Volume 30, Issue 3

Total variation regularization is widely used in geophysical inverse problems for preserving the edges of geological structures. However, it may lead to staircase artifacts and other inversion errors. To overcome this drawback, we present an impedance inversion method by using anisotropic total variation with overlapping group sparsity regularization. This method can not only preserve edges very well but also makes use of the group sparsity feature of acoustic impedance to reduce inversion error. Moreover, we make full use of the properties of the Toeplitz matrix and speed up the inversion process in the frequency domain. The synthetic data and field data inversion results show that the proposed method has the advantages of stronger anti-noise, faster inversion speed, and less error comparing with the conventional total variation regularization inversion method. In addition, our method can effectively alleviate staircase artifacts and therefore improves resolution.

Recently, fractional Landweber and iterative fractional Tikhonov method are well investigated for solving ill-posed problems. In this paper, based on a special discretization scheme of the gradient flow equation, a general iterative scheme is obtained. We call it iterative fractional Tikhonov–Landweber method. Error analysis is conducted for this method. A preliminary numerical test shows that the proposed method has its advantage.

Consider the scattering of electromagnetic waves by a penetrable homogeneous cylinder at oblique incident. The Maxwell equations are then reduced to a system of a pair of the two-dimensional Helmholtz equations for 𝑧-components of the electric and magnetic field through coupled oblique boundary conditions. This paper studies an inverse problem of recovering the penetrable obstacle from the far-field pattern of the electric field. The well-known linear sampling method is used to solve this problem. Compared with the usual inverse scattering problem, the coupled system and the oblique derivative boundary condition bring difficulties in theoretical analysis. Some numerical examples are presented to illustrate the validity and feasibility of the proposed method.

In this paper, we study the backward problem for the stochastic parabolic heat equation driven by a Wiener process. We show that the problem is ill-posed by violating the continuous dependence on the input data. In order to restore stability, we apply a filter regularization method which is completely new in the stochastic setting. Convergence rates are established under different a priori assumptions on the sought solution.

We study an inverse problem involving the restoration of two radiative potentials, not necessarily smooth, simultaneously with initial temperatures in parabolic equations with dynamic boundary conditions. We prove a Lipschitz stability estimate for the relevant potentials using a recent Carleman estimate, and a logarithmic stability result for the initial temperatures by a logarithmic convexity method, based on observations in an arbitrary subdomain.

We consider an inverse scattering problem of recovering the unknown coefficients of a quasi-linearly perturbed biharmonic operator in the three-dimensional case. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove Saito’s formula and uniqueness theorem of recovering some essential information about the unknown coefficients from the knowledge of the high frequency scattering amplitude.

This paper considers a class of nonlinear time-harmonic Maxwell systems at fixed frequency, with nonlinear terms taking the form X ⁢ ( x , | E → ⁢ ( x ) | 2 ) ⁢ E → ⁢ ( x ) \mathscr{X}(x,\lvert\vec{E}(x)\rvert^{2})\vec{E}(x) , Y ⁢ ( x , | H → ⁢ ( x ) | 2 ) ⁢ H → ⁢ ( x ) \mathscr{Y}(x,\lvert\vec{H}(x)\rvert^{2})\vec{H}(x) such that X ⁢ ( x , s ) \mathscr{X}(x,s) , Y ⁢ ( x , s ) \mathscr{Y}(x,s) are both real analytic in 𝑠. Such nonlinear terms appear in nonlinear optics theoretical models. Under certain regularity conditions for 𝒳 and 𝒴, it can be shown that boundary measurements of tangent components of the electric and magnetic fields determine the electric permittivity and magnetic permeability functions as well as the form of the nonlinear terms.

We propose a new concept of noise level: R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level for ill-posed linear integral equations in Tikhonov regularization, which extends the range of regularization parameter. This noise level allows us to choose a more suitable regularization parameter. Moreover, we also analyze error estimates of the approximate solution with respect to this noise level. For ill-posed integral equations, finding fast and effective numerical methods is a challenging problem. For this, we formulate a matrix truncated strategy based on multiscale Galerkin method to generate the linear system of Tikhonov regularization for ill-posed linear integral equations, which greatly reduce the computational complexity. To further reduce the computational cost, a fast multilevel iteration method for solving the linear system is established. At the same time, we also prove convergence rates of the approximate solution obtained by this fast method with respect to the R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level under the balance principle. By numerical results, we show that R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level is very useful and the proposed method is a fast and effective method, respectively.

