\rm such\;that}\;{U^\prime (q^ \dagger, a^\dagger)}^*w^* = (q^\dagger  q^*, a^\dagger  a^*)$$ with ${{(q^\dagger, a^\dagger)}}$ being the (q*, a*)minimum norm solution of the coefficient identification problem, U′(·, ·) the Fréchet derivative of U(·, ·), V the Sobolev space on which the boundary value problem is considered. Our source condition is without the smallness requirement on the source function which is popularized in the theory of regularization of nonlinear illposed problems. Furthermore, some concrete cases of our source condition are proved to be simply the requirement that the sought coefficients belong to certain smooth function spaces.Numerische Mathematik 01/2012; 120(1). · 1.33 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: 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 , and (ii) the coefficient a in the Neumann problem for the elliptic equation , when u is imprecisely given by zδ in . We regularize these problems by correspondingly minimizing the convex functionals and 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 variationminimizing solution in the sense of the Bregman distance under relatively simple source conditions without the smallness requirement on the source functions.Inverse Problems 06/2011; 27(7):075008. · 1.90 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: 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 threedimensional heat equation in a parallelepiped from only one measurement taken at one plane of its boundary (the airsoil interface). Due to the lack of spatial information in the observed data, this problem is extremely illposed. In order to reduce its illposedness, 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.Inverse Problems in Science and Engineering 04/2011; 19(3):281307. · 0.75 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Let H be a Hilbert space with the norm   and A(t) (0 ≤ t ≤ T) be positive selfadjoint unbounded operators from D(A(t))⊂H to H. In the paper, we establish stability estimates of Hölder type and propose a regularization method for the illposed backward parabolic equation with timedependent coefficients 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 timedependent coefficients provides a convergence rate.Inverse Problems 01/2011; 27(2):025003. · 1.90 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: This paper investigates the problem of detection and characterization of shallowly buried landmines (more generally, buried objects) using passive thermal infrared technique. The problem consists of two steps. The first step aims at predicting the evolution of the soil temperature given the thermal properties of the soil and the buried objects using a physical model. In the second step, the forward thermal model and acquired infrared images are used to detect the presence of buried objects and characterize them based on the estimation of their thermal and geometrical properties.01/2011;  [Show abstract] [Hide abstract]
ABSTRACT: We investigate the convergence rates for Tikhonov regularization of the problem of identifying (1) the coefficient q L fty(Ω) in the Dirichlet problem −div(q∇u) = f in Ω, u = 0 on ∂Ω, and (2) the coefficient a L fty(Ω) in the Dirichlet problem −Δu + au = f in Ω, u = 0 on ∂Ω, when u is imprecisely given by zδ H10(Ω), , We regularize these problems by correspondingly minimizing the strictly convex functionals and 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 a). 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 for the regularized solutions.Inverse Problems 11/2010; 26(12):125014. · 1.90 Impact Factor  Journal of Algorithms & Computational Technology 01/2010; 4(1):89120.
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ABSTRACT: Let H be a Hilbert space with norm  , A:D(A) ⊂ H → H a positive definite, selfadjoint operator with compact inverse on H, and T and given positive numbers. The illposed Cauchy problem for elliptic equations is regularized by the wellposed nonlocal boundary value problem with a ≥ 1 being given and α > 0 the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield orderoptimal regularization methods. Numerical results based on the boundary element method are presented and discussed.Inverse Problems 02/2009; 25(5):055002. · 1.90 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we consider a multidimensional inverse heat conduction problem with timedependent coefficients in a box, which is wellknown to be severely illposed, by a variational method. The gradient of the functional to be minimized is obtained by the aid of an adjoint problem, and the conjugate gradient method with a stopping rule is then applied to this illposed optimization problem. To enhance the stability and the accuracy of the numerical solution to the problem, we apply this scheme to the discretized inverse problem rather than to the continuous one. The difficulties with large dimensions of discretized problems are overcome by a splitting method which only requires the solution of easytosolve onedimensional problems. The numerical results provided by our method are very good and the techniques seem to be very promising.Journal of Physics Conference Series 01/2009; 232:361377.  [Show abstract] [Hide abstract]
ABSTRACT: This paper deals with an inverse problem arising in infrared (IR) thermography for buried landmine detection. It is aimed at using a thermal model and measured IR images to detect the presence of buried objects and characterize them in terms of thermal and geometrical properties. The inverse problem is mathematically stated as an optimization one using the wellknown leastsquare approach. The main difficulty in solving this problem comes from the fact that it is severely ill posed due to lack of information in measured data. A twostep algorithm is proposed for solving it. The performance of the algorithm is illustrated using some simulated and real experimental data. The sensitivity of the proposed algorithm to various factors is analyzed. A data processing chain including anomaly detection and characterization is also introduced and discussed.IEEE Transactions on Geoscience and Remote Sensing 01/2009; · 3.47 Impact Factor 
