A non‐linear inverse vibration problem of estimating the time‐dependent stiffness coefficients by conjugate gradient method

An iterative regularization method, i.e. the conjugate gradient method (CGM) is applied to an inverse non‐linear force vibration problem to estimate the unknown time‐dependent stiffness coefficients (or spring constants) in a damped system by using the measured system displacement. It is assumed that no prior information is available on the functional form of the unknown stiffness coefficients in the present study, thus, it is classified as the function estimation in inverse calculation. The accuracy of the inverse analysis is examined by using the simulated exact and inexact displacement measurements. The numerical simulations are performed to test the validity of the present algorithm by using different types of stiffness coefficients and measurement errors. Results show that an excellent estimation on the time‐dependent spring constants can be obtained with any arbitrary initial guesses within a couple of seconds of CPU time at Pentium III‐500 MHz PC. Copyright © 2001 John Wiley & Sons, Ltd.     

Investigation of regularization parameters and error estimating in inverse elasticity problems

The method of Tarantola¹ based on Bayesian statistical theory for solving general inverse problems is applied to inverse elasticity problems and is compared to the spatial regularization technique presented in Schnur and Zabaras.² It is shown that when normal Gaussian distributions are assumed and the error in the data is uncorrelated, the Bayesian statistical theory takes a form similar to the deterministic regularization method presented earlier in Schnur and Zabaras,² As such, the statistical theory can be used to provide a statistical interpretation of regularization and to estimate error in the solution of the inverse problem. Examples are presented to demonstrate the effect of the regularization parameters and the error in the initial data on the solution.     

Structural damage identification via time domain response and Markov Chain Monte Carlo method

The present work addresses the problem of structural damage identification built on the statistical inversion approach. Here, the damage state of the structure is continuously described by a cohesion parameter, which is spatially discretized by the finite element method. The inverse problem of damage identification is then posed as the determination of the posterior probability densities of the nodal cohesion parameters. The Markov Chain Monte Carlo method, implemented with the Metropolis–Hastings algorithm, is considered in order to approximate the posterior probabilities by drawing samples from the desired joint posterior probability density function. With this approach, prior information on the sought parameters can be used and the uncertainty concerning the known values of the material properties can be quantified in the estimation of the cohesion parameters. The assessment of the proposed approach has been performed by means of numerical simulations on a simply supported Euler–Bernoulli beam. The damage identification and assessment are performed considering time domain response data. Different damage scenarios and noise levels were addressed, demonstrating the feasibility of the proposed approach.     

Bayesian decision analysis and reliability certification

Reliability certification is set as a problem of Bayesian Decision Analysis. Uncertainties about the system reliability are quantified by assuming the parameters of the models describing the stochastic behavior of components as random variables. A utility function quantifies the relative value of each possible level of system reliability after having been accepted or the opportunity loss of the same level if the system has been rejected. A decision about accepting or rejecting the system can be made either on the basis of the existing prior assessment of uncertainties or after obtaining further information through testing of the components or the system at a cost. The concepts of value of perfect information, expected value of sample information and the expected net gain of sampling are specialized to the reliability of a multicomponent system to determine the optimum component testing scheme prior to deciding on the system's certification. A component importance ranking is proposed on the basis of the expected value of perfect information about the reliability of each component. The proposed approach is demonstrated on a single component system failing according to a Poisson random process and with natural conjugate prior probability density functions (pdf) for the failure rate and for a multicomponent system under general assumptions.     

Inverse wave scattering in 2-D elastic solids

Summary This paper is concerned with an inverse problem for two-dimensional elastic solids. It seeks to recover the subsurface density profile based on the measurements obtained at the boundary. The method considers a temporal interval for which time dependent measurements are provided. It formulates an optimal estimation problem which seeks to minimize the error difference between the given data and the response from the system. It uses a boundary regularization term to stabilize the inversion. The method leads to an iterative algorithm which, at every iteration, requires the solution to a two-point boundary value problem. Several numerical results are presented which indicate that a close estimate of the unknown density function can be obtained based on the boundary measurements only.     

