One-dimensional Inverse Problem Research Articles

A novel computational procedure based on league championship algorithm for solving an inverse heat conduction problem

Inverse heat conduction problems, which are one of the most important groups of problems, are often ill-posed and complicated problems, and their optimization process has lots of local extrema. This paper provides a novel computational procedure based on finite differences method and league championship algorithm to solve a one-dimensional inverse heat conduction problem. At the beginning, we use the Crank-Nicolson semi-implicit finite difference scheme to discretize the problem domain and solve the direct problem which is a second-order method in time and unconditionally stable. The consistency, stability and convergence of the method are investigated. Then we employ a new optimization method known as league championship algorithm to estimate the unknown boundary condition from some measured temperature on the line. League championship algorithm is a recently proposed probabilistic algorithm for optimization in continuous environments, which tries to simulate a championship environment wherein several teams with different abilities play in an artificial league for several weeks or iterations. To confirm the efficiency and accuracy of the proposed approach, we give some examples for the engineering applications. Results show an excellent agreement between the solution of the proposed numerical algorithm and the exact solution.

An inverse problem for one-dimensional diffusion equation in optical tomography

In this paper, we study the one-dimensional inverse problem for the diffusion equation based optical tomography. The objective of the present work is a mathematical and numerical analysis concerning one-dimensional inverse problem. In the first stage, the forward diffusion equation with boundary conditions is solved using an intermediate elliptic equation. We give the existence and the uniqueness results of the solution. An approximation of the photon density in frequency-domain is proposed using a Splines Galerkin method. In the second stage, we give theoretical results such as the stability and lipschitz-continuity of the forward solution and the Fréchet differentiability of the Dirichlet-to-Neumann nonlinear map with respect to the optical parameters. The Fréchet derivative is used to linearize the considered inverse problem. The Newton method based on the regularization technique will allow us to compute the approximate solutions of the inverse problem. Several test examples are used to verify high accuracy, effectiveness and good resolution properties for smooth and discontinuous optical property solutions.

Optimal Adaptive Designs with Inverse Ordinary Differential Equations.

Many industrial and engineering applications are built on the basis of differential equations. In some cases, parameters of these equations are not known and are estimated from measurements leading to an inverse problem. Unlike many other papers, we suggest to construct new designs in the adaptive fashion 'on the go' using the A-optimality criterion. This approach is demonstrated on determination of optimal locations of measurements and temperature sensors in several engineering applications: (1) determination of the optimal location to measure the height of a hanging wire in order to estimate the sagging parameter with minimum variance (toy example), (2) adaptive determination of optimal locations of temperature sensors in a one-dimensional inverse heat transfer problem and (3) adaptive design in the framework of a one-dimensional diffusion problem when the solution is found numerically using the finite difference approach. In all these problems, statistical criteria for parameter identification and optimal design of experiments are applied. Statistical simulations confirm that estimates derived from the adaptive optimal design converge to the true parameter values with minimum sum of variances when the number of measurements increases. We deliberately chose technically uncomplicated industrial problems to transparently introduce principal ideas of statistical adaptive design.

Multi-model method for solving nonlinear transient inverse heat conduction problems

The multi-model inverse method for nonlinear inverse problems is established based on the multi-model control theory. First the model switching variable is chosen and several typical operating balance points in the workspace of the balance variable are selected. Then for each operating balance point the linear local model is established, and the local controller is designed for each linear local model. Finally, according to the instantaneous matching degree between the actual model and the local models, the inversion results of each local controller are weighted and synthesized to obtain the final inversion result. Numerical tests are implemented to solve the one-dimensional nonlinear inverse heat conduction problem by the multi-model inverse method associated with the dynamic matrix control (DMC) and DMC filter, respectively. Numerical results by the multi-model inverse method based on DMC demonstrate that the multi-model inverse method is a highly computationally efficient and accurate algorithm for inverse problems with complicated direct problems. Numerical results by the multi-model inverse method based on DMC filter show that the presented method can extend the applied field of the complicated linear inverse algorithms such as digital filter to the nonlinear inverse problems and it can obtain satisfactory inversion results.

