Inverse Boundary Problem Research Articles

Numerical Solution of the Cauchy Problem for the Helmholtz Equation Using Nesterov’s Accelerated Method

In this paper, the Cauchy problem for the Helmholtz equation, also known as the continuation problem, is considered. The continuation problem is reduced to a boundary inverse problem for a well-posed direct problem. A generalized solution to the direct problem is obtained and an estimate of its stability is given. The inverse problem is reduced to an optimization problem solved using the gradient method. The convergence of the Landweber method with respect to the functionals is compared with the convergence of the Nesterov method. The calculation of the gradient in discrete form, which is often used in the numerical solutions of the inverse problem, is described. The formulation of the conjugate problem in discrete form is presented. After calculating the gradient, an algorithm for solving the inverse problem using the Nesterov method is constructed. A computational experiment for the boundary inverse problem is carried out, and the results of the comparative analysis of the Landweber and Nesterov methods in a graphical form are presented.

Inverse free boundary problem for degenerate parabolic equation

Inverse free boundary problem for degenerate parabolic equation

On the solvability of an inverse boundary problem for the equation of transverse oscillation of the rod

On the solvability of an inverse boundary problem for the equation of transverse oscillation of the rod

Partial data inverse problems for magnetic Schrödinger operators on conformally transversally anisotropic manifolds

We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n ⩾ 3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.

Inverse problems for real principal type operators

abstract: We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determination methods for general operators, and prove global uniqueness results for determining coefficients in nonlinear real principal type equations. The article presents a unified approach for treating inverse boundary problems for transport and wave equations, and highlights the role of propagation of singularities in the solution of related inverse problems.

Determining a nonlinear hyperbolic system with unknown sources and nonlinearity

AbstractThis paper is devoted to some inverse boundary problems associated with a time‐dependent semilinear hyperbolic equation, where both nonlinearity and sources (including initial displacement and initial velocity) are unknown. It is shown in several generic scenarios that one can uniquely determine the nonlinearity and/or the sources by using passive or active boundary observations. In order to exploit the nonlinearity and the sources simultaneously, we develop a new technique, which combines the observability for linear wave equations and an approximation property with higher order linearization for the semilinear hyperbolic equation.

Numerical Identification of Boundary Conditions for Richards’ Equation

A time stepping quasilinearization approach to the mixed (or coupled) form of one and two dimensional Richards’ equations is developed. For numerical solution of the linear ordinary differential equation (ODE) for 1D case and elliptic for 2D case, obtained after this semidiscretization, a finite volume method is used for direct problems arising on each time level. Next, we propose a version of the decomposition method for the numerical solution of the inverse ODE and 2D elliptic boundary problems. Computational results for some soil types and its related parameters reported in the literature are presented.

The Born approximation in the three-dimensional Calderón problem Ⅱ: Numerical reconstruction in the radial case

In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball $ B_R \subset \mathbb{R}^3 $ of radius $ R $, in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for $ L^{\infty} $ conductivities, and present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary $ \partial B_R $. We then investigate how increasing the radius $ R $ affects the quality of the Born approximation, and the existence of a scattering limit as $ R\to \infty $. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator $ -\Delta +q $, and strong recovery of singularity results are observed in this case.

Coefficient inverse problem for anisotropic time-domain wave equation

In this paper, we investigate an inverse problem of determining coefficient via partial boundary data in anisotropic media. More precisely, we study a scheme for asymptotic reconstruction on coefficient inverse problem for anisotropic time-domain wave equation by a geometric control method. As an initial attempt, it seems natural to treat the case of two dimensional Euclidean space in analogy with many other well-known inverse boundary problems.

