Volume 26, Issue 3

Jin Qinian and Min Zhong [10] considered an iteratively regularized Gauss–Newton method in Banach spaces to find a stable approximate solution of the nonlinear ill-posed operator equation. They have considered a Morozov-type stopping rule (Rule 1) as one of the criterion to stop the iterations and studied the convergence analysis of the method. However, no error estimates have been obtained for this case. In this paper, we consider a modified variant of the method, namely, the simplified Gauss–Newton method under both an a priori as well as a Morozov-type stopping rule. In both cases, we obtain order optimal error estimates under Hölder-type approximate source conditions. An example of a parameter identification problem for which the method can be implemented is discussed in the paper.

In this paper, we study an inverse problem of determining the unknown heat source position of a point-wise heat source in a two-dimensional steady-state heat conduction problem governed by a linear elliptic equation with the Dirichlet boundary conditions. The problem is solved by the hybridization of particle swarm optimization and the gravitational search algorithm with a newly defined mutation operator. Some empirical studies are also performed to gauge the accuracy of the proposed approach.

This paper studies the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional J⁢(k):=(1/2)⁢∥u⁢(0,⋅;k)-f∥L2⁢(0,T)2{J(k):=(1/2)\lVert u(0,\cdot\,;k)-f\rVert^{2}_{L^{2}(0,T)}} corresponding to an inverse coefficient problem for the 1⁢D{1D} parabolic equation ut=(k⁢(x)⁢ux)x{u_{t}=(k(x)u_{x})_{x}} with the Neumann boundary conditions -k⁢(0)⁢ux⁢(0,t)=g⁢(t){-k(0)u_{x}(0,t)=g(t)} and ux⁢(l,t)=0{u_{x}(l,t)=0}. In addition, compactness and Lipschitz continuity of the input-output operator Φ[k]:=u(x,t;k)|x=0+,Φ[⋅]:𝒦⊂H1(0,l)↦H1(0,T),\Phi[k]:=u(x,t;k)\lvert_{x=0^{+}},\quad\Phi[\,\cdot\,]:\mathcal{K}\subset H^{1% }(0,l)\mapsto H^{1}(0,T), as well as solvability of the regularized inverse problem and the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional are proved. Furthermore, relationships between the sufficient conditions for the Lipschitz continuity of the Fréchet gradient and the regularity of the weak solution of the direct problem as well as the measured output f⁢(t):=u⁢(0,t;k){f(t):=u(0,t;k)} are established. One of the derived lemmas also introduces a useful application of the Lipschitz continuity of the Fréchet gradient. This lemma shows that an important advantage of gradient methods comes when dealing with the functionals of class C1,1⁢(𝒦){C^{1,1}(\mathcal{K})}. Specifically, this lemma asserts that if J∈C1,1⁢(𝒦){J\in C^{1,1}(\mathcal{K})} and {k(n)}⊂𝒦{\{k^{(n)}\}\subset\mathcal{K}} is the sequence of iterations obtained by the Landweber iteration algorithm k(n+1)=k(n)+ωn⁢J′⁢(k(n)){k^{(n+1)}=k^{(n)}+\omega_{n}J^{\prime}(k^{(n)})}, then for ωn∈(0,2/Lg){\omega_{n}\in(0,2/L_{g})}, where Lg>0{L_{g}>0} is the Lipschitz constant, the sequence {J⁢(k(n))}{\{J(k^{(n)})\}} is monotonically decreasing and limn→∞⁡∥J′⁢(k(n))∥=0{\lim_{n\to\infty}\lVert J^{\prime}(k^{(n)})\rVert=0}.

A mathematical formulation of an estimation problem of a cavity inside a three-dimensional thermoelastic body by using time domain data is considered. The governing equation of the problem is given by a system of equations in the linear theory of thermoelasticity which is a coupled system of the elastic wave and heat equations. A new version of the enclosure method in the time domain which is originally developed for the classical wave equation is established. For a comparison, the results in the decoupled case are also given.

The problem of reconstructing an unknown input under measuring a part of phase coordinates of a system of ordinary differential equations is considered. We propose a solving algorithm that is stable to perturbations and is based on the combination of ideas from the theory of dynamical inversion and the theory of guaranteed control. The algorithm consists of two blocks: the block of dynamical reconstruction of unmeasured coordinates and the block of dynamical reconstruction of an input.

In this paper we provide a simple proof of a Carleman estimate for a second-order elliptic operator P with Lipschitz leading coefficients. We apply such a Carleman estimate to derive a three-sphere inequality for solutions to equation P⁢u=0{Pu=0}.

In this review we discuss methodology to ascertain the amount of information in given data sets with respect to determination of model parameters with desired levels of uncertainty. We do this in the context of least squares (ordinary, weighted, iterative reweighted weighted or “generalized”, etc.) based inverse problem formulations. The ideas are illustrated with several examples of interest in the biological and environmental sciences.

In this note we correct the proof of [2, Theorem 2.1].