A meshfree method for inverse wave propagation using collocation and radial basis functions

A stable and highly accurate meshfree scheme based on strong form collocation associated with radial basis functions and explicit time integration is proposed to solve ill-posed inverse wave propagation problems. Tikhonov regularization technique and the L-curve criterion are employed to deal with the noise measurement data before solving the governing equation to get a stable solution. Appropriate weights are first derived for imposing boundary conditions and known conditions from the measurement data to yield the optimal convergence for inverse problems. Compared to the required iterative procedures in common methods, no iteration is needed in the proposed method which leads to the time-saving computation and avoids the instability resulting from the iterative methods. Dispersion analysis describes that this method has much less dispersion error than the classical finite element method (FEM) and popular weak form meshfree methods, and stability analysis is performed to predict the critical time step for the explicit time integration. Another outstanding advantage of this method is that it is effortless to extend the solution scheme of one-dimensional inverse problem to high-dimensional inverse problems. Furthermore, the influences of the high noise into measurement data are evaluated, which demonstrate that even handling with very high noise in the input data the proposed algorithm can achieve stable and high-accuracy solutions.

The polynomial Trefftz method for solving backward and inverse source wave problems

The Trefftz method is a truly meshless boundary-type method, because the trial solutions automatically satisfy the governing equation. In order to stably solve the high-dimensional backward wave problems and the one-dimensional inverse source problems, we develop a multiple-scale polynomial Trefftz method (MSPTM), of which the scales are determined a priori by the collocation points. The MSPTM can retrieve the missing initial data and unknown time varying wave source. The present method can also be extended to solve the higher-dimensional wave equations long-term through the introduction of a director in the two-dimensional polynomial Trefftz bases. Several numerical examples reveal that the MSPTM is efficient and stable for solving severely ill-posed inverse problems of wave equations under large noises.

Combined Frequency Domain and Time Domain Airborne Data for Environmental and Engineering Challenges

We describe the main properties of the EQUATOR airborne geophysical system and how the system can be employed to solve some engineering and environmental problems. To date, the EQUATOR system has been used for groundwater search in 2014 and 2015 in Siberia. The name “EQUATOR” originates from the intention to EQUATE, or to balance time and frequency domain approaches in airborne electromagnetics, which is why the system was originally designed. To the best of our knowledge, EQUATOR is the only system to collect electromagnetic data both in time and in frequency domains. We further explain the difference between the two methods on some numerical and field examples. Also we describe some advantages of their coupling. In particular, by virtue of our approach, a numerically more robust solution of the one-dimensional electromagnetic inverse problem may be achieved. The recorded full-time measurements are further converted to in-phase, quadrature and off-time responses, as opposed to standard frequency or time domain solutions, which have been widely adapted within the industry. When inverting the data, our main focus is within the top 100 m of the Earth, which are the most relevant for engineering and environmental studies. We emphasize that a combined use of frequency and time domain data gives more detailed information about geological structures in the near-surface. As an example, we present some case studies obtained over a platform (Siberia) and a crystalline shield (Western Africa) setting. We further suggest using these results in such near-surface studies as sediment thickness estimation, soil state analysis, groundwater search, and geological structural analysis.

Mold Simulator Study of Heat Transfer Phenomenon During the Initial Solidification in Continuous Casting Mold

In this paper, mold simulator trials were firstly carried out to study the phenomena of the initial shell solidification of molten steel and the heat transfer across the initial shell to the infiltrated mold/shell slag film and mold. Second, a one-dimensional inverse heat transfer problem for solidification (1DITPS) was built to determine the temperature distribution and the heat transfer behavior through the solidifying shell from the measured shell thickness. Third, the mold wall temperature field was recovered by a 2DIHCP mathematical model from the measured in-mold wall temperatures. Finally, coupled with the measured slag film thickness and the calculations of 1DITPS and 2DIHCP, the thermal resistance and the thickness of liquid slag film in the vicinity of the meniscus were evaluated. The experiment results show that: the total mold/shell thermal resistance, the mold/slag interfacial thermal resistance, the liquid film thermal resistance, and the solid film thermal resistance is 8.0 to 14.9 × 10−4, 2.7 to 4.8 × 10−4, 1.5 to 4.6 × 10−4, and 3.9 to 6.8 × 10−4 m2 K/W, respectively. The percentage of mold/slag interfacial thermal resistance, liquid film thermal resistance, and solid film thermal resistance over the total mold/shell thermal resistance is 27.5 to 34.4, 17.2 to 34.0, and 38.5 to 48.8 pct, respectively. The ratio of radiation heat flux is around 14.1 to 51.9 pct in the liquid slag film.