Optimal transportation for electrical impedance tomography

This work establishes a framework for solving inverse boundary problems with the geodesic-based quadratic Wasserstein distance ( W 2 W_{2} ). A general form of the Fréchet gradient is systematically derived from the optimal transportation (OT) theory. In addition, a fast algorithm based on the new formulation of OT on S 1 \mathbb {S}^{1} is developed to solve the corresponding optimal transport problem. The computational complexity of the algorithm is reduced to O ( N ) O(N) from O ( N 3 ) O(N^{3}) of the traditional method. Combining with the adjoint-state method, this framework provides a new computational approach for solving the challenging electrical impedance tomography problem. Numerical examples are presented to illustrate the effectiveness of our method.

RNN-Attention Based Deep Learning for Solving Inverse Boundary Problems in Nonlinear Marshak Waves

RNN-Attention Based Deep Learning for Solving Inverse Boundary Problems in Nonlinear Marshak Waves

Calculation of Tissue Oxygenation via an Inverse Boundary Problem for Transcutaneous Oxygenation Wearable Applications.

In this article, we present a toolset to fully leverage a previously developed transcutaneous oxygenation monitor (TCOM) wearable technology to accurately measure skin oxygenation values. We describe numerical models and experimental characterization techniques that allow for the extraction of precise tissue oxygenation measurements. The numerical model is based on an inverse boundary problem of the parabolic equation with Dirichlet boundary conditions. To validate this model and characterize the diffusion of oxygen through the oxygen sensing materials, we designed a series of control/calibration experiments modeled after the device's clinical application using oxygenation values in the physiological range expected for healthy tissue. Our results demonstrate that it is possible to obtain accurate tissue pO2 measurements without the need for long equilibration times with a small wearable device.

Inverse boundary problem for a fourth order Boussinesq equation with integral conditions

Inverse boundary problem for a fourth order Boussinesq equation with integral conditions

Transport of Organic Volatiles through Paper: Physics-Informed Neural Networks for Solving Inverse and Forward Problems

Transport of volatile organic compounds (VOCs) through porous media with active surfaces takes place in many important applications, such as in cellulose-based materials for packaging. Generally, it is a complex process that combines diffusion with sorption at any time. To date, the data needed to use and validate the mathematical models proposed in literature to describe the mentioned processes are scarce and have not been systematically compiled. As an extension of the model of Ramarao et al. (Dry Technol 21(10):2007–2056, 2003) for the water vapor transport through paper, we propose to describe the transport of VOCs by a nonlinear Fisher–Kolmogorov–Petrovsky–Piskunov equation coupled to a partial differential equation (PDE) for the sorption process. The proposed PDE system contains specific material parameters such as diffusion coefficients and adsorption rates as multiplication factors. Although these parameters are essential for solving the PDEs at a given time scale, not all of the required parameters can be directly deduced from experiments, particularly diffusion coefficients and sorption constants. Therefore, we propose to use experimental concentration data, obtained for the migration of dimethyl sulfoxide (DMSO) through a stack of paper sheets, to infer the sorption constant. These concentrations are considered as the outcome of a model prediction and are inserted into an inverse boundary problem. We employ Physics-Informed Neural Networks (PINNs) to find the underlying sorption constant of DMSO on paper from this inverse problem. We illustrate how to practically combine PINN-based calculations with experimental data to obtain trustworthy transport-related material parameters. Finally we verify the obtained parameter by solving the forward migration problem via PINNs and finite element methods on the relevant time scale and show the satisfactory correspondence between the simulation and experimental results.

Simultaneous recoveries for semilinear parabolic systems

In this paper, we study inverse boundary problems associated with semilinear parabolic systems in several scenarios where both the nonlinearities and the initial data can be unknown. We establish several simultaneous recovery results showing that the passive or active boundary Dirichlet-to-Neumann operators can uniquely recover both of the unknowns, even stably in a certain case. It turns out that the nonlinearities play a critical role in deriving these recovery results. If the nonlinear term belongs to a general C 1 class but fulfilling a certain growth condition, the recovery results are established by the control approach via Carleman estimates. If the nonlinear term belongs to an analytic class, the recovery results are established through successive linearization in combination with special complex geometrical optics solutions for the parabolic system.

