Volume 32, Issue 2

In this paper, we consider the inverse scattering problem of identifying a two-layered cavity by internal acoustic measurements under the condition that the interior interface has a mixed transmission boundary condition. We focus on the mathematical analysis of recovering the shape of the interior interface by using the linear sampling method, including reconstructing the surface conductivity by the same measurements.

In this paper, we address stability results in determining the time-dependent scalar and vector potentials appearing in the convection-diffusion equation from the knowledge of the Cauchy data set. We prove Hölder-type stability estimates. The key tool used in this work is the geometric optics solution.

The turbulent exchange in boundary layer models is usually characterized by a scalar eddy viscosity coefficient assumed to be a positive function of the vertical variable. We introduce a more general form for the turbulence exchange description, which includes two functions that describe the turbulence without any assumption about their positivity. We construct a model of the Akerblom–Ekman type, but with a complex coefficient of turbulent exchange. The basic quality criterion for these models and algorithms is the maximal agreement with meteorological observations. We optimize the agreement between the global meteorological archive of high-resolution wind observations that are provided by World Meteorological Organization (WMO) in Binary Universal Form for the Representation (BUFR). The main result of our work is that agreement between model solutions and observations will be much better if the turbulent exchange coefficient is optimized in the space of all complex-valued functions, and not limited to the cone of real positive functions.

Recently in [V. Markaki, D. Kourounis and A. Charalambopoulos, A dual self-monitored reconstruction scheme on the TV \mathrm{TV} -regularized inverse conductivity problem, IMA J. Appl. Math. 86 2021, 3, 604–630], a novel reconstruction scheme has been developed for the solution of the inclusion problem in the inverse conductivity problem on the basis of a weighted self-guided regularization process generalizing the total variation approach. The present work extends this concept by implementing and investigating its applicability in the two-dimensional elasticity setting. To this end, we employ the framework of the reconstruction of linear and isotropic elastic structures described by their Lamé parameters. Numerical examples of increasingly challenging geometric complexities illustrate the enhanced accuracy and efficiency of the method.

This paper is concerned with the inverse scattering of the rough surfaces with multi-frequency phase and phaseless measurements. We present a high-order recursive iteration method based on the homotopy iteration technique to reconstruct the rough surfaces. The convergence for the multi-frequency homotopy iteration method is obtained under some conditions. Some numerical experiments show the effectiveness of the proposed algorithm.

In this paper, we present a new methodology, based on the inverse problem approach, for the determination of an unknown shear force acting on the inaccessible tip of the microcantilever, which is a key component of transverse dynamic force microscopy (TDFM). The mathematical modelling of this phenomenon leads to the inverse problem of determining the shear force g ⁢ ( t ) {g(t)} acting on the inaccessible boundary x = ℓ {x=\ell} in a system governed by the variable coefficient Euler–Bernoulli equation ρ A ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x + κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ⁢ x = 0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ) , \rho_{A}(x)u_{tt}+\mu(x)u_{t}+(r(x)u_{xx}+\kappa(x)u_{xxt})_{xx}=0,\quad(x,t)% \in(0,\ell)\times(0,T), subject to the homogeneous initial conditions and the boundary conditions u ⁢ ( 0 , t ) = u 0 ⁢ ( t ) , u x ⁢ ( 0 , t ) = 0 , ( u x ⁢ x ⁢ ( x , t ) + 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)=u_{0}(t),\quad u_{x}(0,t)=0,\quad(u_{xx}(x,t)+\kappa(x)u_{xxt})_{x=\ell% }=0,\quad\bigl{(}-(r(x)u_{xx}+\kappa(x)u_{xxt})_{x}\bigr{)}_{x=\ell}=g(t), from the final time measured output (displacement) u T ⁢ ( x ) := u ⁢ ( x , T ) {u_{T}(x):=u(x,T)} . We introduce the input-output map ( Φ ⁢ g ) ⁢ ( x ) := u ⁢ ( x , T ; g ) {(\Phi g)(x):=u(x,T;g)} , g ∈ 𝒢 {g\in\mathcal{G}} , and prove that it is a compact and Lipschitz continuous linear operator. Then we introduce the Tikhonov functional J ⁢ ( F ) = 1 2 ⁢ ∥ Φ ⁢ g - u T ∥ L 2 ⁢ ( 0 , ℓ ) 2 J(F)=\frac{1}{2}\lVert\Phi g-u_{T}\rVert_{L^{2}(0,\ell)}^{2} and prove the existence of a quasi-solution of the inverse problem. We derive a gradient formula for the Fréchet gradient of the Tikhonov functional through the corresponding adjoint problem solution and prove that it is a Lipschitz continuous functional. The results of the numerical experiments clearly illustrate the effectiveness and feasibility of the proposed approach.

We propose an inverse problem of determining the mechanical and variable-order parameters of the Euler–Bernoulli beam on viscoelastic foundation. For this goal, we develop a fully-discrete Hermite finite element scheme for this model and analyze the corresponding error estimates. The Levenberg–Marquardt method is then applied to determine the multiple parameters. Extensive numerical experiments are performed under practical settings to demonstrate the behavior of the proposed model and the efficiency of the algorithm.

The second-order mean field games system (MFGS) in a bounded domain with the lateral Cauchy data are considered. This means that both Dirichlet and Neumann boundary data for the solution of the MFGS are given. Two Hölder stability estimates for two slightly different cases are derived. These estimates indicate how stable the solution of the MFGS is with respect to the possible noise in the lateral Cauchy data. Our stability estimates imply uniqueness. The key mathematical apparatus is the apparatus of two new Carleman estimates.

This paper presents classification and analysis of the mathematical models of the spread of COVID-19 in different groups of population such as family, school, office (3–100 people), town (100–5000 people), city, region (0.5–15 million people), country, continent, and the world. The classification covers major types of models (time-series, differential, imitation ones, neural networks models and their combinations). The time-series models are based on analysis of time series using filtration, regression and network methods. The differential models are those derived from systems of ordinary and stochastic differential equations as well as partial differential equations. The imitation models include cellular automata and agent-based models. The fourth group in the classification consists of combinations of nonlinear Markov chains and optimal control theory, derived by methods of the mean-field game theory. COVID-19 is a novel and complicated disease, and the parameters of most models are, as a rule, unknown and estimated by solving inverse problems. The paper contains an analysis of major algorithms of solving inverse problems: stochastic optimization, nature-inspired algorithms (genetic, differential evolution, particle swarm, etc.), assimilation methods, big-data analysis, and machine learning.