Inverse Boundary Value Problem Research Articles

Domain sampling methods for an inverse boundary value problem of the heat equation

Abstract We investigate the reconstruction of an unknown cavity inside a heat conductor with only single boundary measurement, which is a typical non-destructive testing of defects in material science using thermography. Mathematically, it can be formulated as an inverse boundary value problem for the heat equation. Two domain sampling methods, i.e., the range test (RT) and no-response test (NRT), are developed to realize the reconstruction. We first justify the convergence of the RT, which is based on the analytical extension property of solutions to the heat equation. Then we prove the duality of the RT and NRT by looking at the relation between their indicator functions. Consequently, the unknown cavity can be reconstructed by using either the RT or NRT if the solution associated to the single measurement cannot be analytically extended across its boundary. Next, we design two efficient algorithms for the RT and NRT to reconstruct the unknown cavity numerically. Each algorithm consists of two steps. The first step is to determine an approximation of the unknown cavity by implementing the sampling methods in a coarse grid, while the second step is to improve the reconstruction by using a fine grid in the approximate domain. The numerical results suggest that both algorithms can generate some rough information on the cavity with only one boundary measurement. Finally, by using this information as an initial guess for Newton's method, we improve the accuracy of the mentioned reconstruction results.

An optimal transport approach for 3D electrical impedance tomography

Abstract This work solves the three-dimensional inverse boundary value problem with the quadratic Wasserstein distance (W2), which originates from the optimal transportation (OT) theory. The computation of the W2 distance on the manifold surface is boiled down to solving the generalized Monge-Ampère equation, whose solution is directly related to the gradient of the W2 distance. An efficient first-order method based on iteratively solving Poisson's equation is introduced to solve the fully nonlinear elliptic equation. Combining with the adjoint-state technique, the optimization framework based on the W2 distance is developed to solve the three-dimensional electrical impedance tomography (EIT) problem. The proposed method is especially suitable for severely ill-posed and highly nonlinear inverse problems. Numerical experiments demonstrate that our method improves the stability and outperforms the traditional regularization methods.

Fading regularization method for an inverse boundary value problem associated with the biharmonic equation

In this paper, we propose a numerical algorithm that combines the fading regularization method with the method of fundamental solutions (MFS) to solve a Cauchy problem associated with the biharmonic equation. We introduce a new stopping criterion for the iterative process and compare its performance with previous criteria. Numerical simulations using MFS validate the accuracy of this stopping criterion for both compatible and noisy data and demonstrate the convergence, stability, and efficiency of the proposed algorithm, as well as its ability to deblur noisy data.

Solving inverse boundary value problem of Poisson equation by LS-SVM

Abstract In this paper, a new method based on least squares support vector machines (LS-SVM) is presented for solving the inverse boundary value problem of Poisson equation. The closed form analytical solution is obtained by optimizing the regression parameters. The core problem is to transform the parametric regression problem into a quadratic programming problem. To demonstrate the efficiency of the proposed algorithm, numerical experiments are conducted. The proposed method is found to be feasible for the inverse boundary value problem of Poisson equation.

Linearized inverse problem for biharmonic operators at high frequencies

In this paper, we study the phenomenon of increasing stability in the inverse boundary value problems for the biharmonic equation. The inverse problem is to determine the potential from DtN map. It is a kind of nonlinear inverse problem. By considering a linearized form, we obtain an increasing Lipschitz‐like stability when is large. Furthermore, we extend the discussion to the linearized inverse biharmonic potential problem with attenuation, where an exponential dependence of the attenuation constant is traced in the stability estimate.

High‐frequency stability estimates for the linearized inverse boundary value problem for the biharmonic operator with attenuation on some bounded domains

In this article, high‐frequency stability estimates are explored for the determination of the zeroth‐order perturbation of the biharmonic operator with constant attenuation from the linearized partial Dirichlet‐to‐Neumann map when part of the boundary is inaccessible and flat. The results obtained suggest improvement of the stability with an appropriate choice of frequency.

