Alemdar Hasanov

An analysis of inverse source problems with final time measured output data for the heat conduction equation: A semigroup approach

Abstract This paper presents a semigroup approach for inverse source problems for the abstract heat equation u t = A u + F , when the measured output data is given in the form the final overdetermination u T ( x ) ≔ u ( x , T ) . A representation formula for a solution of the inverse source problem is proposed. This representation shows a non-uniqueness structure of the inverse problem solution, and also permits one to derive a sufficient condition for uniqueness. Some examples related to identifying the unknown spacewise and time-dependent heat sources f ( x ) and h ( t ) of the heat equation u t = u x x + f ( x ) h ( t ) , from the final overdetermination or from a single point time measurement are presented.

An inverse source problem with single Dirichlet type measured output data for a linear parabolic equation

Abstract The problem of determining an unknown source term in a linear parabolic equation u t = ( k ( x ) u x ) x + F ( x , t ) , ( x , t ) ∈ Ω T , from the Dirichlet type measured output data h ( t ) : = u ( 0 , t ) is studied. A formula for the Frechet gradient of the cost functional J ( F ) = ‖ u ( 0 , t ; F ) − h ( t ) ‖ 2 is derived via the solution of the corresponding adjoint problem, within the weak solution theory for PDEs and the quasi-solution approach. The Lipschitz continuity of the gradient is proved. Based on the obtained results the convergence theorem for the gradient method is proposed.

Inverse heat conduction problems with boundary and final time measured output data

This article presents a systematic study of inverse problems of identifying the unknown source term F(x, t) of the heat conduction (or linear parabolic) equation u t  = (k(x)u x ) x  + F(x, t) from measured output data in the form of Dirichlet h(t) ≔ u(0, t), Neumann f(t) ≔ −k(0)u x (0, t) types of boundary conditions, also in the form the final time overdetermination u T (x) ≔ u(x, T). In the first part of this article the adjoint problem approach is used to derive formulas for the Fréchet gradient of cost functionals via solutions of the corresponding adjoint problems. It is proved that all these gradients are Lipschitz continuous. A necessary conditions for unicity and hence distinguishablity of solutions of all the three types of inverse source problems are derived. In the second part of this article the semigroup theory is used to obtain a general representation of a solution of the inverse source problem for the abstract evolution equation u t  = Au + F with final data overdetermination. This representation shows non-uniqueness structure of the inverse problem solution, and also permits one to derive a sufficient condition for unicity.

Identification of an unknown source term in a vibrating cantilevered beam from final overdetermination

Inverse problems of determining the unknown source term F(x, t) in the cantilevered beam equation utt = (EI(x)uxx)xx + F(x, t) from the measured data μ(x) := u(x, T) or ν(x) := ut(x, T) at the final time t = T are considered. In view of weak solution approach, explicit formulae for the Frechet gradients of the cost functionals J1(F) = ||u(x, T; w) − μ(x)||20 and J2(F) = ||ut(x, T; w) − ν(x)||20 are derived via the solutions of corresponding adjoint (backward beam) problems. The Lipschitz continuity of the gradients is proved. Based on these results the gradient-type monotone iteration process is constructed. Uniqueness and ill-conditionedness of the considered inverse problems are analyzed.

Semi-analytic inversion method for the determination of elastoplastic properties of power hardening materials from limited torsional experiment

An inverse problem related to the determination of elastoplastic properties of a beam is considered within J 2 deformation theory. A new fast algorithm is proposed for the identification of elastoplastic properties of engineering materials. This algorithm is based on finding the three main parameters of the unknown curve g(ξ2) (plasticity function), namely, the elasticity limit , the modulus of rigidity G and the strain hardening exponent κ. The advantage of this algorithm is that only two values of measured output data are required for the reconstruction of the unknown function g(ξ2). Note that all parameterization algorithms use numerous torsional experiments as necessary data for the determination of only some part of the unknown stress–strain curve. Numerical examples related to applicability and high accuracy of the proposed approach are presented for the cases of noise free and noise data.

Comparative analysis of inverse coefficient problems for parabolic equations. Part I: adjoint problem approach

This article presents a mathematical and computational analysis of the adjoint problem approach for parabolic inverse coefficient (or inverse heat conduction) problems based on boundary measured data. In Part I the mathematical analysis is given for three classes of typical inverse coefficient problems with various Neumann or/and Dirichlet types of measured output data. Although all three types of considered inverse coefficient problems are severely ill-posed, comparative numerical analysis show that the ill-posedness depends also on where the Neumann and Dirichlet conditions are given: in the direct problem or as an output data. For all these types of inverse problems the integral identities relating solutions of direct problems and appropriate adjoint problems solutions are derived. These integral identities permit proof of monotonicity, Lipschitz continuity, and hence invertibility of the corresponding input–output mappings. Based on these results solvability of all three types of inverse coefficient problems are proved. The degree of ill-posedness of inverse problems is demonstrated on numerical test examples.

