Volume 32, Issue 1

In this study, singular Sturm–Liouville operators on a star graph with edges are investigated. First, the behavior of sufficiently large eigenvalues is learned. Then the solution of the inverse problem is given to determine the potential functions and parameters of the boundary condition on the star graph with the help of a dense set of nodal points. Lastly, a constructive solution to the inverse problems of this class is obtained.

Let X be a Banach space with norm ∥ ⋅ ∥ {\|\cdot\|} . Let A : D ⁢ ( A ) ⊂ X → X {A:D(A)\subset X\rightarrow X} be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that ε > 0 {\varepsilon>0} and T > 0 {T>0} are two given constants. The backward parabolic equation of finding a function u : [ 0 , T ] → X {u:[0,T]\rightarrow X} satisfying u t + A ⁢ u = 0 , 0 < t < T , ∥ u ⁢ ( T ) - φ ∥ ⩽ ε , u_{t}+Au=0,\quad 0<t<T,\;\|u(T)-\varphi\|\leqslant\varepsilon, for φ in X , is regularized by the generalized Sobolev equation u α ⁢ t + A α ⁢ u α = 0 , 0 < t < T , u α ⁢ ( T ) = φ , u_{\alpha t}+A_{\alpha}u_{\alpha}=0,\quad 0<t<T,\;u_{\alpha}(T)=\varphi, where 0 < α < 1 {0<\alpha<1} and A α = A ⁢ ( I + α ⁢ A b ) - 1 {A_{\alpha}=A(I+\alpha A^{b})^{-1}} with b ⩾ 1 {b\geqslant 1} . Error estimates of the method with respect to the noise level are proved.

We propose novel reverse time migration (RTM) methods for the imaging of periodic obstacles using only measurements from the lower or upper side of the obstacle arrays at a fixed frequency. We analyze the resolution of the lower side and upper side RTM methods in terms of propagating modes of the Rayleigh expansion, Helmholtz–Kirchhoff equation and the distance of the measurement surface to the obstacle arrays, where the periodic structure leads to novel analysis. We give some numerical experiments to justify the competitive efficiency of our imaging functionals and the robustness against noises. Further, numerical experiments show sharp images especially for the vertical part of the periodic obstacle in the lower-RTM case, which is not shared by results for imaging bounded compactly supported obstacles.

It is well known that the backward heat conduction problem of recovering the temperature u ⁢ ( ⋅ , t ) {u(\,\cdot\,,t)} at a time t ≥ 0 {t\geq 0} from the knowledge of the temperature at a later time, namely g := u ⁢ ( ⋅ , τ ) {g:=u(\,\cdot\,,\tau)} for τ > t {\tau>t} , is ill-posed, in the sense that small error in g can lead to large deviation in u ⁢ ( ⋅ , t ) {u(\,\cdot\,,t)} . However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for t > 0 {t>0} and ill-posed for t = 0 {t=0} . We use this observation to obtain stable approximate solutions as solutions for t ∈ ( 0 , τ ] {t\in(0,\tau]} with t as regularization parameter for approximating the solution at t = 0 {t=0} , and derive error estimates under suitable source conditions. We shall also provide some numerical examples to illustrate the approximation properties of the regularized solutions.

This paper investigates the inverse problem for the non-stationary radiation transfer equation, which involves finding the attenuation coefficient using the data of serial irradiation of the medium with pulses of various durations. In the framework of single and double scattering approximations, we obtain asymptotic estimates of the scattered radiation flux density for a short duration of the probing pulse. We propose extrapolation procedures for the ballistic component of the radiation transfer equation solution using the data of multiple irradiations of the medium by pulsed radiation sources, which allows us to obtain approximate formulas for finding the attenuation coefficient. The results of numerical experiments with a well-known digital phantom confirm the effectiveness of the extrapolation algorithm for improving the quality of tomographic images of scattering media.

