[Show abstract][Hide abstract] ABSTRACT: We consider the problem of identifying a nonlinear heat transfer law at the boundary, or of the temperature-dependent 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 time-dependent 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 ill-posed because small errors in the input integral observations cause large errors in the output source. For a stable reconstruction a variational least-squares 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 07/2014; 264:82–98. DOI:10.1016/j.cam.2014.01.005 · 1.27 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We prove stability estimates of Hölder-type for Burgers-type 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.
Journal of Inverse and Ill-Posed Problems 01/2014; DOI:10.1515/jiip-2013-0050 · 0.88 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We investigate semismooth Newton and quasi-Newton 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 quasi-Newton 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.
Journal of Inverse and Ill-Posed Problems 10/2013; 21(5). DOI:10.1515/jip-2013-0031 · 0.88 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The determination of the space- or time-dependent heat transfer coefficient which links the boundary temperature to the heat flux through a third-kind Robin boundary condition in transient heat conduction is investigated. The reconstruction uses average surface temperature measurements. In both cases of the space- or time-dependent unknown heat transfer coefficient the inverse problems are nonlinear and ill posed. Least-squares 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 time-dependent ambient temperature entering a third-kind 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 time-dependent unknown ambient temperature
the inverse problems are linear and ill-posed. Least-squares 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.
IMA Journal of Applied Mathematics 01/2013; 80(1). DOI:10.1093/imamat/hxt012 · 0.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The ill-posed 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 well-posed non-local 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 1-w(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.
Journal of Inverse and Ill-Posed Problems 12/2012; 20(5). DOI:10.1515/jip-2012-0046 · 0.88 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
International Journal of Computer Mathematics 07/2012; 89(11):1-15. DOI:10.1080/00207160.2012.668891 · 0.82 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 L2-norm under relatively simple source conditions without the smallness requirement on the source functions.
Journal of Mathematical Analysis and Applications 04/2012; 388(1). DOI:10.1016/j.jmaa.2011.11.008 · 1.12 Impact Factor
[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 LL-curve 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
${{{z^\delta} \in {H^1}(\Omega), \|u-z^\delta\|_{H^1(\Omega)}\le\delta, \delta > 0}}$
. We regularize this problem by minimizing the strictly convex functional of (q, a)
$$\begin{array}{lll}\int\limits_{\Omega}\left(q| \nabla (U(q,a)-z^{\delta})|^2 + a(U(q,a)-z^{\delta})^2\right)dx\\\quad+\rho\left(\|q-q^*\|^2_{L^2(\Omega)} + \|a-a^*\|^2_{L^2(\Omega)}\right)\end{array}$$
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 ill-posed 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 variation-minimizing 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 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 two-step method for solving it. The performance of the proposed algorithm is illustrated with numerical examples.
Inverse Problems in Science and Engineering 04/2011; 19(3-3):281-307. DOI:10.1080/17415977.2011.551829 · 0.87 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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 regularization method for the ill-posed backward parabolic equation with time-dependent 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 time-dependent 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 infra-red 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 ill-posedness 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 gradient-based 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 second-order 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.
Journal of Algorithms & Computational Technology 03/2010; 4(1):89-120. DOI:10.1260/1748-3018.4.1.89
[Show abstract][Hide abstract] ABSTRACT: In this paper we consider a multi-dimensional inverse heat conduction problem with time-dependent coefficients in a box, which is well-known to be severely ill-posed, 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 ill-posed 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 easy-to-solve one-dimensional problems. The numerical results provided by our method are very good and the techniques seem to be very promising.
Journal of Physics Conference Series 10/2009; 232(2):361-377. DOI:10.1016/j.cam.2009.06.037