Reconstruction of multiplicative space- and time-dependent sources

This paper presents a numerical regularization approach to the simultaneous determination of multiplicative space- and time-dependent source functions in a nonlinear inverse heat conduction problem with homogeneous Neumann boundary conditions together with specified interior and final time temperature measurements. Under these conditions a unique solution is known to exist. However, the inverse problem is still ill-posed since small errors in the input interior temperature data cause large errors in the output heat source solution. For the numerical discretisation, the boundary element method combined with a regularized nonlinear optimization are utilized. Results obtained from several numerical tests are provided in order to illustrate the efficiency of the adopted computational methodology.     

Helmholtz-Type Regularization Method for Permittivity Reconstruction Using Experimental Phantom Data of Electrical Capacitance Tomography

Electrical capacitance tomography (ECT) attempts to image the permittivity distribution of an object by measuring the electrical capacitance between sets of electrodes placed around its periphery. Image reconstruction in ECT is a nonlinear ill-posed inverse problem, and regularization methods are needed to stabilize this inverse problem. The reconstruction of complex shapes (sharp edges) and absolute permittivity values is a more difficult task in ECT, and the commonly used regularization methods in Tikhonov minimization are unable to solve these problems. In the standard Tikhonov regularization method, the regularization matrix has a Laplacian-type structure, which encourages smoothing reconstruction. A Helmholtz-type regularization scheme has been implemented to solve the inverse problem with complicated-shape objects and the absolute permittivity values. The Helmholtz-type regularization has a wavelike property and encourages variations of permittivity. The results from experimental data demonstrate the advantage of the Helmholtz-type regularization for recovering sharp edges over the popular Laplacian-type regularization in the framework of Tikhonov minimization. Furthermore, this paper presents examples of the reconstructed absolute value permittivity map in ECT using experimental phantom data.      

Logical inference for inverse problems

Estimating a deterministic single value for model parameters when reconstructing the system response has a limited meaning if one considers that the model used to predict its behaviour is just an idealization of reality, and furthermore, the existence of measurements errors. To provide a reliable answer, probabilistic instead of deterministic values should be provided, which carry information about the degree of uncertainty or plausibility of those model parameters providing one or more observations of the system response. This is widely-known as the Bayesian inverse problem, which has been covered in the literature from different perspectives, depending on the interpretation or the meaning assigned to the probability. In this paper, we revise two main approaches: the one that uses probability as logic, and an alternative one that interprets it as information content. The contribution of this paper is to provide an unifying formulation from which both approaches stem as interpretations, and which is more general in the sense of requiring fewer axioms, at the time the formulation and computation is simplified by dropping some constants. An extension to the problem of model class selection is derived, which is particularly simple under the proposed framework. A numerical example is finally given to illustrate the utility and effectiveness of the method.     

Identifying Lamé parameters from time-dependent elastic wave measurements

In many sectors of today’s industry it is of utmost importance to detect defects in elastic structures contained in technical devices to guarantee their failure-free operation. As currently used signal processing techniques have natural limits with respect to accuracy and significance, modern mathematical methods are crucial to improve current algorithms. We consider in this paper a parameter identification approach for isotropic and linear elastic structures described by their Lamé parameters and a material density. This approach can be employed for non-destructive defect detection, location and characterization from time-dependent measurements of one elastic wave. To this end, we show that the operator linking the static parameters with the wave measurements is Fréchet differentiable, which allows to set up Newton-like methods for the non-linear parameter identification problem. We indicate the performance of a specific inexact Newton-like regularization method by numerical examples for a testing problem of a thin plate from measurements of the normal component of the displacement field on the boundary. As an extension, we further augment this method with a total variation regularization and thereby improve reconstructed parameters that feature edges.     

