One-dimensional Inverse Problem Research Articles

A collocation heat polynomials method for one-dimensional inverse Stefan problems

The inverse one-phase Stefan problem in one dimension, aimed at identifying the unknown time-dependent heat flux P(t) with a known moving boundary position s(t), is investigated. A previous study (Kassabek et al., 2021) attempted to reconstruct the unknown heat flux P(t) using the Variational Heat Polynomials Method (VHPM). In this paper, we develop the Collocation Heat Polynomials Method (CHPM) for the reconstruction of the time-dependent heat flux P(t). This method constructs an approximate solution as a linear combination of heat polynomials, which satisfies the heat equation, with the coefficients determined using the collocation method. To address the resulting ill-posed problem, Tikhonov regularization is applied. As an application, we demonstrate the effectiveness of the method on benchmark problems. Numerical results show that the proposed method accurately reconstructs the time-dependent heat flux P(t), even in the presence of significant noise. The results are also compared with those obtained in Kassabek et al. (2021) using the VHPM.

Application of inverse heat transfer to fatigue fracture

Cyclic actuation tends to cause self-heating in the material as the structure experiences fatigue, where the movement and coalescence of defects lead to crack formation, propagation, and eventual fracture. This study explores the modeling aspects of the self-heating phenomenon using a one-dimensional inverse heat problem to analyze heat generation and dissipation to quantify the plastic work rates needed for predicting fatigue life. The approach is based on analyzing the surface thermography obtained using an infrared camera and numerically solving an inverse Fourier heat conduction equation. Formulation of the inverse problem via constrained optimization and method of solution for analyzing the fatigue behavior of CS 1018 flat dog bone specimens subjected to fully reversed bending fatigue are presented. The proposed model demonstrates superior accuracy in predicting the location of maximum heat generation and identifying potential fracture zones compared to traditional methods.

An one-dimensional inverse problem for the wave equation

For the wave equation with inhomogeneity σ(x)um t + q(x)up a forward and an one-dimensional inverse problems are studied. Here m > 1 and p > 1 are real numbers. The forward problem is considered in the domain x > 0,t > 0 with zero initial data and Dirichlet boundary condition at x = 0. An unique solvability theorem of this problem is proved. The inverse problem is devoted to determining the coefficients σ(x) and q(x). As an additional information for recovering this coefficients, two forward problems with different Dirichlet data are considered and traces of the derivative of their solutions with respect to x are given at x =0 on a finite interval. For the inverse problem a local existence and uniqueness theorem is established.

AN INVERSE PROBLEM FOR A NONLINEAR HYPERBOLIC EQUATION

For a second-order hyperbolic equation with inhomogeneity |u|m−1u, m > 1, a forward and an one-dimensional inverse problems are studied. The inverse problem is devoted to determining the coefficient under heterogeneity. As an additional information, the trace of the derivative with respect to x of the solution to the forward initial-boundary value problem is given at x = 0 on a finite interval. Conditions for the unique solvability of the forward problem are found. For the inverse problem a local existence and uniqueness theorems are established and a stability estimate of its solutions is found.

Solvability of a One-Dimensional Dynamic Direct and Inverse Problem for a System of Hopf Type Equations

Solvability of a One-Dimensional Dynamic Direct and Inverse Problem for a System of Hopf Type Equations

One-Dimensional Inverse Problem for Nonlinear Equations of Electrodynamics

For the system of nonlinear electrodynamics equations, we consider the problem of determining the medium conductivity coefficient multiplying the nonlinearity. It is assumed that the permittivity and permeability are constant and the conductivity depends only on one spatial variable, with this conductivity being zero on the half-line x . For a mode in which only two electromagnetic field components participate, the wave propagation process caused by the incidence of a plane wave with a constant amplitude from the domain x0 onto an inhomogeneity localized on the half-line x0 is considered. With a given conductivity coefficient, the conditions for the solvability of the direct problem and the properties of its solution are studied. To solve the inverse problem, the trace of the electrical component of the solution of the direct problem is specified on a finite segment of the axis x=0. A theorem on the local existence and uniqueness of the solution of the inverse problem is established, and a global estimate of the conditional stability of its solutions is found.