In this paper, we study the inverse problems of determining the unknown transverse shear force g ⁢ ( t ) {g(t)} in a system governed by the damped Euler–Bernoulli equation ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x + ( κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ⁢ x = 0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ] , \rho(x)u_{tt}+\mu(x)u_{t}+(r(x)u_{xx})_{xx}+(\kappa(x)u_{xxt})_{xx}=0,\quad(x,% t)\in(0,\ell)\times(0,T], subject to the boundary conditions u ⁢ ( 0 , t ) = 0 , u x ⁢ ( 0 , t ) = 0 , [ r ⁢ ( x ) ⁢ u x ⁢ x + κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ] x = ℓ = 0 , - [ ( r ⁢ ( x ) ⁢ u x ⁢ x + κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ] x = ℓ = g ⁢ ( t ) , u(0,t)=0,\quad u_{x}(0,t)=0,\quad[r(x)u_{xx}+\kappa(x)u_{xxt}]_{x=\ell}=0,% \quad-[(r(x)u_{xx}+\kappa(x)u_{xxt})_{x}]_{x=\ell}=g(t), for t ∈ [ 0 , T ] {t\in[0,T]} , from the measured deflection ν ⁢ ( t ) := u ⁢ ( ℓ , t ) {\nu(t):=u(\ell,t)} , t ∈ [ 0 , T ] {t\in[0,T]} , and from the bending moment ω ⁢ ( t ) := - ( r ⁢ ( 0 ) ⁢ u x ⁢ x ⁢ ( 0 , t ) + κ ⁢ ( 0 ) ⁢ u x ⁢ x ⁢ t ⁢ ( 0 , t ) ) , t ∈ [ 0 , T ] , \omega(t):=-(r(0)u_{xx}(0,t)+\kappa(0)u_{xxt}(0,t)),\quad t\in[0,T], where the terms ( κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ⁢ x {(\kappa(x)u_{xxt})_{xx}} and μ ⁢ ( x ) ⁢ u t {\mu(x)u_{t}} account for the Kelvin–Voigt damping and external damping, respectively. The main purpose of this study is to analyze the Kelvin–Voigt damping effect on determining the unknown transverse shear force (boundary input) through the given boundary measurements. The inverse problems are transformed into minimization problems for Tikhonov functionals, and it is shown that the regularized functionals admit unique solutions for the inverse problems. By suitable regularity on the admissible class of shear force g ⁢ ( t ) {g(t)} , we prove that these functionals are Fréchet differentiable, and the derivatives are expressed through the solutions of corresponding adjoint problems posed with measured data as boundary data associated with the direct problem. The solvability of these adjoint problems is obtained under the minimal regularity of the boundary data g ⁢ ( t ) {g(t)} , which turns out to be the regularizing effect of the Kelvin–Voigt damping in the direct problem. Furthermore, using the Fréchet derivative of the more regularized Tikhonov functionals, we obtain remarkable Lipschitz stability estimates for the transverse shear force in terms of the given measurement by a feasible condition only on the Kelvin–Voigt damping coefficient.

Finding the history of a groundwater contaminant plume from final measurements is an ill-posed problem and, consequently, its solution is extremely sensitive to errors in the input data. In this paper, we study this problem mathematically. So, firstly, existence and uniqueness theorems of a quasi-solution in an appropriate class of admissible initial data are given. Secondly, in order to overcome the ill-posedness of the problem and also approximate the quasi-solution, two approaches (computational and iterative algorithms) are provided. In the computational algorithm, the finite element method and TSVD regularization are applied. This method is tested by two numerical examples. The results reveal the efficiency and applicability of the proposed method. Also, in order to construct the iterative methods, an explicit formula for the gradient of the cost functional J is given. This result helps us to construct two iterative methods, i.e., the conjugate gradient algorithm and Landweber iteration algorithm. We prove the Lipschitz continuity of the gradient of the cost functional, monotonicity and convergence of the iterative methods. At the end of the paper, a numerical example is given to show the validation of the iterative algorithms.

In this paper, we study the well-posedness of the solution of an optimal control problem related to a multi-parameters identification problem in degenerate parabolic equations. Problems of this type have important applications in several fields of applied science. Unlike other inverse coefficient problems for classical parabolic equations, the mathematical model discussed in the paper is degenerate on both lateral boundaries of the domain. Moreover, the status of the two unknown coefficients are different, namely that the reconstruction of the source term is mildly ill-posed, while the inverse initial value problem is severely ill-posed. On the basis of optimal control framework, the problem is transformed into an optimization problem. The existence of the minimizer is proved and the necessary conditions which must be satisfied by the minimizer are also established. Due to the difference between ill-posedness degrees of the two unknown coefficients, the extensively used conjugate theory for parabolic equations cannot be directly applied for our problem. By carefully analyzing the necessary conditions and the direct problem, the uniqueness, stability and convergence of the minimizer are obtained. The results obtained in the paper are interesting and useful, and can be extended to more general parabolic equations with degenerate coefficients.

We consider a problem of dynamic 2D vector tomography, i.e. the object under investigation changes during the data acquisition. More precisely, we consider the case when the object motion is a combination of rotation and shifting. The task is then to reconstruct the searched-for vector field by known values of the dynamic ray transforms. In order to solve this dynamic inverse problem, we first study properties of the dynamic ray transforms operators. In particular, the singular value decompositions of the operators are constructed using classic orthogonal polynomials. Following from this study, a numerical algorithm for solving the dynamic problem is proposed based on the truncated singular value decomposition method.