Volume 29, Issue 1

Consider the direct and inverse scattering problem of time harmonic acoustic waves by a two-dimensional elastic obstacle which contains an unknown impenetrable object inside. We apply the boundary integral equation method to solve the direct scattering problem. Since the obtained boundary integral system is a mixed form of scalar and vector equations, we consider the existence of the solution in the sense of a weak formulation. The factorization method is employed to solve the corresponding inverse scattering problem, that is, we try to recover the shape of the interaction surface from the knowledge of acoustic far-field patterns. With the help of a special data to pattern operator G , we can reconstruct the shape of the interaction surface by using the classical factorization method. Furthermore, a new mixed reciprocity relation is proposed to show the relation between the interior elastic field and unitary matrix 𝒮{\mathcal{S}}.

We prove that if the Neumann eigenvalues of the impulsive Sturm–Liouville operator -D2+q{-D^{2}+q} in L2⁢(0,π){L^{2}(0,\pi)} coincide with those of the Neumann Laplacian, then q=0{q=0}.

In 2012, Jin Qinian considered an inexact Newton–Landweber iterative method for solving nonlinear ill-posed operator equations in the Banach space setting by making use of duality mapping. The method consists of two steps; the first one is an inner iteration which gives increments by using Landweber iteration, and the second one is an outer iteration which provides increments by using Newton iteration. He has proved a convergence result for the exact data case, and for the perturbed data case, a weak convergence result has been obtained under a Morozov type stopping rule. However, no error bound has been given. In 2013, Kaltenbacher and Tomba have considered the modified version of the Newton–Landweber iterations, in which the combination of the outer Newton loop with an iteratively regularized Landweber iteration has been used. The convergence rate result has been obtained under a Hölder type source condition. In this paper, we study the modified version of inexact Newton–Landweber iteration under the approximate source condition and will obtain an order-optimal error estimate under a suitable choice of stopping rules for the inner and outer iterations. We will also show that the results proved in this paper are more general as compared to the results proved by Kaltenbacher and Tomba in 2013. Also, we will give a numerical example of a parameter identification problem to support our method.

The study of a time-periodic solution of the multidimensional wave equation ∂2∂⁡t2⁢u~-Δx⁢u~=f~⁢(x,t){\frac{\partial^{2}}{\partial t^{2}}\widetilde{u}-\Delta_{x}\widetilde{u}=% \widetilde{f}(x,t)}, u~⁢(x,t)=ei⁢k⁢t⁢u⁢(x){\widetilde{u}(x,t)=e^{ikt}u(x)}, over the whole space ℝ3{\mathbb{R}^{3}} leads to the condition of the Sommerfeld radiation at infinity. This is a problem that describes the motion of scattering stationary waves from a source that is in a bounded area. The inverse problem of finding this source is equivalent to reducing the Sommerfeld problem to a boundary value problem for the Helmholtz equation in a finite domain. Therefore, the Sommerfeld problem is a special inverse problem. It should be noted that in the work of Bezmenov [I. V. Bezmenov, Transfer of Sommerfeld radiation conditions to an artificial boundary of the region based on the variational principle, Sb. Math. 185 1995, 3, 3–24] approximate forms of such boundary conditions were found. In [T. S. Kalmenov and D. Suragan, Transfer of Sommerfeld radiation conditions to the boundary of a limited area, J. Comput. Math. Math. Phys. 52 2012, 6, 1063–1068], for a complex parameter λ, an explicit form of these boundary conditions was found through the boundary condition of the Helmholtz potential given by the integral in the finite domain Ω: ($*$)u⁢(x,λ)=∫Ωε⁢(x-ξ,λ)⁢ρ⁢(ξ,λ)⁢𝑑ξu(x,\lambda)=\int_{\Omega}\varepsilon(x-\xi,\lambda)\rho(\xi,\lambda)\,d\xi{} where ε⁢(x-ξ,λ){\varepsilon(x-\xi,\lambda)} are fundamental solutions of the Helmholtz equation, -Δx⁢ε⁢(x)-λ⁢ε=δ⁢(x),-\Delta_{x}\varepsilon(x)-\lambda\varepsilon=\delta(x), ρ⁢(ξ,λ){\rho(\xi,\lambda)} is a density of the potential, λ is a complex number, and δ is the Dirac delta function. These boundary conditions have the property that stationary waves coming from the region Ω to ∂⁡Ω{\partial\Omega} pass ∂⁡Ω{\partial\Omega} without reflection, i.e. are transparent boundary conditions. In the present work, in the general case, in ℝn{\mathbb{R}^{n}}, n≥3{n\geq 3}, we have proved the problem of reducing the Sommerfeld problem to a boundary value problem in a finite domain. Under the necessary conditions for the Helmholtz potential (**), its density ρ⁢(ξ,λ){\rho(\xi,\lambda)} has also been found.