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

In this article, the solvability of an inverse boundary problem for the 2D equation of transverse oscillation of the rod from knowledge of additional condition is studied. To investigate the solvability of the inverse problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. Using the contraction mappings principle existence and uniqueness of the solution of an equivalent problem is proved. Further, using the equivalency, the existence and uniqueness theorems of the classical solution of the original problem are proved.

FDM-based alternating iterative algorithms for inverse BVPs in 2D steady-state anisotropic heat conduction with heat sources

In the framework of steady-state inhomogeneous anisotropic heat conduction with heat sources, we study the convergent and stable numerical reconstruction of the missing boundary conditions (temperatures and normal heat fluxes) on an inaccessible boundary and the thermal field in a 2D solid from the knowledge of over-prescribed data on an accessible portion of the boundary and temperature data on the remaining boundary. This inverse boundary value problem is approached by employing the alternating iterative algorithms of Kozlov et al. (1991), in conjunction with the standard finite-difference method and a novel modified one, for both exact and perturbed data. For noisy data available on the over-prescribed boundary, the numerical solution is retrieved by using three regularising/stabilising stopping criteria. Numerical results are presented for 2D simply and doubly connected domains bounded by a (piecewise) smooth curve, as well as exact and noisy boundary data, whilst appropriate errors in the fractional Sobolev spaces corresponding to the boundary temperatures and normal heat fluxes, respectively, are computed efficiently.

Nonintrusive thermal contact conductance estimation in double-layered pipelines: A reciprocity functional method perspective

Thermal contact conductance (TCC) is a critical factor in various engineering applications involving heat transfer between solid surfaces. Accurate knowledge of TCC enables a more precise thermal system analysis, leading to improved efficiency, reduced energy consumption, and prevention of thermal damages. Moreover, TCC estimation can help to identify material discontinuities or failures. A recent tool for TCC estimation is the Reciprocity Functional Method (RFM) coupled with the Classical Integral Transform Technique (CITT). This method allows the solution of inverse boundary value problems without the need of iterative methods or intrusive measurements, making it valuable for nondestructive testing of interfacial flaws between solid materials. Furthermore, it enables the computation of TCCs using analytical expressions, resulting in a computationally fast procedure. The technique involves the solution of two auxiliary problems: one for the temperature discontinuity estimation, and the other for the interfacial heat flux estimation, both at the inaccessible interface between the two contacting materials. In this study, we applied the RFM and the CITT to estimate different types of TCCs, using a three-dimensional double-layer hollow cylinder geometry as a representative model, like double-layer pipelines used in the oil industry. Temperature measurements for solving the inverse problem were considered available on the outer surface of the pipe which, in the case of a real experiment, could be obtained via an infrared camera. The results demonstrated a good estimate for various analyzed test-cases, even when the measurements were contaminated with Gaussian noises.

Inversion of the attenuated momenta ray transform of planar symmetric tensors

We present a reconstruction method that stably recovers the real valued, symmetric tensors compactly supported in the Euclidean plane, from knowledge of their attenuated momenta ray transform. The problem is recast as an inverse boundary value problem for a system of transport equations, which we solve by an extension of Bukhgeim’s A-analytic theory. The method of proof is constructive. To illustrate the reconstruction method, we present results obtained in the numerical implementation for the non-attenuated case of one-tensors.

Training RBF neural networks for solving nonlinear and inverse boundary value problems

Radial basis function neural networks (RBFNN) have been increasingly employed to solve boundary value problems (BVPs). In the current study, we propose such a technique for nonlinear (apparently for the first time) elliptic BVPs of orders two and four in 2-D and 3-D. The method is also extended, in a natural way, to solving 2-D and 3-D inverse BVPs. The RBFNN is trained via the least squares minimization of a nonlinear functional using the MATLAB®routine lsqnonlin. In this way, as well as the solution, appropriate values of the RBF approximation parameters are automatically delivered. The efficacy of the proposed RBFNN is demonstrated through several numerical experiments.

