Volume 24, Issue 2

Inverse problem of identifying the unknown spacewise dependent source f ( x ) in a nonlinear damped wave equation with a memory term is investigated. The missing coefficient f ( x ) is reconstructed from the final time observation u ( x , T ) = ψ T ( x ). Uniqueness of a solution to the inverse source problem is proved. We also propose a Landweber-type algorithm for reconstruction. Convergence of approximations towards the exact solution is established for a linear equation.

We study the Cauchy problem for nonlinear (semilinear) elliptic partial differential equations in Hilbert spaces. The problem is severely ill-posed in the sense of Hadamard. Under a weak a priori assumption on the exact solution, we propose a new regularization method for stabilising the ill-posed problem. These new results extend some earlier works on Cauchy problems for nonlinear elliptic equations. Numerical results are presented and discussed.

In ℓ 1 -regularization, which is an important tool in signal and image processing, one usually is concerned with signals and images having a sparse representation in some suitable basis, e.g., in a wavelet basis. Many results on convergence and convergence rates of sparse approximate solutions to linear ill-posed problems are known, but rate results for the ℓ 1 -regularization in case of lacking sparsity had not been published until 2013. In the last two years, however, two articles appeared providing sufficient conditions for convergence rates in case of non-sparse but almost sparse solutions. In the present paper, we suggest a third sufficient condition, which unifies the existing two and, by the way, also incorporates the well-known restricted isometry property.

Under the assumption that the solution of a linear operator equation is presented in the form of a sum of several components with various smoothness properties, a modified Tikhonov regularization method is studied. The stabilizer of this method is the sum of three functionals, where each one corresponds to only one component. Each such functional is either the total variation of a function or the total variation of its derivative. For every component, the convergence of approximate solutions in a corresponding normed space is proved and a general discrete approximation scheme for the regularizing algorithm is justified.

We present the results of numerical testing on determination of the sound speed c in the acoustic equation u tt - c 2 Δ u = 0 by the boundary control method . The inverse data is a response operator (a hyperbolic Dirichlet-to-Neumann map) given on controls, which are supported on a part of the boundary. The speed is determined in the subdomain covered by acoustic rays, which are emanated from the points of this part orthogonally to the boundary. The determination is time-optimal : the longer the observation time is, the larger the subdomain is, in which c is recovered. The numerical results are preceded with a brief exposition of the relevant variant of the BC-method.

Material properties such as thermal conductivity, magnetic permeability, electric permittivity, modulus of elasticity, Poisson's ratio, thermal expansion coefficient, etc. can vary spatially throughout a given solid object as it is the case in functionally graded materials. Finding this spatial variation is an inverse problem that requires boundary values of the field quantity such as temperature, magnetic field potential or electric field potential and its derivatives normal to the boundaries. In this paper, we solve the direct problem of predicting the spatial distribution of the field variable based on its measured boundary values and on the assumed spatial distribution of the diffusion coefficient using radial basis functions, the finite volume method and the finite element method, whose accuracies are verified against analytical solutions. Minimization of the sum of normalized least-squares differences between the calculated and measured values of the field quantity at the boundaries then leads to the correct parameters in the analytic model for the spatial distribution of the spatially varying material property.

The problem of determining the initial condition in parabolic equations from boundary observations is studied. It is reformulated as a variational problem and then a formula for the gradient of the functional to be minimized is derived via an adjoint problem. The variational problem is discretized by finite difference splitting methods and solved by the conjugate gradient method. Some numerical examples are presented to show the efficiency of the method. Also as a by-product of the variational method, we propose a numerical scheme for numerically estimating singular values of the solution operator in the inverse problem.