Volume 25, Issue 5

Recently, Semenova [12] considered a derivative free iterative method for nonlinear ill-posed operator equations with a monotone operator. In this paper, a modified form of Semenova’s method is considered providing simple convergence analysis under more realistic nonlinearity assumptions. The paper also provides a stopping rule for the iteration based on an a priori choice of the regularization parameter and also under the adaptive procedure considered by Pereverzev and Schock [11].

In this article, the well-known conjugate gradient (CG) method for linear systems in Hilbert spaces is extended to a reflexive Banach space setting. In this setting, the Riesz isomorphism has to be replaced by the duality mapping. Due to the nonlinearity of the duality mapping, the short term recursion and conjugacy of search directions cannot be maintained simultaneously. The well-posedness of the proposed iteration and its global convergence are shown under appropriate conditions. Error bounds and stopping criteria are presented as well. The results extend to a limited-memory variant of the algorithm. The behavior of the method is demonstrated by numerical examples.

The aim of this work is an analysis of some geometrical inverse problems related to the identification of cavities in linear elasticity framework. We rephrase the inverse problem into a shape optimization one using an energetic least-squares functional. The shape derivative of this cost functional is combined with the level set method in a steepest descent algorithm to solve the shape optimization problem. The efficiency of this approach is illustrated by several numerical results.

Consider the heat conduction process with heat flux exchanges on the boundary of a 2-dimensional domain. The aim is to identify both the boundary heat exchange coefficient and the initial heat distribution simultaneously from the final measurement data of the heat field. We prove the uniqueness for this nonlinear inverse problem for the strictly positive exchange coefficient and initial heat distribution in terms of the maximal principle and the eigenfunction expansions. Then a regularizing scheme combining the data mollification and quasi-reversibility method along time direction together is established to recover two unknowns, with the choice strategies for the regularizing parameters and the error estimates on the regularizing solutions. The reconstruction implementations are carried out in terms of the potential representation of the heat field, with numerical examples showing the validity of the proposed scheme.

Via Carleman estimates we prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the conormal derivative of the temperature on a relatively open subset of the lateral boundary of the space-time domain.

An inverse problem for the two-dimensional Schrödinger equation with Lcomp{L^{p}_{\rm com}}-potential, p>1{p>1}, is considered. Using the ∂¯{\bar{\partial}}-method, the potential is recovered from the Dirichlet-to-Neumann map on the boundary of a domain containing the support of the potential. We do not assume that the potential is small or that the Faddeev scattering problem does not have exceptional points. The paper contains a new estimate on the Faddeev Green function that immediately implies the absence of exceptional points near the origin and infinity when v∈Lcomp{v\in L^{p}_{\rm com}}.

Classical compartmental epidemic models of infectious diseases track the dynamic transition of individuals between different epidemiological states or risk groups. Reliable quantification of various transmission pathways in these models is paramount for optimal resource allocation and successful design of public health intervention programs. However, with limited epidemiological data available in the case of an emerging disease, simple phenomenological models based on a smaller number of parameters can play an important role in our quest to make forward projections of possible outbreak scenarios. In this paper, we employ the generalized Richards model for stable numerical estimation of the epidemic size (defined as the total number of infections throughout the epidemic) and its turning point using case incidence data of the early epidemic growth phase. The minimization is carried out by what we call the Reduced Iteratively Regularized Gauss–Newton (RIRGN) algorithm, a problem-oriented numerical scheme that takes full advantage of the specific structure of the non-linear operator at hand. The convergence analysis of the RIRGN method is suggested and numerical simulations are conducted with real case incidence data for the 2014–15 Ebola epidemic in West Africa. We show that the proposed RIRGN provides a stable algorithm for early estimation of turning points using simple phenomenological models with limited data.

By our definition, “restricted Dirichlet-to-Neumann (DN) map” means that the Dirichlet and Neumann boundary data for a coefficient inverse problem (CIP) are generated by a point source running along an interval of a straight line. On the other hand, the conventional DN data can be generated, at least sometimes, by a point source running along a hypersurface. CIPs with restricted DN data are non-overdetermined in the n -dimensional case, with n≥2{n\geq 2}. We develop, in a unified way, a general and radically new numerical concept for CIPs with restricted DN data for a broad class of PDEs of second order, such as, e.g., elliptic, parabolic and hyperbolic ones. Namely, using Carleman weight functions, we construct globally convergent numerical methods. Hölder stability and uniqueness are also proved. The price we pay for these features is a well-acceptable one in the numerical analysis, that is, we truncate a certain Fourier-like series with respect to some functions depending only on the position of the point source. At least three applications are imaging of land mines, crosswell imaging and electrical impedance tomography.