One-Dimensional Inverse Problem for Nonlinear Equations of Electrodynamics

One-Dimensional Inverse Problem for Nonlinear Equations of Electrodynamics

One-dimensional inverse problems of determining the kernel of the integro-differential heat equation in a bounded domain

Abstract The integro-differential equation of heat conduction with the time-convolution integral on the right side is considered. The direct problem is the initial-boundary problem for this integro-differential equation. Two inverse problems are studied for this direct problem consisting in determining a kernel of the integral member on two given additional conditions with respect to the solution of the direct problems, respectively. The problems are replaced with the equivalent system of the integral equations with respect to unknown functions and on the basis of contractive mapping the unique solvability inverse problem.

Solutions of one-dimensional inverse heat conduction problems: a review

Estimations using the inverse conduction approach to predict temperature and heat flux at the exposed surface lead to indirect measurements away from the exposed surface within the solid. The approach is extremely useful when access to direct measurements is not possible due to various working conditions, and thereby provides estimates without disturbing the flow under the real flow condition over the surface. The approach is useful not only for heat transfer applications but also for numerous engineering applications, including fluid mechanics and furnace applications. The approach requires the time history of effective parameters at the strategic locations away from the exposed surface to be known. In the present paper, a review of sequential development of various inverse heat conduction methods is presented to get solutions in different geometries.

Shannon-Taylor technique for solving one-dimensional inverse heat conduction problem

The Shannon-Taylor interpolation technique was introduced by Butzer and Engels in 1983. In this work, the sinc-function is replaced by a Taylor approximation polynomial. In this work, we implement the Shannon-Taylor approximations to solve a one-dimensional heat conduction problem. One of the major advantages of this approach is that the resulting linear system of equations of the approximation procedure has an explicit coefficient matrix. This is not the case of the classical sinc methods due to finite integrals involving e^{-x^2}. We establish rigorous error estimates involving an additional Taylor’s series tail. Numerical illustrations are depicted.

Weighted Radial Basis Collocation Method for the Nonlinear Inverse Helmholtz Problems

In this paper, a meshfree weighted radial basis collocation method associated with the Newton’s iteration method is introduced to solve the nonlinear inverse Helmholtz problems for identifying the parameter. All the measurement data can be included in the least-squares solution, which can avoid the iteration calculations for comparing the solutions with part of the measurement data in the Galerkin-based methods. Appropriate weights are imposed on the boundary conditions and measurement conditions to balance the errors, which leads to the high accuracy and optimal convergence for solving the inverse problems. Moreover, it is quite easy to extend the solution process of the one-dimensional inverse problem to high-dimensional inverse problem. Nonlinear numerical examples include one-, two- and three-dimensional inverse Helmholtz problems of constant and varying parameter identification in regular and irregular domains and show the high accuracy and exponential convergence of the presented method.

One-Dimensional Inverse Problem of Passive Acoustic Thermometry Using the Heat Conductivity Equation: Computer and Physical Simulation

An algorithm for reconstructing the time-varying one-dimensional distribution of the deep temperature of the human body under local heating is proposed and experimentally tested on a model. The algorithm requires that the temperature obey the heat conduction equation, the integration of which with a weight that takes into account absorption in the object, makes it possible to obtain the time dependence of the acoustic brightness temperature (measured signal), which in turn is determined by the parameters of the equation. The desired temperature is obtained by solving the heat conduction equation with the found parameters. The algorithm reconstructs two parameters: blood flow and the amplitude of the heating source, which are not determined each time anew, but only refined. In this case, the integration time increases, but the temporal resolution does not suffer: new results can be obtained after any period of time. After 2 min of heating, it is possible to reconstruct the temperature and size of the heated region with an accuracy acceptable for medical applications: 0.5°C and 0.5 mm, respectively.

Solution of Inverse Problem of Non-Stationary Heat Conduction Using a Laplace Transform

This paper presents some solutions to the one-dimensional Cauchy-type transient inverse heat conduction problem. Based on known courses of temperature and the heat flux on one of the boundaries of the region, courses of temperature and heat flux on the opposite boundary of the region were determined with the use of the Laplace’s transform. Solution of the inverse problem does not require determination of temperature distribution on subsequent time moments inside the region. Stability of the solution to the inverse problem of the Cauchy type was investigated by means of determining the minimal time step ensuring the stability of the solution. Two examples of one-dimensional transient inverse problems were solved. The first one is related to the reconstruction of shock temperature change on the boundary of the region based on known courses of temperature and heat flux on the opposite boundary. The second one concerned the reconstruction of the heat flux acting on one of the boundaries through a period of time. In both considered problems the reconstructed boundary conditions were discontinuous. Stable results were obtained for both investigated examples, however, the stability of the inverse problem depended on the length of time step.