Identification of unknown temporal and spatial load distributions in a vibrating Euler-Bernoulli beam from Dirichlet boundary measured data

Source identification problems in a system governed by Euler-Bernoulli beam equation ź ( x ) u t t + ( r ( x ) u x x ) x x = F ( x ) H ( t ) , ( x , t ) ź ( 0 , l ) × ( 0 , T ) , from available boundary observation (measured data), namely, from measured slope ź ( t ) : = u x ( 0 , t ) at x = 0 , are considered. We propose a new approach to identifying the unknown temporal ( H ( t ) ) and spatial ( F ( x ) ) loads. This novel approach is based on weak solution theory for PDEs and quasi-solution method for inverse problems combined with the adjoint method. It allows to construct not only a mathematical theory of inverse source problems for Euler-Bernoulli beam, but also an effective numerical algorithm for reconstruction of unknown loads. Introducing the input-output operators ( ź H ) ( t ) : = u x ( 0 , t ; H ) and ( Ψ F ) ( t ) : = u x ( 0 , t ; F ) , t ź ( 0 , T ) , we show that both operators are compact. Based on this result and general regularization theory, we prove an existence of unique solutions of the regularized normal equations ( ź ź ź + α I ) H α = ź ź ź and ( Ψ ź Ψ + α I ) F α = Ψ ź ź . Then we develop the adjoint problem approach to prove Frechet differentiability of the corresponding cost functionals and Lipschitz continuity of the Frechet gradients. Derived explicit gradient formulas via the adjoint problem solution and known load, allow use of gradient type convergent iterative algorithms. Results of numerical simulations for benchmark problems illustrate robustness and high accuracy of the algorithm based on the proposed approach.

Identification of unknown spatial load distributions in a vibrating Euler–Bernoulli beam from limited measured data

Two types of inverse source problems of identifying asynchronously distributed spatial loads governed by the Euler–Bernoulli beam equation , with hinged–clamped ends ( ), are studied. Here are linearly independent functions, describing an asynchronous temporal loading, and are the spatial load distributions. In the first identification problem the values , of the deflection , are assumed to be known, as measured output data, in a neighbourhood of the finite set of points , corresponding to the internal points of a continuous beam, for all . In the second identification problem the values , of the slope , are assumed to be known, as measured output data in a neighbourhood of the same set of points P for all . These inverse source problems will be defined subsequently as the problems ISP1 and ISP2. The general purpose of this study is to develop mathematical concepts and tools that are capable of providing effective numerical algorithms for the numerical solution of the considered class of inverse problems. Note that both measured output data and contain random noise. In the first part of the study we prove that each measured output data and can uniquely determine the unknown functions . In the second part of the study we will introduce the input–output operators , and , , corresponding to the problems ISP1 and ISP2, and then reformulate these problems as the operator equations: and , where and . Since both measured output data contain random noise, we use the most prominent regularisation method, Tikhonov regularisation, introducing the regularised cost functionals and . Using a priori estimates for the weak solution of the direct problem and the Tikhonov regularisation method combined with the adjoint problem approach, we prove that the Frechet gradients and of both cost functionals can explicitly be derived via the corresponding weak solutions of adjoint problems and the known temporal loads . Moreover, we show that these gradients are Lipschitz continuous, which allows the use of gradient type iteration convergent algorithms. Two applications of the proposed theory are presented. It is shown that solvability results for inverse source problems related to the synchronous loading case, with a single interior measured data, are special cases of the obtained results for asynchronously distributed spatial load cases.

Identification of an unknown spatial load distribution in a vibrating cantilevered beam from final overdetermination

Abstract The inverse problem of determining the unknown spatial load distribution F(x) in the cantilever beam equation m(x)utt = -(EI(x)uxx)xx + F(x)H(t), with arbitrary but separable source term, from the measured data uT(x) := u(x,T), x ∈ (0,l), at the final time T > 0 is considered. Some a priori estimates of the weak solution u ∈ H°2,1(ΩT) of the forward problem are obtained. Introducing the input-output map, it is proved that this map is a compact operator. The adjoint problem approach is then used to derive an explicit gradient formula for the Fréchet derivative of the cost functional J(F) = ∥ u(·,T;F) - uT(·) ∥L2(0,l)2. The Lipschitz continuity of the gradient is proved. The collocation algorithm combined with the truncated singular value decomposition (TSVD) is used to estimate the degree of ill-posedness of the considered inverse source problem. The conjugate gradient algorithm (CGA), based on the explicit gradient formula, is proposed for numerical solution of the inverse problem. The algorithm is examined through numerical examples related to reconstruction of various spatial loading distributions F(x). The numerical results illustrate bounds of applicability of proposed algorithm, also its efficiency and accuracy.

Comparative analysis of inverse coefficient problems for parabolic equations. Part II: Coarse-fine grid algorithm

This article presents a computational analysis of the adjoint problem approach for parabolic inverse coefficient problems based on boundary measured data. The proposed coarse-fine grid algorithm constructed on the basis of this approach is an effective computational tool for the numerical solution of inverse coefficient problems with various Neumann or/and Dirichlet type measured output data. In the previous Part I paper it was shown that the ill-posedness also depends on where Neumann and Dirichlet conditions are given: in the direct problem or as an output data. Based on integral identities relating solutions of direct problems to appropriate adjoint problems solutions, a coarse-fine grid algorithm for parabolic coefficient identification problems is constructed. It is shown that use of a coarse grid for the interpolation of the unknown coefficient and a fine grid for the numerical solution of the well-posed forward and backward parabolic problems guarantees an optimal compromise between the accuracy and stability in numerically solving the inverse problems. The efficiency and applicability of this method is demonstrated on various numerical examples with noisy free and noisy data.