One-dimensional Inverse Problem Research Articles

Numerical solution of a one-dimensional inverse problem by the discontinuous Galerkin method

In this paper, we combine a Tikhonov regularization with a discontinuous Galerkin method to solve an inverse problem in one-dimension. We show that the regularization is simpler than in the case of the inversion using continuous finite elements. We numerically demonstrate that there exist optimal step sizes and polynomial degrees for inversion using the DG method. Numerical results are compared with those obtained by applying the standard finite element method with B-splines as a basis.

A PRIORI MODELING FOR GRADIENT BASED INVERSE SCATTERING ALGORITHMS

This paper presents a Fisher information based Bayesian approach to analysis and design of the regularization and preconditioning parameters used with gradient based inverse scattering algorithms. In particular, a one-dimensional inverse problem is considered where the permittivity and conductivity profiles are unknown and the input data consist of the scattered field over a certain bandwidth. A priori parameter modeling is considered with linear, exponential and arctangential parameter scalings and robust preconditioners are obtained by choosing the related scaling parameters based on a Fisher information analysis of the known background. The Bayesian approach and a principal parameter (singular value) analysis of the stochastic Cramer-Rao bound provide a natural interpretation of the regularization that is necessary to achieve stable inversion, as well as an indicator to predict the feasibility of achieving successful reconstruction in a given problem set-up. In particular, the Tikhonov regularization scheme is put into a Bayesian estimation framework. A time-domain least-squares inversion algorithm is employed which is based on a quasi-Newton algorithm together with an FDTD-electromagnetic solver. Numerical examples are included to illustrate and verify the analysis.

Finding a control parameter in a one-dimensional parabolic inverse problem by using the Bernstein Galerkin method

An inverse problem concerning a diffusion equation with source control parameter is considered. The approximation of the problem is based on the Bernstein polynomial basis. The properties of Bernstein polynomials are first presented. The Bernstein polynomial basis vanishes except the first polynomial at x = 0, which is equal to 1 and the last polynomial at x = R, which is also equal to 1 over the interval [0, R]. This provides greater flexibility in which the boundary conditions are imposed at the end points of the interval. These properties together with the Galerkin method are then utilized to reduce the inverse problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Solution of the one-dimensional inverse acoustic problem with allowance for velocity dispersion and frequency-dependent wave attenuation

The solution of this inverse problem determines the dependence of the reflection coefficient of echo signals that an acoustic flaw detector receives and processes on distance. This problem is considered in a medium with velocity dispersion and frequency-dependent wave attenuation. The results of numerical experiments are given.

Inverse problem for the strongly degenerate heat equation in a domain with free boundary

In a domain with free boundary, we establish conditions for the existence and uniqueness of a solution of the inverse problem of finding the time-dependent coefficient of heat conductivity. We study the case of strong degeneration where the unknown coefficient tends to zero as t → +0 as a power function t β , where β ≥ 1. Problems for degenerate parabolic equations, free-boundary problems, and inverse problems arise in the course of investigation of many processes important from the practical point of view. Each of these types of problems is fairly well investigated, but their combination within a single problem, in fact, has not been studied. The inverse problem for the one-dimensional heat equation in a domain with free boundary was investigated in [1]. Conditions for the existence and uniqueness of a solution of the inverse problem for a parabolic equation with strong power degeneration were established in [2]. The inverse problems for hyperbolic and elliptic equations with degeneration were studied in [3] and [4], respectively. The free-boundary problem for a degenerate parabolic system was investigated in [5]. In the present paper, we study the one-dimensional inverse problem for the heat equation with strong power degeneration in the case where a part of the boundary is unknown. An analogous problem for the case of weak degeneration was considered in [6].

The inverse problem of acoustothermography with correlation reception of thermal acoustic radiation

For the one-dimensional inverse problem of acoustothermography with correlation reception of thermal acoustic radiation, an integral equation is presented and experimentally verified. A method of solving the inverse problem is proposed. The method is based on combining the correlation functions of thermal acoustic radiation that were obtained for different distances between the receivers.

Inverse reconstruction of boundary condition coefficients in one-dimensional transient heat conduction

This paper describes the restoration of boundary conditions in one-dimensional transient inverse heat conduction problems (IHCP). In the formulation, convective boundary conditions are represented by linear relations between the temperature and the heat flux, together with an initial condition as a function of space. The temperature inside the solution domain together with the space-dependent ambient temperature or heat transfer coefficient are found from additional boundary temperature or average boundary temperature measurements. The determination of the spacewise dependent ambient temperature is a linear IHCP, whilst the determination of the spacewise dependent heat transfer coefficient is a nonlinear IHCP. For both problems uniqueness theorems are available. Numerical results obtained using the boundary element method are presented and discussed.

