A unified framework for constructing globally convergent algorithms for multidimensional coefficient inverse problems

We present a unified framework for constructing the globally convergent algorithms for a broad class of multidimensional coefficient inverse problems arising in natural science and industry. Based on the convexification approach, the unified framework substantiates the numerical solution of the corresponding problem of nonconvex optimization. A globally convergent iterative algorithm for an inverse problem of diffuse optical mammography is constructed. It utilizes the contraction property of a nonlinear operator resulting from applying the convexification approach. The effectiveness of this algorithm is demonstrated in computational experiments.     

Existence results for equilibrium problems with applications to evolution equations

In this paper, we consider evolution equations with time-dependent pseudomonotone and quasimonotone operators by a new approach based on equilibrium problems theory. We establish new existence results for equilibrium problems associated to pseudomonotone and quasimonotone bifunctions in the topological sense. The results obtained are then applied to derive existence of solutions for evolution equations. The approach is new and leads to improve and unify most of the results obtained in this direction.     

A globally convergent numerical method for an inverse elliptic problem of optical tomography

A new globally convergent numerical method is developed for an inverse problem for the elliptic equation with the unknown potential. The boundary data simulating measurements in optical tomography are generated by the running source. Global convergence analysis is presented along with numerical experiments.     

Asymptotic analysis of contact problems between an elastic material and thin-rigid plates

We consider an elastic material in contact with a three-dimensional rigid plate of varying thickness. We suppose that a perfect adhesion occurs along thin zones disposed in a self-similar way on the interface between the two materials. We suppose that the elasticity coefficients in the plate depend on its thickness and tend to infinity as this thickness tends to zero. We derive the effective material properties using Γ-convergence methods.     

The unknown object problem in a sea with sloping bottom

This article considers the acoustic unknown object problem for a shallow ocean with a sloping seabed. The incident waves are sent from point sources along a s raight line parallel to the sea surface, and the corresponding scattered fields are measured from a line above the unknown object. We prove a uniqueness theorem for the inverse problem,and describe a generalizeddual space indicator method for numerical solution.Numerical results are given in Section 4.     

Investigation of hyperbolic systems with order degeneration arising in I. Vekua's hierarchical models

We study the well-posedness of an initial-boundary value problem corresponding to the zeroth approximation of I. Vekua's hierarchical models for elastic cusped prismatic shells. The mathematical model is described by a two-dimensional order-degenerating hyperbolic system. We formulate the problem in the weak setting and prove the uniqueness and existence theorems. We show that the sequence of corresponding explicit Galerkin approximations converges to the exact solution in an appropriate weighted Lebesgue space.     

An inverse problem of Bilayer textile thickness determination in dynamic heat and moisture transfer

An inverse problem of bilayer textile thickness determination in dynamic heat and moisture transfer is presented satisfying the heat–moisture comfort level of human body. Heat and mass transfer law in bilayer textiles is displayed by proving the existence and uniqueness of solution to the coupled partial differential equations with initial-boundary value conditions. The finite difference method is employed to derive the numerical solution to partial differential equations. The regularized solution of the inverse problem is reformulated into solving function minimum problem through the Tikhonov regularization method. The golden section method is applied to solve the direct search problem and achieve the optimal solution to the inverse problem. Numerical algorithm and its numerical results provide theoretical explanation for textile materials research and development.     

Stability of equilibrium solution to inhomogeneous heat equation under a 3-point boundary condition

We consider a one-dimensional heat equation with inhomogeneous term, satisfying three-point boundary conditions, such that the temperature at the end is controlled by a sensor at the point η. We show that the integral solution, in the space of continuous functions satisfying the boundary values, converges to the equilibrium solution. This answers a question posed for nonlinear Laplacians, but in the linear case only.