Determination of an optimal regularization factor in system identification with Tikhonov regularization for linear elastic continua

This paper presents a geometric mean scheme (GMS) to determine an optimal regularization factor for Tikhonov regularization technique in the system identification problems of linear elastic continua. The characteristics of non‐linear inverse problems and the role of the regularization are investigated by the singular value decomposition of a sensitivity matrix of responses. It is shown that the regularization results in a solution of a generalized average between the a priori estimates and the a posteriori solution. Based on this observation, the optimal regularization factor is defined as the geometric mean between the maximum singular value and the minimum singular value of the sensitivity matrix of responses. The validity of the GMS is demonstrated through two numerical examples with measurement errors and modelling errors. Copyright © 2001 John Wiley & Sons, Ltd.     

不适定热传导反问题的两种求解方法及其比较

本文以不适定热传导反问题为对象，采用两种方法进行了求解。一种方法基于对具有测量误差的边界条件进行适当的微扰，使之化为适定问题；另一种方法基于Ｔｉｋｈｏｎｏｖ的正则平滑思想，对反问题中的输入数据进行平滑处理，以便使函数及其一阶导数均实现一致逼近。通过计算与求解表明，两种方法均能得到具有一定精度与稳定性的结果，其中以正则化法更为理想     

Adaptive error estimation technique of the Trefftz method for solving the over-specified boundary value problem

In this paper, numerical solutions are investigated based on the Trefftz method for an over-specified boundary value problem contaminated with artificial noise. The main difficulty of the inverse problem is that divergent results occur when the boundary condition on over-specified boundary is contaminated by artificial random errors. The mechanism of the unreasonable result stems from its ill-posed influence matrix. The accompanied ill-posed problem is remedied by using the Tikhonov regularization technique and the linear regularization method, respectively. This remedy will regularize the influence matrix. The optimal parameter λ of the Tikhonov technique and the linear regularization method can be determined by adopting the adaptive error estimation technique. From this study, convergent numerical solutions of the Trefftz method adopting the optimal parameter can be obtained. To show the accuracy of the numerical solutions, we take the examples as numerical examination. The numerical examination verifies the validity of the adaptive error estimation technique. The comparison of the Tikhonov regularization technique and the linear regularization method was also discussed in the examples.     

A deforming finite element method analysis of inverse Stefan problems

A deforming FEM (DFEM) analysis of one-dimensional inverse Stefan problems is presented. Specifically, the problem of calculating the position and velocity of the moving interface from the temperature measurements of two or more sensors located inside the solid phase is addressed. Since the interface velocity is considered to be the primary variable of the problem, the DFEM formulation is found to have many advantages over other traditional front tracking methods. The present inverse formulation is based on a minimization of the error between the calculated and measured temperatures, utilizing future temperature data to calculate current values of the unknown parameters. Also, the use of regularization is found to be useful in obtaining more accurate results, especially when the interface is located far away from the sensors. The method is illustrated with several examples. The effects of the location of the sensors, of the error in the sensor measurements and of several computational parameters were examined.     

Interconnection between statistical weights of test data and their duration,completeness of the prior information,and errors in measurements

The problem of combining data of fault-free tests of different durations is considered. An analysis of dependence between their statistical weights and the scatter of the mean lifetime, completeness of the prior information about mean lifetime, and errors in measurements is presented.Translated from Izmeritel'naya Tekhnika, No. 8, pp. 17–21, August, 1993.     

Simulation of processes in heat sensors based on solution of inverse problems in heat conduction

The paper deals with nonstationary problems in heat conduction, which arise in connection with the determination of the heat flux density and temperature on the surface of a model in intermittent high-enthalpy wind-tunnel facilities by the results of temperature measurements using intramodel heat sensors. The solution of inverse problem in heat conduction in a one-dimensional formulation with an arbitrary time dependence of the heat flux density is obtained by two methods, namely, by iterations and by integral transformations with finite limits. In the former method, the inverse problem is reduced to a system of two coupled integral and integro-differential equations of the Volterra type relative to the temperature and heat flux density on the external boundary. Calculations demonstrate that the numerical solution asymptotically approaches the exact solution, and the iteration method exhibits smoothing properties and is stable with respect to random errors of measurement. In the integral method, an inverse problem for the class of boundary functions satisfying the Dirichlet conditions and represented by a partial sum of the Fourier series reduces to a set of algebraic equations which has a unique solution. In the absence of measurement errors, the solution of inverse problem is exact. Examples are given of constructing solutions in the presence of random noise; it is demonstrated that, in the case of reasonable restriction of the range of frequencies to be analyzed, the errors in the solution do not exceed the mean-square level of noise.Translated from Teplofizika Vysokikh Temperatur, Vol. 43, No. 1, 2005, pp. 071–085.Original Russian Text Copyright © 2005 by E. P. Stolyarov.     