This article proposes a unified approach to the issues of uniqueness and Lipschitz stability for the final data inverse source problems of determining the unknown spatial load F ⁢ ( x ) {F(x)} in the evolution equations. The approach is based on integral identities outlined here for the one-dimensional and multidimensional heat equations ρ ⁢ ( x ) ⁢ u t - ( k ⁢ ( x ) ⁢ u x ) x = F ⁢ ( x ) ⁢ G ⁢ ( t ) , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ] , \rho(x)u_{t}-(k(x)u_{x})_{x}=F(x)G(t),\quad(x,t)\in(0,\ell)\times(0,T], and ρ ⁢ ( x ) ⁢ u t - div ⁡ ( k ⁢ ( x ) ⁢ ∇ ⁡ u ) = F ⁢ ( x ) ⁢ G ⁢ ( x , t ) , ( x , t ) ∈ Ω × ( 0 , T ] , Ω ∈ ℝ n , \rho(x)u_{t}-\operatorname{div}(k(x)\nabla u)=F(x)G(x,t),\quad(x,t)\in\Omega% \times(0,T],\,\Omega\in\mathbb{R}^{n}, for the damped wave, and the Euler–Bernoulli beam and Kirchhoff plate equations ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t - ( r ⁢ ( x ) ⁢ u x ) x = F ⁢ ( x ) ⁢ G ⁢ ( x , t ) , ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ ( x ) ⁢ G ⁢ ( t ) , \rho(x)u_{tt}+\mu(x)u_{t}-(r(x)u_{x})_{x}=F(x)G(x,t),\quad\rho(x)u_{tt}+\mu(x)% u_{t}+(r(x)u_{xx})_{xx}=F(x)G(t), for ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ] {(x,t)\in(0,\ell)\times(0,T]} , and ρ ⁢ ( x ) ⁢ h ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( D ⁢ ( x ) ⁢ ( u x 1 , x 1 + ν ⁢ u x 2 , x 2 ) ) x 1 , x 1 + ( D ⁢ ( x ) ⁢ ( u x 2 , x 2 + ν ⁢ u x 1 , x 1 ) ) x 2 , x 2 + 2 ⁢ ( 1 - ν ) ⁢ ( D ⁢ ( x ) ⁢ u x 1 , x 2 ) x 1 , x 2 \displaystyle\rho(x)h(x)u_{tt}+\mu(x)u_{t}+(D(x)(u_{x_{1},x_{1}}+\nu u_{x_{2},% x_{2}}))_{x_{1},x_{1}}+(D(x)(u_{x_{2},x_{2}}+\nu u_{x_{1},x_{1}}))_{x_{2},x_{2% }}+2(1-\nu)(D(x)u_{x_{1},x_{2}})_{x_{1},x_{2}} = F ⁢ ( x ) ⁢ G ⁢ ( t ) , ( x , t ) ∈ Ω T := Ω × ( 0 , T ) , Ω := ( 0 , ℓ 1 ) × ( 0 , ℓ 2 ) , \displaystyle =F(x)G(t),\quad(x,t)\in\Omega_{T}:=\Omega\times(0,T),\,% \Omega:=(0,\ell_{1})\times(0,\ell_{2}), and allows us to prove the uniqueness and stability of the solutions for all considered inverse problems under the same conditions imposed on the load G ⁢ ( t ) {G(t)} or G ⁢ ( x , t ) {G(x,t)} .

An approach to solving the problem of determining the thermal conductivity of a substance based on the results of observing the dynamics of the temperature field is proposed. The consideration is based on the Dirichlet boundary value problem for the three-dimensional nonstationary heat equation. The effectiveness of the proposed approach is based on the application of the Fast Automatic Differentiation technique. The required thermal conductivity coefficient is determined as the solution to the formulated optimal control problem.