Volume 30, Issue 5

In this paper, we consider Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. In part I, we establish Carleman estimates for the coupled quantitative thermoacoustic equations by assuming that the coefficients satisfy suitable conditions and taking the usual weight function φ ⁢ ( x , t ) = e λ ⁢ ψ ⁢ ( x , t ) , ψ ⁢ ( x , t ) = | x - x 0 | 2 - β ⁢ | t - t 0 | 2 + β ⁢ t 0 2 \varphi(x,t)={\mathrm{e}}^{\lambda\psi(x,t)},\quad\psi(x,t)=\lvert x-x_{0}% \rvert^{2}-\beta\lvert t-t_{0}\rvert^{2}+\beta t_{0}^{2} for x in a bounded domain in ℝ n {\mathbb{R}^{n}} with C 3 {C^{3}} -boundary and t ∈ ( 0 , T ) {t\in(0,T)} , where t 0 = T 2 {t_{0}=\frac{T}{2}} . We will discuss applications of the Carleman estimates to some inverse problems for the coupled quantitative thermoacoustic equations in the succeeding part II paper [M. Cristofol, S. Li and Y. Shang, Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. Part II: Inverse problems, preprint 2020, https://hal.archives-ouvertes.fr/hal-02863385].

In this article, we study the inverse problem of recovering a space-dependent coefficient of the Moore–Gibson–Thompson (MGT) equation from knowledge of the trace of the solution on some open subset of the boundary. We obtain the Lipschitz stability for this inverse problem, and we design a convergent algorithm for the reconstruction of the unknown coefficient. The techniques used are based on Carleman inequalities for wave equations and properties of the MGT equation.

In this paper, we discuss the Legendre spectral projection method for solving Fredholm integral equations of the first kind using Tikhonov regularization. First, we discuss the convergence analysis under an a priori parameter strategy for the Tikhonov regularization using Legendre polynomial basis functions, and we obtain the optimal convergence rates in the uniform norm. Next, we discuss Arcangeli’s discrepancy principle to find a suitable regularization parameter and obtain the optimal order of convergence in uniform norm. We present numerical examples to illustrate the theoretical results.

The means to obtain the adsorption isotherms is a fundamental open problem in competitive chromatography. A modern technique of estimating adsorption isotherms is to solve a nonlinear inverse problem in a partial differential equation so that the simulated batch separation coincides with actual experimental results. However, this identification process is usually ill-posed in the sense that the uniqueness of adsorption isotherms cannot be guaranteed, and moreover, the small noise in the measured response can lead to a large fluctuation in the traditional estimation of adsorption isotherms. The conventional mathematical method of solving this problem is the variational regularization, which is formulated as a non-convex minimization problem with a regularized objective functional. However, in this method, the choice of regularization parameter and the design of a convergent solution algorithm are quite difficult in practice. Moreover, due to the restricted number of injection profiles in experiments, the types of measured data are extremely limited, which may lead to a biased estimation. In order to overcome these difficulties, in this paper, we develop a new inversion method – the virtual injection promoting double feed-forward neural network (VIP-DFNN). In this approach, the training data contain various types of artificial injections and synthetic noisy measurement at outlet, generated by a conventional physics model – a time-dependent convection-diffusion system. Numerical experiments with both artificial and real data from laboratory experiments show that the proposed VIP-DFNN is an efficient and robust algorithm.

This paper deals with an inverse source problem governed by the Poisson equation. The aim is to reconstruct multiple anomalies from internal observation data within an arbitrary sub-region. The considered model problem is motivated by various applications such as the identification of geological anomalies underneath the Earth’s surface. To overcome the ill-posedness, the inverse problem is formulated as a self-regularized topology optimization one. A least-square functional measuring the misfit between the observed quantities and the values provided by the model problem is introduced. The misfit function involves a regularization term penalizing the relative perimeter of the unknown domain. Finally, it is minimized with respect to a finite number of circular-shaped anomalies. The existence and stability of the solution to the inverse problem are demonstrated. The reconstruction process is based on the topological derivative method. The variation of the shape functional with respect to a small geometric perturbation is studied, and the leading terms of a second-order topological asymptotic expansion are derived. A non-iterative reconstruction algorithm is developed via the minimization of the obtained asymptotic formula with respect to the location and size parameters of the unknown set of anomalies. The efficiency and accuracy of the proposed approach are justified by some numerical experiments.

Let 𝒫 {\mathcal{P}} be an n -dimensional convex polytope and let 𝒮 {\mathcal{S}} be a hypersurface in ℝ n {\mathbb{R}^{n}} . This paper investigates potentials to reconstruct 𝒫 {\mathcal{P}} , or at least to compute significant properties of 𝒫 {\mathcal{P}} , if the modulus of the Fourier transform of 𝒫 {\mathcal{P}} on 𝒮 {\mathcal{S}} with wave length λ, i.e., | ∫ 𝒫 e - i ⁢ 1 λ ⁢ 𝐬 ⋅ 𝐱 ⁢ 𝐝𝐱 |   for  ⁢ 𝐬 ∈ 𝒮 , \biggl{\lvert}\int_{\mathcal{P}}e^{-i\frac{1}{\lambda}\mathbf{s}\cdot\mathbf{x% }}\,\mathbf{dx}\biggr{\rvert}\quad\text{for }\mathbf{s}\in\mathcal{S}, is given, λ is sufficiently small and 𝒫 {\mathcal{P}} and 𝒮 {\mathcal{S}} have some well-defined properties. The main tool is an asymptotic formula for the Fourier transform of 𝒫 {\mathcal{P}} with wave length λ when λ → 0 {\lambda\rightarrow 0} . The theory of X-ray scattering of nanoparticles motivates this study, since the modulus of the Fourier transform of the reflected beam wave vectors is approximately measurable on a half sphere in experiments.

This paper is concerned with the two-dimensional inverse acoustic scattering by an unbounded, sound-soft rough surface. We propose a nonlinear integral equation method using multi-frequency phaseless near-field data associated with point sources to reconstruct the shape and location of the rough surface, yielding a fast imaging algorithm. Numerical examples are presented to show the effectiveness of the inverse algorithm.

In this paper, we propose a method for inpainting regular textures composed of oriented patterns and completion of images distorted by long stripes. We have developed a model fit to describe a regular texture as a sum of a few ridge functions. The ridge functions are known in tomography as a result of a back-projection operator applied to projections, or samples of the Radon transform. A computational scheme for the singular value decomposition of a regular texture into the sum of informative ridge functions is presented. The numerical experiments on various images demonstrate the efficiency of the techniques proposed.

One of the crucial challenges in applying the theory of compressed sensing in practice is to cope with the perturbation caused by different types of unknown noise, which may arise from the physical implementation or the human mis-modeling. In this paper, we study the robust recovery from a completely perturbed model via a recently popular L 1 - 2 L_{1-2} method. By using the powerful restricted isometry constant of order t ⁢ k tk for t > 1 t>1 , we first obtain a series of perturbation analysis results for this L 1 - 2 L_{1-2} method, which shows that this method is also able to guarantee a robust recovery for any (nearly) sparse signals when both the ideal observations and the exact measurement matrix are perturbed by the unknown noise. Moreover, one of the established recovery conditions under the noise-free settings is also demonstrated to be much better than the state-of-the-art one. Finally, some simulation experiments are further carried out to verify the effectiveness of this L 1 - 2 L_{1-2} method.