A boundary element-based inverse-problem in estimating transient boundary conditions with conjugate gradient method

A Boundary Element Method (BEM)-based inverse algorithm utilizing the iterative regularization method, i.e. the conjugate gradient method (CGM), is used to solve the Inverse Heat Conduction Problem (IHCP) of estimating the unknown transient boundary temperatures in a multi-dimensional domain with arbitrary geometry. The results obtained by the CGM are compared with that obtained by the standard Regularization Method (RM). The error estimation based on the statistical analysis is derived from the formulation of the RM. A 99 per cent confidence bound is thus obtained. Finally, the effects of the measurement errors to the inverse solutions are discussed. Results show that the advantages of applying the CGM in the inverse calculations lie in that (i) the major difficulties in choosing a suitable quadratic norm, determining a proper regularization order and determining the optimal smoothing (or regularization) coefficient in the RM are avoided and (ii) it is less sensitive to the measurement errors, i.e. more accurate solutions are obtained. © 1998 John Wiley & Sons, Ltd.     

Pairwise combination of methods of detecting radiation sources

The pairwise combination of methods for radio measurements when detecting radiation sources is considered. The statistical parameters of two-dimensional detection with a linear resolving limit are investigated and it is shown that it is possible to achieve a considerable reduction in the probability of missing a radiation source when using such a combination of detection methods. __________ Translated from Izmeritel'naya Tekhnika, No. 8, pp. 68–73, 2008.     

A boundary elements pseudo heat source method formulation for inverse analysis of solidification problems

 An efficient methodology is presented to solve inverse solidification problems. In the procedure, the latent heat effects are implemented by introducing pseudo heat sources near the moving interface. The material properties can be temperature dependent. To account for the nonlinear part of the governing differential equations, a finite-boundary element formulation is employed. To reduce the oscillations in the solution, a sequential regularization scheme is used. A procedure for proper selection of regularization parameters is presented. To smooth the solutions further, a secondary regularization scheme is introduced and employed. Two complete examples are presented to demonstrate the applicability and the accuracy of the methods. Received: 1 March 2002 / Accepted: 10 February 2003     

The inverse problem of the phenomenological theory of the optical properties of thin films

For thin metal films the solution of the inverse problem of the phenomenological theory of the optical properties of thin films is incorrect when the ratio of the film thickness to the incident light wavelength becomes less than 0.05 because the set of equations describing the relation between the measured optical characteristics and the optical constants of the film is virtually a set of linear equations with determinant equal to zero. For this reason the usual methods of solving the inverse problem give ambiguous solutions (optical constants) which are unstable to small errors in the measured optical characteristics. If the method of continuous differential descent is used as the regularization method for solution of this inverse problem unambiguous and stable solutions can be obtained.     

An inverse problem in simultaneously estimating boundary moisture fluxes in a porous annular cylinder

Summary Based on the conjugate gradient method, this study presents a means of solving the inverse boundary value problem of coupled heat and moisture transport in a porous annular cylinder. While knowing the moisture history at the measuring positions, the unknown time-dependent inner-and-outer boundary moisture fluxes can be simultaneously determined. It is assumed that no prior information is available on the functional form of the unknown moisture fluxes. The accuracy of this inverse heat and moisture transport problem is examined by using the simulated exact and inexact moisture measurements in the numerical experiments. Results show that excellent estimation on the time-dependent boundary moisture fluxes can be obtained with any arbitrary initial guesses. Moreover, the methodology presented in this paper can also be used to calculate the cutting forces in nanomachining by atomic force microscopy (AFM), and to determine the heat sources in an X-ray lithographic process.