Stability estimates for an inverse boundary value problem for biharmonic operators with first order perturbation from partial data

In this paper we study an inverse boundary value problem for the biharmonic operator with first order perturbation. Our geometric setting is that of a bounded simply connected domain in the Euclidean space of dimension three or higher. Assuming that the inaccessible portion of the boundary is flat, and we have knowledge of the Dirichlet-to-Neumann map on the complement, we prove logarithmic type stability estimates for both the first and the zeroth order perturbation of the biharmonic operator.

On an inverse boundary-value problem for the pseudohyperbolic equation with nonclassical boundary conditions

In this paper, we consider an inverse boundary-value problem for a fourth-order pseudohyperbolic equation with nonclassical boundary conditions. The primary purpose of the work is to study the existence and uniqueness of the classical solution of the considered inverse boundary-value problem. To investigate the solvability of the considered problem, we carried out a transformation from the original problem to some auxiliary equivalent problem with trivial boundary conditions. Furthermore, we prove the existence and uniqueness theorem for the auxiliary problem by the contraction mappings principle. Based on the equivalency of these problems, the existence and uniqueness of the classical solution of the original problem are shown.

INVERSE BOUNDARY PROBLEM FOR THE EQUATION OF FORCED VIBRATIONS OF A CANTILEVER BEAM WITH INTEGRAL CONDITIONS

The paper studies the solvability of an inverse boundary value problem with an unknown time-dependent coefficient for the equation of forced vibrations of a cantilever beam with the integral. Bending transverse vibrations of a homogeneous beam under the action of an external force in the absence of rotational motion during bending are described by a fourth-order differential equation. The purpose of the work is to determine the unknown coefficient and solve the problem under consideration. THIS problem under consideration is reduced to an auxiliary equivalent problem. Next, the existence and uniqueness of a solution to the equivalent problem is proved using the contraction mapping principle. As a result, using equivalence, the uniqueness of the existence of the classical solution is proved.

Increasing stability of a linearized inverse boundary value problem for a nonlinear Schrödinger equation on transversally anisotropic manifolds

We consider the problem of recovering a nonlinear potential function in a nonlinear Schrödinger equation on transversally anisotropic manifolds from the linearized Dirichlet-to-Neumann map at a large wavenumber. By calibrating the complex geometric optics solutions according to the wavenumber, we prove the increasing stability of recovering the coefficient of a cubic term as the wavenumber becomes large.

SIMULTANEOUS IDENTIFICATION OF THE RIGHT-HAND SIDE AND TIME-DEPENDENT COEFFICIENTS IN A TWO-DIMENSIONAL PARABOLIC EQUATION

This paper investigates the simultaneous identification of time-dependent lowest and source terms in a two-dimensional (2D) parabolic equation from the additional measurements. To investigate the solvability of the inverse problem, we first examine an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. Then, applying the contraction mappings principle existence and uniqueness of the solution of an equivalent problem is proved. Furthermore, using the equivalency, the existence and uniqueness theorem for the classical solution of the original problem is obtained and some discussions on the numerical solutions for this inverse problem are presented including numerical examples.

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.

Mathematical Modeling on a Physics-Informed Radial Basis Function Network

The article is devoted to approximate methods for solving differential equations. An approach based on neural networks with radial basis functions is presented. Neural network training algorithms adapted to radial basis function networks are proposed, in particular adaptations of the Nesterov and Levenberg-Marquardt algorithms. The effectiveness of the proposed algorithms is demonstrated for solving model problems of function approximation, differential equations, direct and inverse boundary value problems, and modeling processes in piecewise homogeneous media.

Increasing Stability in an Inverse Boundary Value Problem—Bayesian Viewpoint

Increasing Stability in an Inverse Boundary Value Problem—Bayesian Viewpoint

On determining and breaking the gauge class in inverse problems for reaction-diffusion equations

Abstract We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well-known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on ${\mathbb R}^n$ . Moreover, we present cases where it is possible to exploit the nonlinear interaction to break the gauge symmetry. This leads to full determination results of the nonlinear term. As an application, we show that it is possible to give a full resolution to classes of inverse source problems of determining a source term and nonlinear terms simultaneously. This is in strict contrast to inverse source problems for corresponding linear equations, which always have the gauge symmetry. We also consider a Carleman estimate with boundary terms based on intrinsic properties of parabolic equations.