The monitoring, analysis and prediction of epidemic spread in the region require the construction of mathematical model, big data processing and visualization because the amount of population and the size of the region could be huge. One of the important steps is refinement of mathematical model, i.e. determination of initial data and coefficients of system of differential equations of epidemiologic processes using additional information. We analyze numerical method for solving inverse problem of epidemiology based on genetic algorithm and traditional optimization approach. Our algorithms are applied to analysis and prediction of epidemic situation in regions of Russian Federation, Republic of Kazakhstan and People’s Republic of China. Due to a great amount of data we use a special software ”Digital Earth” for visualization of epidemic.

This paper deals with an inverse coefficient problem of simultaneously identifying the thermal conductivity k⁢(x)k(x) and radiative coefficient q⁢(x)q(x) in the 1D heat equation ut=(k⁢(x)⁢ux)x-q⁢(x)⁢uu_{t}=(k(x)u_{x})_{x}-q(x)u from the most available Dirichlet and Neumann boundary measured outputs. The Neumann-to-Dirichlet and Neumann-to-Neumann operators Φ⁢[k,q]⁢(t):=u⁢(ℓ,t;k,q)\Phi[k,q](t):=u(\ell,t;k,q), Ψ⁢[k,q]⁢(t):=-k⁢(0)⁢ux⁢(0,t;k,q)\Psi[k,q](t):=-k(0)u_{x}(0,t;k,q) are introduced, and main properties of these operators are derived. Then the Tikhonov functional J⁢(k,q)=12⁢∥Φ⁢[k,q]-ν∥L2⁢(0,T)2+12⁢∥Ψ⁢[k,q]-φ∥L2⁢(0,T)2J(k,q)=\tfrac{1}{2}\lVert\Phi[k,q]-\nu\rVert^{2}_{L^{2}(0,T)}+\tfrac{1}{2}\lVert\Psi[k,q]-\varphi\rVert^{2}_{L^{2}(0,T)} of two functions k⁢(x)k(x) and q⁢(x)q(x) is introduced, and an existence of a quasi-solution of the inverse coefficient problem is proved. An explicit formula for the Fréchet gradient of the Tikhonov functional is derived through the weak solutions of two appropriate adjoint problems.

In this paper, we study two inverse problems for stochastic parabolic equations with additive noise. One is to determinate the history of a stochastic heat process and the random heat source simultaneously by the observation at the final time 𝑇. For this inverse problem, we obtain a conditional stability result. The other one is an inverse source problem to determine two kinds of sources simultaneously by the observation at the final time and on the lateral boundary. The main tool for solving the inverse problems is a new global Carleman estimate for the stochastic parabolic equation.

PSF (point spread function) estimation plays an important role in blind image deconvolution. It has been shown in our previous work that minimization of the Stein’s unbiased risk estimate (SURE) – unbiased estimate of mean squared error (MSE) – could yield an accurate PSF estimate. In this paper, we show that the PSF estimation error is upper bounded by the deconvolution accuracy and the mismatch between the assumed PSF parametric form and the underlying true one. For this reason, we incorporate the ℓ1{\ell_{1}}-penalized sparse deconvolution into the SURE instead of previously used Wiener filter. In particular, we apply the iterative soft-thresholding algorithms to solve ℓ1{\ell_{1}}-minimization, and develop recursive evaluations of SURE, which is then shown to converge to the existing theoretical result. In practical implementations with large-scale data, we apply the Monte-Carlo simulation to avoid the explicit matrix operation. Numerical examples demonstrate the improvements of PSF estimate, and the resulting deconvolution performance.

In the paper, for systems described by ordinary differential equations a review of algorithms of dynamical input reconstruction by results of inaccurate observations of its solutions is given. The problem under discussion is referred to the class of dynamical inverse problems. The proposed algorithms are stable with respect to informational noises and computational errors. They are based on the combination of methods of the theory of ill-posed problems and the theory of feedback control. The essence of the methodology underlying the algorithms suggested in the paper consists in the representation of a reconstruction algorithm in the form of a feedback control algorithm for a certain artificial dynamical system, a model; such an algorithm, whose output is the realization of the control in the model, is dynamical by its definition.