Publications (73)64.33 Total impact
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ABSTRACT: Descent gradient methods are the most frequently used algorithms for computing regularizers of inverse problems. They are either directly applied to the discrepancy term, which measures the difference between operator evaluation and data or to a regularized version incorporating suitable penalty terms. In its basic form, gradient descent methods converge slowly. We aim at extending different optimization schemes, which have been recently introduced for accelerating these approaches, by addressing more general penalty terms. In particular we use a general setting in infinite Hilbert spaces and examine accelerated algorithms for regularization methods using total variation or sparsity constraints. To illustrate the efficiency of these algorithms, we apply them to a parameter identification problem in an elliptic partial differential equation using total variation regularization.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the problem of identifying a nonlinear heat transfer law at the boundary, or of the temperaturedependent heat transfer coefficient in a parabolic equation from boundary observations. As a practical example, this model applies to the heat transfer coefficient that describes the intensity of heat exchange between a hot wire and the cooling water in which it is placed. We reformulate the inverse problem as a variational one which aims to minimize a misfit functional and prove that it has a solution. We provide a gradient formula for the misfit functional and then use some iterative methods for solving the variational problem. Thorough investigations are made with respect to several initial guesses and amounts of noise in the input data. Numerical results show that the methods are robust, stable and accurate.  [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.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
ABSTRACT: We prove stability estimates of Höldertype for Burgerstype equations ut = (a(x,t)ux)x  d(x,t)uux + f(x,t), (x,t) ∈ (0,1)×(0,T), u(0,t) = g0(t), u(1,t) = g1(t), 0 ≤ t ≤ T, backward in time, with a(x,t), d(x,t), g0(t), g1(t), f(x,t) being smooth functions, under relatively weak conditions on the solutions.  [Show abstract] [Hide abstract]
ABSTRACT: We investigate semismooth Newton and quasiNewton methods for minimization problems arising from weighted ℓ 1 regularization. We give proofs of the local convergence of these methods and show how their interpretation as active set methods leads to the development of efficient numerical implementations of these algorithms. We also propose and analyze Broyden updates for the semismooth quasiNewton method. The efficiency of these methods is analyzed and compared with standard implementations. The paper concludes with some numerical examples that include both linear and nonlinear operator equations.  [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.  [Show abstract] [Hide abstract]
ABSTRACT: The restoration of the space or timedependent ambient temperature entering a thirdkind convective Robin boundary condition in transient heat conduction is investigated. The temperature inside the solution domain together with the ambient temperature are determined from additional boundary measurements. In both cases of the space or timedependent unknown ambient temperature the inverse problems are linear and illposed. Leastsquares penalized variational formulations are proposed and new formulae for the gradients are derived. Numerical results obtained using the conjugate gradient method combined with a boundary element direct solver are presented and discussed.  [Show abstract] [Hide abstract]
ABSTRACT: The illposed backward parabolic equation u t +Au=0,0<t<T,∥u(T)f∥⩽ε with a densely defined linear operator A such that A generates an analytic semigroup {S(t)} t⩾0 in a Banach space X and ε>0 being given is stabilized by the Tikhonov regularization method and by the wellposed nonlocal boundary value problem v αt +Av α =0,0<t<T,αv α (0)+v α (T)=f,α>0 A priori and a posteriori parameter choice rules for these regularization methods are suggested which yield the error estimate ∥u(t)v α (t)∥⩽cε w(t) E 1w(t) for all t∈[0,T], where c, k are computable constants, E is a bound for ∥u(0)∥ and w(τ) is a defined harmonic function.  [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.  [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.  [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.  [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.  [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.  [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.  [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.  [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.  [Show abstract] [Hide abstract]
ABSTRACT: This paper introduces a mathematical formulation of the problem of detection and characterization of shallowly buried landmines (more generally, buried objects) using the passive thermal infrared technique. The problem consists of two steps. In the first step, referred to as thermal modeling which aims at predicting the soil temperature provided by the thermal properties of the soil and the buried objects, a parabolic partial differential equation based model is formulated. The proposed model is validated using experimental data. For solving the model, a splitting finite difference scheme is used. In the second step, referred to as inverse problem setting for landmine detection, the forward thermal model and acquired infrared images are used to detect the presence of buried objects and to characterize them based on the estimation of their thermal and geometrical properties. Mathematically, this inverse problem is stated as the estimation of a piecewise constant coefficient of the heat transfer equation. To reduce the illposedness of this problem, which is due to the lack of spatial information in the measured data, we make use of a parametrization of the coefficient which needs only a small number of unknowns. The problem is then solved by gradientbased optimization methods. Numerical results both validate the proposed thermal model and illustrate the performance of the suggested algorithm for the inverse problem.  [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.  [Show abstract] [Hide abstract]
ABSTRACT: A Cauchy problem for general elliptic secondorder linear partial differential equations in which the Dirichlet data in H 1/2 (Γ 1 ∪Γ 3 ) is assumed available on a larger part of the boundary Γ of the bounded domain Ω than the boundary portion Γ 1 on which the Neumann data is prescribed, is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.
Publication Stats
732  Citations  
64.33  Total Impact Points  
Top Journals
Institutions

20092014

Institute of Mathematics, Vietnam
Hà Nội, Ha Nội, Vietnam


20022014

University of Leeds
 Department of Applied Mathematics
Leeds, England, United Kingdom


20052009

Free University of Brussels
 Electronics and Informatics (ETRO)
Bruxelles, Brussels Capital Region, Belgium


2003

Institut National des Sciences Appliquées de Rouen
Rouen, Upper Normandy, France


19941998

Universität Siegen
 Department of Mathematics
Siegen, North RhineWestphalia, Germany


19911995

Freie Universität Berlin
 Institute of Mathematics
Berlín, Berlin, Germany
