Volume 30, Issue 4

For Stokes equations under divergence-free and mixed boundary conditions, the inverse problem of shape identification from boundary measurement is investigated. Taking the least-square misfit as an objective function, the state-constrained optimization is treated by using an adjoint state within the Lagrange approach. The directional differentiability of a Lagrangian function with respect to shape variations is proved within the velocity method, and a Hadamard representation of the shape derivative by boundary integrals is derived explicitly. The application to gradient descent methods of iterative optimization is discussed.

Let 𝒫 and P ′ \mathcal{P}^{\prime} be 3-dimensional convex polytopes in R 3 \mathbb{R}^{3} and S ⊆ R 3 S\subseteq\mathbb{R}^{3} be a non-empty intersection of an open set with a sphere. As a consequence of a somewhat more general result it is proved that 𝒫 and P ′ \mathcal{P}^{\prime} coincide up to translation and/or reflection in a point if | ∫ P e - i ⁢ s ⋅ x ⁢ dx | = | ∫ P ′ e - i ⁢ s ⋅ x ⁢ dx | \bigl{\lvert}\int_{\mathcal{P}}e^{-i\mathbf{s}\cdot\mathbf{x}}\,\mathbf{dx}\bigr{\rvert}=\bigl{\lvert}\int_{\mathcal{P}^{\prime}}e^{-i\mathbf{s}\cdot\mathbf{x}}\,\mathbf{dx}\bigr{\rvert} for all s ∈ S \mathbf{s}\in S . This can be applied to the field of crystallography regarding the question whether a nanoparticle modelled as a convex polytope is uniquely determined by the intensities of its X-ray diffraction pattern on the Ewald sphere.

We investigate a globally convergent method for solving a one-dimensional inverse medium scattering problem using backscattering data at a finite number of frequencies. The proposed method is based on the minimization of a discrete Carleman weighted objective functional. The global convexity of this objective functional is proved.

We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a symmetry assumption on the metric, we obtain uniqueness and stability results in the inverse scattering problem for a potential with data generated by two incident waves from opposite directions. Further, similar results are given using one measurement provided the potential also satisfies a symmetry assumption. This work extends the results of [Rakesh and M. Salo, Fixed angle inverse scattering for almost symmetric or controlled perturbations, SIAM J. Math. Anal. 52 2020, 6, 5467–5499] and [Rakesh and M. Salo, The fixed angle scattering problem and wave equation inverse problems with two measurements, Inverse Problems 36 2020, 3, Article ID 035005] from the Euclidean case to certain Riemannian metrics.

We study some inverse problems involving elasticity models by assuming the knowledge of measurements of a function of the displaced field. In the first case, we have a linear model of elasticity with a semi-linear type forcing term in the solution. Under the hypothesis the fluid is incompressible, we recover the displaced field and the second Lamé parameter from power density measurements in two dimensions. A stability estimate is shown to hold for small displacement fields, under some natural hypotheses on the direction of the displacement, with the background pressure fixed. On the other hand, we prove in dimensions two and three a stability result for the second Lamé parameter when the displacement field follows the (nonlinear) Saint-Venant model when we add the knowledge of displaced field solution measurements. The Saint-Venant model is the most basic model of a hyperelastic material. The use of over-determined elliptic systems is new in the analysis of linearization of nonlinear inverse elasticity problems.

Transient electromagnetic method (TEM), as one of the essential time-domain electromagnetic prospecting approaches, has the advantage of expedition, efficiency and convenience. In this paper, we study the transient electromagnetic inversion problem of different geological anomalies. First, Maxwell’s differential equations are discretized by the staggered finite-difference (FD) method; then we propose to solve the TEM inversion problem by minimizing the Wasserstein metric, which is related to the optimal transport (OT). Experimental tests based on the layered model and a 3D model are performed to demonstrate the feasibility of our proposed method.

In this paper, we are concerned with the identification of the piecewise constant Robin coefficient in an elliptic equation. The iterative regularization method is one of the very effective methods for solving this kind of nonlinear ill-posed inverse problems. But it usually requires to solve numerous amounts of forward solutions during the iterative process, which will cost a lot of computational time in high-dimensional spaces. A reduced basis method is considered to reduce the computational time for solving the forward problems, and its error estimate is also studied. Finally, we propose a reduced basis Landweber algorithm to solve the elliptic inverse Robin problem and present several numerical experiments to demonstrate the accuracy and efficiency of the algorithm.

In this paper, we discuss the role of the damping term μ ⁢ u t {\mu u_{t}} in unique determination of unknown spatial load F ⁢ ( x ) {F(x)} in a damped Euler–Bernoulli beam equation u t ⁢ t + μ ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ ( x ) ⁢ G ⁢ ( t ) {u_{tt}+\mu u_{t}+(r(x)u_{xx})_{xx}=F(x)G(t)} from the measured final time displacement u T ⁢ ( x ) := u ⁢ ( x , T ) {u_{T}(x):=u(x,T)} or velocity ν T ⁢ ( x ) := u t ⁢ ( x , T ) {\nu_{T}(x):=u_{t}(x,T)} . In [A. Hasanov Hasanoğlu and V. G. Romanov, Introduction to Inverse Problems for Differential Equations, Springer, Cham, 2017] it was shown that the unique determination of the spatial load F ⁢ ( x ) {F(x)} in the undamped wave equation u t ⁢ t - ( k ⁢ ( x ) ⁢ u x ) x = F ⁢ ( x ) ⁢ G ⁢ ( t ) {u_{tt}-(k(x)u_{x})_{x}=F(x)G(t)} from final data (displacement or velocity) is not possible. This result is also valid for the undamped beam equation u t ⁢ t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ ( x ) ⁢ G ⁢ ( t ) {u_{tt}+(r(x)u_{xx})_{xx}=F(x)G(t)} . We show that in the presence of the damping term, the spatial load can be uniquely determined from the final time output under some acceptable conditions with respect to the final time T > 0 {T>0} and the damping coefficient μ > 0 {\mu>0} . The cases of pure spatial load G ⁢ ( t ) ≡ 1 {G(t)\equiv 1} , and the exponentially decaying load G ⁢ ( t ) = e - η ⁢ t {G(t)=e^{-\eta t}} are analyzed separately, leading also to sufficient conditions for the uniqueness of the inverse problem solution. The results presented in this paper clearly demonstrate the key role of the damping term in the unique determination of the unknown spatial load F ⁢ ( x ) {F(x)} in the damped Euler–Bernoulli beam equation.

It is demonstrated in this paper that the propagation of the electric wave field in a heterogeneous medium in 3D can sometimes be governed well by a single scalar second order PDE, which is derived from Maxwell’s equations. The corresponding component of the electric field dominates two other components. This justifies some past results of the second author with coauthors about numerical solutions of coefficient inverse problems with experimental electromagnetic data. In addition, since it is simpler to work in applications with a single scalar second order PDE rather than with the complete Maxwell system, the result of this paper might be useful to researchers working on applied issues of the propagation of electromagnetic waves in inhomogeneous media.

This article gives a brief overview of the research in microlocal analysis, tomography, and integral geometry of Professor Eric Todd Quinto, Robinson Professor of Mathematics at Tufts University, along with the collaborators and colleagues who influenced his work.