Reconstruction of boundary condition laws in heat conduction using the boundary element method

Physical problems involving heat exchange between the ends of a rod and the surrounding environment can be formulated as a set of equations representing the heat equation and boundary conditions relating the heat fluxes to the difference between the boundary temperatures and the temperature of the surrounding fluid through a function f . When the heat transfer is purely convective, or solely radiative, then one assumes that f is a linear functional (Newton’s law of cooling), or obeys a fourth-order power law (Stefan’s law), respectively. However, there are many practical heat transfer situations in which either the governing equation does not take a simple form or the actual method of heat transfer is unknown. In such cases the heat transfer coefficient depends on the boundary temperature and the dependence has a complicated or unknown structure. In this study, we investigate a one-dimensional inverse heat conduction problem with unknown nonlinear boundary conditions. We develop the boundary element method to construct and solve numerically the missing terms involving the boundary temperature, the heat flux and the function f . To stabilise the solution we employ an engineering approach based on approximating the function f by a polynomial function of temperature. Numerical results are presented and discussed.

Determination of a control parameter in a one-dimensional parabolic equation using the moving least-square approximation

In this paper the approximation of moving least-square (MLS) is used for finding the solution of a one-dimensional parabolic inverse problem with source control parameter. Comparing with other numerical methods based on meshes such as finite difference method, finite element method and boundary element method, etc. the MLS approximation has merits of simpler numerical procedures, lower computation cost and arbitrary nodes. The result of a numerical example is presented.

Filter digital form of two future temperature methods for the inverse heat conduction: A spectral comparison

AbstractSequential methods that use future times in order to solve an inverse heat conduction problem can be re‐formulated as a sequential digital filter algorithm. The aim of this paper is to explore the filter effect of the function specification method and the truncated singular value decomposition method, in a sequential form. For this purpose, the power spectral densities of the corresponding filter coefficients are evaluated and it is shown that they act as band‐pass filter. Three numerical tests with different frequency spectra are considered in order to solve a linear one‐dimensional transient inverse problem. Furthermore, three models of crescent complexity are used in each method. It is shown that both methods provide similar results and in most cases, the optimum estimations require the same number of future times. As it is expected, similar estimations correspond to a similar band‐pass filter. The effect of the number of future times, the influence of the frequency spectrum of the input and the complexity of the model used can be clearly interpreted in the spectral space. The most complex models act as a highly selective band‐pass filter, but they require greater number of future times. Copyright © 2008 John Wiley & Sons, Ltd.

On the inverse problem of soil profile reconstruction: a comparison of time-domain approaches

We discuss a one-dimensional inverse material profile reconstruction problem that arises in layered media underlain by a rigid bottom, when total wavefield surficial measurements are used to guide the reconstruction. To tackle the problem, we adopt the systematic framework of PDE-constrained optimization and construct an augmented misfit functional that is further endowed by a regularization scheme. We report on a comparison of spatial regularization schemes such as Tikhonov and total variation against a temporal scheme that treats the model parameters as time-dependent. We study numerically the effects of inexact initial estimates, data noise, and regularization parameter choices for all three schemes, and report inverted profiles for the modulus, and for simultaneous inversion of both the modulus and viscous damping. Our numerical experiments demonstrate comparable or superior performance of the time-dependent regularization over the Tikhonov and total variation schemes for both smooth and sharp target profiles, albeit at increased computational cost.

A linearized compact difference scheme for an one-dimensional parabolic inverse problem

The parabolic equation with the control parameter is a class of parabolic inverse problems and is nonlinear. While determining the solution of the problems, we shall determinate some unknown control parameter. These problems play a very important role in many branches of science and engineering. The article is devoted to the following parabolic initial-boundary value problem with the control parameter: ∂u/∂t=∂2u/∂x2+p(t)u+ϕ(x,t),0<x<1,0<t⩽T satisfying u(x,0)=f(x),0<x<1; u(0,t)=g0(t), u(1,t)=g1(t), u(x∗,t)=E(t),0⩽t⩽T where ϕ(x,t),f(x),g0(t),g1(t) and E(t) are known functions, u(x,t) and p(t) are unknown functions. A linearized compact difference scheme is constructed. The discretization accuracy of the difference scheme is two order in time and four order in space. The solvability of the difference scheme is proved. Some numerical results and comparisons with the difference scheme given by Dehghan are presented. The numerical results show that the linearized difference scheme of this article improve the accuracy of the space and time direction and shorten computation time largely. The method in this article is also applicable to the two-dimensional inverse problem.

Reconstruction of heat transfer coefficients using the boundary element method

This paper describes the reconstruction of the heat transfer coefficient (space, Problem I, or time dependent, Problem II) in one-dimensional transient inverse heat conduction problems from surface temperature or average temperature measurements. Since the inverse problem posed does not involve internal temperature measurements, this means that non-destructive testing of materials can be performed. In the formulation, convective boundary conditions relate the boundary temperature to the heat flux. Numerical results obtained using the boundary element method are presented and discussed.