The Born approximation in the three-dimensional Calderón problem

Uniqueness and reconstruction in the three-dimensional Calderón inverse conductivity problem can be reduced to the study of the inverse boundary problem for Schrödinger operators −Δ+q. We study the Born approximation of q in the ball, which amounts to studying the linearization of the inverse problem. We first analyze this approximation for real and radial potentials in any dimension d≥3. We show that this approximation satisfies a closed formula that only involves the spectrum of the Dirichlet-to-Neumann map associated to −Δ+q, which is closely related to a particular moment problem. We then turn to general real and essentially bounded potentials in three dimensions and introduce the notion of averaged Born approximation, which captures the exact invariance properties of the inverse problem. We obtain explicit formulas for the averaged Born approximation in terms of the matrix elements of the Dirichlet to Neumann map in the basis of spherical harmonics. To show that the averaged Born approximation does not destroy information on the potential, we also study the high-energy behavior of the matrix elements of the Dirichlet to Neumann map.

Inverse Boundary Problem for the Two Photon Absorption Transport Equation

We study the inverse boundary problem for the nonlinear two photon absorption radiative transport equation. We show that the absorption coefficients and the scattering coefficient can be uniquely determined from the albedo operator. If the scattering is absent, we do not require smallness of the incoming source, and the reconstruction of the absorption coefficients is explicit.

Effective Medium Theory for Embedded Obstacles in Elasticity with Applications to Inverse Problems

We consider the time-harmonic elastic wave scattering from a general (possibly anisotropic) inhomogeneous medium with an embedded impenetrable obstacle. We show that the impenetrable obstacle can be effectively approximated by an isotropic elastic medium with a particular choice of material parameters. We derive sharp estimates to rigorously verify such an effective approximation. Our study is strongly motivated by the related studies of two challenging inverse elastic problems including the inverse boundary problem with partial data and the inverse scattering problem of recovering mediums with buried obstacles. The proposed effective medium theory readily yields some interesting applications of practical significance to these inverse problems.

Inverse boundary problem in estimating heat transfer coefficient of a round pulsating bubbly jet: design of experiment

ABSTRACT An inverse algorithm is developed to estimate the transient convective coefficient distribution of a pulsating bubbly jet impinging on a cylindrical thermal mass. Cooling of the thermal mass is simulated by solving the two-dimensional transient heat equation in a cylindrical coordinate system using the finite difference method. The sum of squared differences between calculated and measured temperature data is the objective functional. As this is a nonlinear IHCP, the adjoint method is employed to derive the gradient components of the objective function, needed for the implementation of CGM. The procedure is carried out for six gas Reynolds numbers changing continuously through ten seconds of the jet impingement. The inverse scheme is validated using noiseless temperature data. The accuracy and convergence rate of the algorithm is then assessed by simulating the cooling process of thermal mass with four different materials. Moreover, the effects of sensors’ location, as well as initial guess on the estimation, are evaluated. Time-varying heat transfer coefficients are then estimated in the presence of Gaussian noise, having four standard deviations. Results demonstrate that, despite the estimation is affected by the noise level, the adjoint method is most efficient and rapidly convergent for the thermal mass with moderate thermal conductivity having a noisy temperature field of 1 .

A hybrid algorithm for solving inverse boundary problems with respect to intermediate masses on a beam

The inverse problem of determining the weight of three intermediate masses on a uniform beam from the known three natural frequencies has been solved. The performed numerical analysis allows restoring the value of only the second mass in a unique way. The inverse problem of determining the weight of three intermediate masses is solved uniquely except in the case when the first and the third masses are located geometrically symmetric relative to the middle of the beam. The hybrid algorithm for the unique solving inverse problem of determining the weight of three intermediate masses has been developed. The first three natural frequencies of the beam are calculated numerically by using the Maple computer package. Analytical relations between the masses are found.