A regularization method for inverse heat transfer problems using dynamic Bayesian networks with variable structure

In this paper, dynamic Bayesian networks (DBNs) were employed to solve one-dimensional inverse heat transfer problems (IHTPs) with temperature history data. Temperature-dependent conductivities and surface heat flux histories were discretized into a finite number of parameters and used as parent nodes in the DBN. Parameter sensitivity analyses were adopted to identify the dominant parameters at each time step. Non-dominant parameters were removed from the corresponding frames for the sparsification of the DBN, which reduces the number of parameters to be identified simultaneously. Thus, the structure of the DBN was dynamically optimized for the regularization of inverse problems. Numerical and experimental tests for IHTPs were conducted to validate the proposed method. Both thermal conductivities and surface heat flux were identified with relatively small error, even if considering the measurement error. Sensitivities of the remaining parameters increased significantly with the parameter identification process, and insensitive parameters in the initial DBN frames were also effectively identified in the subsequent frames. The accuracy of parameter identification can be improved, and the cost of the parameter sampling combination can be reduced. The proposed method is not subject to heat transfer problems and can be used for a wide range of applications.

A computational source modeling of brain activity: An inverse problem

Source determination in an inverse problem from the over-specified data plays a crucial role in cognitive modeling. In this paper, an accurate and fast method is proposed for solving the one-dimensional inverse problem concerning diffusion equation with source control parameter. The proposed method is based on applying a compact finite difference scheme for spatial components and solving the system that arose from this scheme by multigrid.

One-Dimensional Inverse Problems of Finding the Kernel of Integrodifferential Heat Equation in a Bounded Domain

One-Dimensional Inverse Problems of Finding the Kernel of Integrodifferential Heat Equation in a Bounded Domain

Gelfand-Levitan-Krein method in one-dimensional elasticity inverse problem

In this article we propose the numerical solution of the one dimensional inverse coefficient problem for seismic equation. We use a dynamical version of Gelfand-Levitan-Krein approach for reducing a nonlinear inverse problem for recovering the shear wave’s velocity and the density of the medium to two sequences of the linear integral equations. We propose numerical algorithm for solving these equations based on a fast inversion of a Toeplitz matrix. The proposed numerical methods base on the structure of the problem and therefore improve the efficiency of the algorithms, compared with standard approaches. We present numerical results for solving considered integral equations.

Some inverse problems for the Burgers equation and related systems

In this article we deal with one-dimensional inverse problems concerning the Burgers equation and some related nonlinear systems (involving heat effects and/or variable density). In these problems, the goal is to find the size of the spatial interval from some appropriate boundary observations of the solution. Depending on the properties of the initial and boundary data, we prove uniqueness and non-uniqueness results. In addition, we also solve some of these inverse problems numerically and compute approximations of the interval sizes.

Simulation study for cross-sectional absorption distribution in turbid medium using spatially resolved backscattered light with lateral scanning and one-dimensional solution of nonlinear inverse problem

For a practical technique of cross-sectional imaging of animal bodies, we developed a new method using spatially resolved backscattered light. This method is based on the solution of the one-dimensional nonlinear inverse problem, and on the lateral scan of the source–detector pair along the object surface. Using this method, unknown variables in inverse problems can be reduced more greatly than when using conventional methods. A stable solution for the inverse problem becomes possible. The possibility of using the proposed method was assessed using simulation analysis. The results verified that cross-sectional imaging from several to 10 millimeter depths is possible for animal tissue. This analysis clarified the specific spatial resolution and accuracy in the estimated absorption coefficient. Distortionless imaging was confirmed. Results suggest that the proposed method represents new options as a stable and practical method for biological cross-sectional imaging.

Influence of Errors in Known Constants and Boundary Conditions on Solutions of Inverse Heat Conduction Problem

This work examines the effects of the known boundary conditions on the accuracy of the solution in one-dimensional inverse heat conduction problems. The failures in many applications of these problems are attributed to inaccuracy of the specified constants and boundary conditions. Since the boundary conditions and material properties in most thermal problems are imposed with uncertainty, the effects of their inaccuracy should be understood prior to the inverse analyses. The deviation from the exact solution has been examined for each case according to the errors in material properties, boundary location, and known boundary conditions. The results show that the effects of such errors are dramatic. Based on these results, the applicability and limitations of the inverse heat conduction analyses have been evaluated and discussed.