A Numerical Algorithm for Solving a One-Dimensional Inverse Heat Conduction Problem

A Numerical Algorithm for Solving a One-Dimensional Inverse Heat Conduction Problem

Estimation of heat-transfer characteristics on a fin under wet conditions

The finite difference method in conjunction with the least-squares scheme and experimental temperature data is proposed to solve the one-dimensional (1-D) and two-dimensional (2-D) inverse heat conduction problems in order to predict the average overall heat-transfer coefficient h ¯ on a fin and wet fin efficiency η f for various air speeds under wet conditions. The sensitive and overall heat-transfer coefficients on the fin, the Lewis number and the functional form between the relative humidity and fin temperature are unknown a priori in the present study. In addition, the sensitive and overall heat-transfer coefficients on the fin are also assumed to be non-uniform. Thus the whole fin is divided into several sub-fin regions in order to predict the h ¯ and η f values. Variations of these two predicted values with the relative humidity (RH) for various air speeds can be obtained using the present inverse scheme in conjunction with measured fin temperatures. In order to validate the accuracy and reliability of the present inverse scheme, a comparison between the present estimates obtained from the 1-D and 2-D models and exact values is made using simulated temperature data. The results show that the present estimates of the h ¯ value obtained from the 1-D and 2-D models agree well with the exact values even for the case with the measurement errors. Variations of the h ¯ and η f values obtained from 2-D model with the RH value are similar to those obtained from 1-D model for various air speeds. However, the present estimates of the h ¯ and η f values obtained from the 2-D model an slightly deviate from those obtained from the 1-D model. It is worth mentioning that the deviation between the h ¯ value in the downstream region and that in the upstream region can be observed using the 2-D model. This phenomenon cannot be obtained from the 1-D model.

A Numerical Algorithm for Solving a One-Dimensional Inverse Heat Conduction Problem

A Numerical Algorithm for Solving a One-Dimensional Inverse Heat Conduction Problem

Possibility of the Analysis of Two-Dimensional Inverse Problems in Hydrodynamic Lubrication (An Examination Using a Plane Inclined Bearing with Finite Width)

Inverse problems in hydrodynamic lubrication mean that the film pressures are treated as known and Reynolds equation is solved to give the film thickness distribution. In a previous paper, we presented numerical methods for one-dimensional inverse problems. This paper presents the formulation for two-dimensional inverse problems using the finite element method and CIP method. These numerical methods have been applied to a plane inclined bearing with finite width. It is shown that inverse solution for film thickness is correct over almost region, however, on and around inlet corner points, correct solutions are not obtained. Both analysis using the finite element method and CIP method give the same results. The cause of incorrect solutions should further be pursued.

Estimation of piecewise constant coefficients of parabolic equations: applications to the detection of buried objects

A one-dimensional inverse problem arising in infrared thermography for the detection and characterization of buried objects is introduced. Mathematically, the problem is to reconstruct a piecewise constant coefficient of a scalar heat equation in a finite rod from measurements taken at one of its extremities. The problem is posed in the well known least-squares setting and solved by a quasi-Newton method. The contributions of this article include: (i) the parameterization of a piecewise constant function by a small number of unknown parameters which represent its constant values and locations of discontinuities; (ii) the application of the adjoint field technique in the calculation of the gradient of a discretized objective function and (iii) the application of the considered inverse problem in the detection and characterization of buried objects. Numerical results illustrate the good performance of the proposed algorithm.

Magnetic Field-Based Eddy-Current Modeling for Multilayered Specimens

Eddy-current inspection for nondestructive evaluation has traditionally been investigated in terms of coil impedance signals via theoretical and experimental methods. However, advanced eddy-current techniques use solid-state sensors such as Hall devices, giant magnetoresistive sensors, anisotropic magnetoresistive sensors, and superconducting quantum interference devices for magnetic field measurement to achieve better sensitivity and high temporal and spatial resolution in material evaluation and characterization. Here, we review the Dodd and Deeds integral model and use the truncated region eigenfunction expansion (TREE) method for computation of the magnetic field. This results in series expressions instead of integral ones. Thus, the computation is both simplified and speeded up so that it becomes convenient for solving one-dimensional eddy-current inverse problems. We compare the theoretical results from the analytical model with the results from a numerical simulation based on the finite-element method in terms of accuracy and computation time.

Restoring boundary conditions in heat conduction

The restoration of boundary conditions in one-dimensional transient inverse heat-conduction problems (IHCP) is described. In the formulation, the boundary conditions are represented by linear relations between the temperature and the heat flux, together with an initial condition as a function of space. The temperature inside the solution domain, together with the space or time-dependent ambient temperature of the environment surrounding the heat conductor, are found from additional boundary-temperature or average boundary-temperature measurements. Numerical results obtained using the boundary-element method are presented and discussed.