Conference Paper: A Nonlinear Probabilistic Curvature Motion Filter for Positron Emission Tomography Images.
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ABSTRACT: Positron Emission Tomography (PET) is an important nuclear medicine imaging technique which enhances the effectiveness of diagnosing many diseases. The rawprojection data, i.e. the sinogram, from which the PET is reconstructed, contains a very high level of Poisson noise. The latter complicates the PET image’s interpretation which may lead to erroneous diagnoses. Suitable denoising techniques prior to reconstruction can significantly alleviate the problem. In this paper, we propose filtering the sinogram with a constraint curvature motion diffusion for which we compute the edge stopping function in terms of edge probability under the assumption of contamination by Poison noise. We demonstrate through simulations with images contaminated by Poisson noise that the performance of the proposed method substantially surpasses that of recently published methods, both visually and in terms of statistical measures.Scale Space and Variational Methods in Computer Vision, Second International Conference, SSVM 2009, Voss, Norway, June 15, 2009. Proceedings; 01/2009  [Show abstract] [Hide abstract]
ABSTRACT: For the heat equation backward in timeut=uxx,x∈R,t∈(0,T),‖u(⋅,T)−φ(⋅)‖Lp(R)⩽ϵ subject to the constraint‖u(⋅,0)‖Lp(R)⩽E with T>0, φ∈Lp(R), 0ϵE, 1p∞ being given, we prove that if u1 and u2 are two solutions of the problem, then there is a constant c>0 such that‖u1(⋅,t)−u2(⋅,t)‖Lp(R)⩽cϵt/TE1−t/T,∀t∈[0,T]. In case p=2 we establish stability estimates of Hölder type for all derivatives with respect to x and t of the solutions. We suggest a useful strategy of choosing mollification parameters which provides a continuity at t=0 when an additional condition on the smoothness of u(x,0) is given. Furthermore, we propose a stable marching difference scheme for this illposed problem and test several related numerical methods for it.Journal of Mathematical Analysis and Applications  J MATH ANAL APPL. 01/2009; 353(2):627641.  [Show abstract] [Hide abstract]
ABSTRACT: This chapter introduces the application of infrared (IR) thermography in land mine detection. IR thermography in general and for remotely detecting buried land mines in particular, seems to be a promising diagnostic tool. Due to the difference in thermophysical properties of mines and the soil (mines retain or release heat at a rate different from the soil), soilsurface thermal contrasts above the mines are formed. These contrasts are captured by IR cameras to show the changes in temperature over the mines, which can be used for detecting them. Clearly, the degree of success of such detection technology depends on the factors that affect the formation of the thermal contrasts (signatures), such as the depth of burial; soil properties and attributes, including mine properties (size); as well as the time of day during which the measurement is carried out. Another important factor that strongly influences the viability of IR detection method is the rate of false alarms. Indeed, IR sensors may detect any thermally transmitting objects, not only land mines. It is therefore necessary to develop parameter estimation and decisionmaking tools that enable the IR technology to distinguish signals resulting from a land mine and unrelated clutter signals. This chapter consists of four sections. The first section introduces the physical principles of IR thermography and gives an overview of its literature. The flowchart of the technique is also given in this section. In the second one, we summarize a thermal model of the soil with the presence of shallowly buried objects. This model is used for studying the influence of soil and land mine properties on the temperature distribution of the soil, especially on its surface. The third section aims at detecting possible anomalies in the soil using IR images and classifying them as mines or nonmine objects. The classification is based on the estimation of the thermal and geometric properties of the detected anomalies. In the fourth section, the performance, in terms of probability of detection and false alarm rates, of the proposed approach for an experimental data set is presented. The processing chain of IR thermography, including data acquisition, data preprocessing, anomaly detection, and estimation of thermal and geometric properties of the detected anomalies, is presented and illustrated using an experimental data set measured in an outdoor minefield. Finally, conclusions on the statistical reliability of the IR technique are drawn.12/2008: pages 336;  [Show abstract] [Hide abstract]
ABSTRACT: A onedimensional inverse problem arising in infrared thermography for the detection and characterization of buried objects is introduced. Mathematically, the problem is to reconstruct a piecewise constant coefficient of a scalar heat equation in a finite rod from measurements taken at one of its extremities. The problem is posed in the well known leastsquares setting and solved by a quasiNewton method. The contributions of this article include: (i) the parameterization of a piecewise constant function by a small number of unknown parameters which represent its constant values and locations of discontinuities; (ii) the application of the adjoint field technique in the calculation of the gradient of a discretized objective function and (iii) the application of the considered inverse problem in the detection and characterization of buried objects. Numerical results illustrate the good performance of the proposed algorithm.Inverse Problems in Science and Engineering 10/2008; 16(7):903925. · 0.75 Impact Factor
Publication Stats
239  Citations  
42.59  Total Impact Points  
Top Journals
Institutions

2005–2009

Free University of Brussels
 Electronics and Informatics (ETRO)
Brussels, BRU, Belgium


1994–1998

Universität Siegen
 Department of Mathematics
Siegen, North RhineWestphalia, Germany


1991–1992

Freie Universität Berlin
 Institute of Mathematics
Berlín, Berlin, Germany