Identifying Initial Condition in Degenerate Parabolic Equation with Singular Potential

A hybrid algorithm and regularization method are proposed, for the first time, to solve the one-dimensional degenerate inverse heat conduction problem to estimate the initial temperature distribution from point measurements. The evolution of the heat is given by a degenerate parabolic equation with singular potential. This problem can be formulated in a least-squares framework, an iterative procedure which minimizes the difference between the given measurements and the value at sensor locations of a reconstructed field. The mathematical model leads to a nonconvex minimization problem. To solve it, we prove the existence of at least one solution of problem and we propose two approaches: the first is based on a Tikhonov regularization, while the second approach is based on a hybrid genetic algorithm (married genetic with descent method type gradient). Some numerical experiments are given.

Inverse analysis of heat conduction problems with relatively long heat treatment

This paper proposes a solution method for one-dimensional inverse heat conduction problems that require a relatively long time. A hybrid technique is applied to analyze laser surface heating and spray cooling on a hot surface. In the present study, the unknown temperature in half-range expansions form is assumed, and the shifting function method is used to obtain an analytic solution. The coefficients of the half-range expansions could be determined with the least-squares method in conjunction with the analytic solution and measured temperatures. Mathematical and experimental examples are given to illustrate the analyses.

The meshfree strong form methods for solving one dimensional inverse Cauchy-Stefan problem

In this paper, we extend the application of meshfree node based schemes for solving one-dimensional inverse Cauchy-Stefan problem. The aim is devoted to recover the initial and boundary conditions from some Cauchy data lying on the admissible curve s(t) as the extra overspecifications. To keep matters simple, the problem has been considered in one dimensional, however the physical domain of the problem is supposed as an irregular bounded domain in $$\mathbb {R}^2$$R2. The methods provide the space-time approximations for the heat temperature derived by expanding the required approximate solutions using collocation scheme based on radial point interpolation method (RPIM). The proposed method makes appropriate shape functions which possess the important Delta function property to satisfy the essential conditions automatically. In addition, to conquer the ill-posedness of the problem, particular optimization technique has been applied for solving the system of equations $$Ax=b$$Ax=b in which A is a nonsymmetric stiffness matrix. As the consequences, reliable approximate solutions are obtained which continuously depend on input data.

Одномерные обратные задачи электромагнитного зондирования слоистых диэлектрических сред

Одномерные обратные задачи электромагнитного зондирования слоистых диэлектрических сред

Application of meshless local radial point interpolation (MLRPI) on a one-dimensional inverse heat conduction problem

Abstract In this paper, the meshless local radial point interpolation (MLRPI) method is applied to one-dimensional inverse heat conduction problems. The meshless LRPIM is one of the truly meshless methods since it does not require any background integration cells. In this case, all integrations are carried out locally over small quadrature domains of regular shapes, such as lines in one dimensions, circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using radial basis functions. These shape functions which are constructed by point interpolation method using the radial basis functions have delta function property. The time derivatives are approximated by the time-stepping method. Numerical experiments are carried out and compared with implicit Finite difference method.

Calculation of Heat Flux Across the Hot Surface of Continuous Casting Mold Through Two-Dimensional Inverse Heat Conduction Problem

A novel method for the estimation of the mold hot surface heat flux based on the measured responding temperatures from two columns of thermocouples that embedded inside the mold during continuous casting has been developed. The method includes a Two-Dimensional Inverse transient Heat Conduction Problem (2D-IHCP) model that was solved by the conjugate gradient method with Adjoint Equation. The model was validated by comparing the results with those calculated by a robust One-Dimensional Inverse transient Heat Conduction Problem (1D-IHCP). The solution of a test problem indicated that the Mean Absolute Percentage Error of the estimated heat flux calculated by the new method is about 9 to 40 pct of those calculated by the 1D-IHCP. Then, the method is applied to compute the heat flux for a mold simulator experiment. The results indicated that the heat fluxes and temperatures across mold hot surface calculated by 2D-IHCP show the same variation tendency as those calculated by 1D-IHCP. However, the heat fluxes calculated by 2D-IHCP are about 1.2 to 2 times larger than those calculated by 1D-IHCP for the locations below the liquid mold flux surface and are about 50 to 90 pct of those calculated by 1D-IHCP for the locations above the liquid mold flux surface.

