Volume 29, Issue 3

We consider the zeta function ζΩ\zeta_{\Omega} for the Dirichlet-to-Neumann operator of a simply connected planar domain Ω bounded by a smooth closed curve of perimeter 2⁢π2\pi. We name the difference ζΩ-ζD\zeta_{\Omega}-\zeta_{\mathbb{D}} the normalized Steklov zeta function of the domain Ω, where 𝔻 denotes the closed unit disk. We prove that (ζΩ-ζD)′′⁢(0)≥0(\zeta_{\Omega}-\zeta_{\mathbb{D}})^{\prime\prime}(0)\geq 0 with equality if and only if Ω is a disk. We also provide an elementary proof that, for a fixed real 𝑠 satisfying s≤-1s\leq-1, the estimate (ζΩ-ζD)′′⁢(s)≥0(\zeta_{\Omega}-\zeta_{\mathbb{D}})^{\prime\prime}(s)\geq 0 holds with equality if and only if Ω is a disk. We then bring examples of domains Ω close to the unit disk where this estimate fails to be extended to the interval (0,2)(0,2). Other computations related to previous works are also detailed in the remaining part of the text.

Let (Ω,g){(\Omega,g)} be a smooth compact two-dimensional Riemannian manifold with boundary and let Λg:f↦∂ν⁡u|∂⁡Ω{\Lambda_{g}:f\mapsto\partial_{\nu}u|_{\partial\Omega}} be its DN map, where u obeys Δg⁢u=0{\Delta_{g}u=0} in Ω and u|∂⁡Ω=f{u|_{\partial\Omega}=f}. The Electric Impedance Tomography Problem is to determine Ω from Λg{\Lambda_{g}}. A criterion is proposed that enables one to detect (via Λg{\Lambda_{g}}) whether Ω is orientable or not. The algebraic version of the BC-method is applied to solve the EIT problem for the Moebius band. The main instrument is the algebra of holomorphic functions on the double covering 𝕄{{\mathbb{M}}} of M , which is determined by Λg{\Lambda_{g}} up to an isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy (M′,g′){(M^{\prime},g^{\prime})} of (M,g){(M,g)}. This copy is conformally equivalent to the original, provides ∂⁡M′=∂⁡M{\partial M^{\prime}=\partial M}, Λg′=Λg{\Lambda_{g^{\prime}}=\Lambda_{g}}, and thus solves the problem.

If the Radon transform of a compactly supported distribution f≠0{f\neq 0} in ℝn{\mathbb{R}^{n}} is supported on the set of tangent planes to the boundary ∂⁡D{\partial D} of a bounded convex domain D , then ∂⁡D{\partial D} must be an ellipsoid. The special case of this result when the domain D is symmetric was treated in [J. Boman, A hypersurface containing the support of a Radon transform must be an ellipsoid. I: The symmetric case, J. Geom. Anal. 2020, 10.1007/s12220-020-00372-8]. Here we treat the general case.

We introduce numerical algorithms for solving the inverse and direct scattering problems for the Manakov model of vector nonlinear Schrödinger equation. We have found an algebraic group of 4-block matrices with off-diagonal blocks consisting of special vector-like matrices for generalizing the scalar problem’s efficient numerical algorithms to the vector case. The inversion of block matrices of the discretized system of Gelfand–Levitan–Marchenko integral equations solves the inverse scattering problem using the vector variant the Toeplitz Inner Bordering algorithm of Levinson’s type. The reversal of steps of the inverse problem algorithm gives the solution of the direct scattering problem. Numerical tests confirm the proposed vector algorithms’ efficiency and stability. We also present an example of the algorithms’ application to simulate the Manakov vector solitons’ collision.

We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles, the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node takes place only up to a logarithmic factor and the notion of smoothness. By using mini-batching technique, we show that the proposed methods with stochastic oracle can be additionally parallelized at each node. The considered algorithms can be applied to many data science problems and inverse problems.

The inverse wave problem of tomographic type is considered. It consists in reconstruction of several scatterer’s characteristics in the form of spatial distributions for sound speed, medium density, absorption coefficient and power index of its frequency dependence, as well as vector of flow velocity. In the form of a survey material (based on several publications), a sequence of steps is discussed that leads to reconstruction of each individual spatial distribution in the presence of different combinations of the mentioned characteristics. The minimum number of frequencies required for reconstruction is discussed when the complete set of scattering data is available at each of the frequencies. For the first time, two possible approaches to reconstruct the scatterer characteristics in the presence of inhomogeneous spatial distributions of the density and the flow velocity vector are compared, and attention is drawn to the perspectives of reconstruction by functional algorithms in this case. The possibility of separating the sought spatial distributions during the inverse problem solution is illustrated by numerical modeling.

We prove Hölder-logarithmic stability estimates for the problem of finding an integrable function v on ℝd{{\mathbb{R}}^{d}} with a super-exponential decay at infinity from its Fourier transform ℱ⁢v{\mathcal{F}v} given on the ball Br{B_{r}}. These estimates arise from a Hölder-stable extrapolation of ℱ⁢v{\mathcal{F}v} from Br{B_{r}} to a larger ball. We also present instability examples showing an optimality of our results.

In this work we investigate numerically the reconstruction approach proposed in [F. O. Goncharov and R. G. Novikov, An analog of Chang inversion formula for weighted Radon transforms in multidimensions, Eurasian J. Math. Comput. Appl. 4 2016, 2, 23–32] for weighted ray transforms (weighted Radon transforms along oriented straight lines) in 3D. In particular, the approach is based on a geometric reduction of the data modeled by weighted ray transforms to new data modeled by weighted Radon transforms along two-dimensional planes in 3D. Such reduction could be seen as a preprocessing procedure which could be further completed by any preferred reconstruction algorithm. In a series of numerical tests on modelized and real SPECT (single photon emission computed tomography) data we demonstrate that such procedure can significantly reduce the impact of noise on reconstructions.

The paper is devoted to numerical research of the medical ultrasound tomography problem. This problem consists in finding small inclusions in the breast tissue by boundary measurements of the acoustic waves generated by sources located on the boundary. For medical diagnostic, it is important to recover the image of the acoustical medium and to determine the values of velocity, attenuation and density. In the paper, we describe a numerical experiment of visualization of several inclusions using the energy version of the reverse time migration (RTM). Certainly, the RTM image does not separate velocity and attenuation inclusions. However, kinematic and amplitude analysis gives a possibility to estimate values of the velocity and attenuation. As a result, we split the RTM image into two ones. Note that in this paper we consider the scalar value of sound velocity.

An inverse extremum problem (optimal control problem) for a quasi-linear radiative-conductive heat transfer model of endovenous laser ablation is considered. The problem is to find the powers of the source spending on heating the fiber tip and on radiation. As a result, it provides a given temperature distribution in some part of the model domain. The unique solvability of the initial-boundary value problem is proved, on the basis of which the solvability of the optimal control problem is shown. An iterative algorithm for solving the optimal control problem is proposed. Its efficiency is illustrated by a numerical example.