A one-dimensional inverse problem in composite materials: Regularization and error estimates

In this paper we investigate an inverse one-dimensional heat conduction problem in multi-layer medium. The inverse problem is first formulated in the frequency domain via Fourier transform technique. An effective regularization method for the stable reconstruction of solution is given with proven error estimates. Several numerical examples are constructed to demonstrate the effectiveness of the proposed method.

Inverse problems of the inhomogeneous theory of elasticity for thick-walled shells

The inhomogeneous theory of elasticity considers bodies, the mechanical characteristics of which (the modulus of elasticity and Poisson’s ratio) are functions of the coordinates. If indirect problems of the inhomogeneous theory of elasticity are identifi ed, and the stress-strain state of the body has well-known functions of mechanical characteristics, the essence of inverse problems is to determine the functions of the inhomogeneity for a given stress state of the body. One of the fi rst solutions to such an inverse problem was published in the work of Lekhnitskii (“Radial distribution of stresses in the wedge and half-plane with variable modulus of elasticity”. PMM, XXVI(1), pp. 146–151, 1962). In this article, we consider one-dimensional inverse problems for thick-walled cylindrical and spherical shells that are subjected to internal and external pressures in a non-varying temperature fi eld. The aim of this work is to identify the dependence of the elastic modulus on the radial co-ordinate for which the equivalent stress according to a particular theory of strength will be constant at all points of the body (such structures are called equal stress), or the equivalent stress in all points will be equal to the strength of the material (such structures are called equal strength). For example, the author has proven that the limit

Estimation of unknown boundary functionsin an inverse heat conduction problem using a mollified marching scheme

In this article, a one-dimensional inverse heat conduction problem with unknown nonlinear boundary conditions is studied. In many practical heat transfer situations, the heat transfer coefficient depends on the boundary temperature and the dependence has a complicated or unknown structure. For this reason highly nonlinear boundary conditions are imposed involving both the flux and the temperature. A numerical procedure based on the mollification method and the space marching scheme is developed to solve numerically the proposed inverse problem. The stability and convergence of numerical solutions are investigated and the numerical results are presented and discussed for some test problems.

A meshless method for an inverse two-phase one-dimensional nonlinear Stefan problem

We extend a meshless method of fundamental solutions recently proposed by the authors for the one-dimensional two-phase inverse linear Stefan problem, to the nonlinear case. In this latter situation the free surface is also considered unknown which is more realistic from the practical point of view. Building on the earlier work, the solution is approximated in each phase by a linear combination of fundamental solutions to the heat equation. The implementation and analysis are more complicated in the present situation since one needs to deal with a nonlinear minimization problem to identify the free surface. Furthermore, the inverse problem is ill-posed since small errors in the input measured data can cause large deviations in the desired solution. Therefore, regularization needs to be incorporated in the objective function which is minimized in order to obtain a stable solution. Numerical results are presented and discussed.

Moving Least Squares Method for a One-Dimensional Parabolic Inverse Problem

This paper investigates the numerical solution of a class of one-dimensional inverse parabolic problems using the moving least squares approximation; the inverse problem is the determination of an unknown source term depending on time. The collocation method is used for solving the equation; some numerical experiments are presented and discussed to illustrate the stability and high efficiency of the method.

On one-dimensional inverse problems arising from polarimetric measurements of nematic liquid crystals

We revisit the problem of determining dielectric parameters in layered nematic liquid crystals from polarimetric measurements originally introduced by Lionheart and Newton. After a detailed analysis of the model, of the scales involved, and of natural obstacles to the reconstruction of more than one dielectric parameters, we produce two simple one-dimensional inverse problems which can be studied without any expertise in liquid crystals. We then confirm that very little can be recovered about the internal configuration of smooth dielectric parameters from these measurements, and give a uniqueness result for one of the two problem, when the unknown parameter satisfies a monotonicity property. In that case, the available data can be expressed in terms of Laplace and Hankel transforms.