Tikhonov Regularization Technique Research Articles

The inverse methods based on S-FEMs with an adaptive SVD regularization technique for solving Cauchy inverse heat transfer problems

This paper presents three inverse models based on S-FEMs with adaptive singular value decomposition (SVD) regularization technique for solving inverse heat transfer problems of Cauchy type with noisy input data “measured” on the domain boundary. The smoothed finite element methods (S-FEMs) are used as forward solvers, including edge-based S-FEM (ES-FEM), node-based S-FEM (NS-FEM) and cell-based S-FEM (CS-FEM that is the same as the standard FEM using T3 elements). First, the full-size S-FEM system equation is created based on the S-FEM theory. A set of local nodal temperature equations is then extracted from the full-size equations through a matrix partitioning operation, considering the nodes related to the heat flux on the Cauchy boundary. The Fourier heat convection theory is used to establish a relationship between the local nodal temperatures and the known heat fluxes on the Cauchy boundary. This procedure effectively converts a Cauchy type inverse problem to a forward-like problem with a set of ill-posed equations for inverse analyses. In order to mitigate the ill-posedness and to achieve a reliable and accurate inverse solution, we propose an adaptive SVD technique to regularize the solution, through minimizing the error between the approximate heat fluxes of the inverse model and the given (“measured”) flux values on the Neumann boundaries. The proposed adaptive procedure effectively determines the number of small singular values to be deleted, and hence the noises in the input data cannot be magnified. The present techniques are tested using a number of Cauchy type heat transfer problems, and it is concluded that our inverse procedure is much more effective compared to the widely used Tikhonov regularization technique, and it is a systematic procedure for accurate, reliable and stable inverse solutions with noisy input data.

On global iterative schemes based on Hessenberg process for (ill-posed) Sylvester tensor equations

This paper is concerned with studying the performance of some global iterative schemes based on Hessenberg process to solve the Sylvester tensor equation (STE). Furthermore, we briefly mention the implementation of flexible versions. Under certain conditions, we devote part of the work to derive upper bounds for condition number of matrices with a special Kronecker product structure which can be helpful to obtain error bounds related to perturbation analysis of the mentioned STE. In addition, we propose using the Tikhonov regularization technique in conjunction with the tensor form of the flexible Hessenberg process when the STE is ill-posed. Numerical examples are examined for two test problems arising from discretization of three-dimensional partial differential equations. We also present some experimental results related to restoration of color images.

A radial basis function (RBF)-finite difference (FD) method for the backward heat conduction problem

In this paper a numerical scheme based on the idea of radial basis function finite difference (RBF-FD) technique is considered to solve the backward heat conduction problems (BHCP). In the meshless numerical method of RBF-FD, according to the finite difference technique we approximate the required derivatives for every point xi ∈ Ω in the corresponding local-support domain Ωi. Then the partial differential equation problem is transformed into the problem of a linear system of algebraic equations. This method also belongs to localized radial basis function method or the closest point method. To compare RBF-FD method with another RBF technique, radial basis function collocation method (RBFCM) and the method of approximate particular solutions (MAPS) are also considered to solve such inverse problem, and in the computation the standard Tikhonov regularization technique with L-curve method for choose optional regularized parameter is used for solving the highly ill condition system of linear equations. Several numerical examples are presented to demonstrate the ability of the present approach for solving the backward heat conduction problem.

The generalized finite difference method for an inverse boundary value problem in three-dimensional thermo-elasticity

In this study, a new framework for the efficient and accurate solutions for an inverse problem associated with three-dimensional (3D) coupled thermo-elasticity equation is presented. The ill-conditioned problem is solved here with the generalized finite difference method (GFDM) together with the first-order Tikhonov regularization technique and the L-curve criterion. The GFDM uses the Taylor series expansions and the moving least squares approximation to derive explicit formulae for the required partial derivatives of unknown variables. The method is truly meshless that can be applied for solving problems merely defined over irregular clouds of points. In addition, for problems involving complex geometries, a new distance criterion for adaptive selection of nodes in the GFDM simulations is proposed. Preliminary numerical experiments show that the regularized GFDM proposed in this study are very promising for accurate and efficient inverse thermo-elasticity simulations, even with a comparatively large level of noise added into the input data.

Analysis of the High-Frequency Response of Thin Wires Irradiated by Electromagnetic Waves Through Tikhonov Regularization Technique

The formula for analyzing the response characteristics of the induced current on a transmission line (TL) excited by high-frequency electromagnetic (EM) waves is a mixed integral-differential equation. The integral part is the Fredholm integral equation of the first kind with serious ill-posedness, which causes difficulty in solving it stably. The regularization technique is a useful method for the analysis of the ill-posed problems. Therefore, in this paper, a method based on the Tikhonov regularization technique is proposed to analyze the high-frequency electromagnetic response of a finite length TL illuminated by external EM waves. The integral operator is discretized by the Simpson formula. The validity and correctness of the proposed method have been verified by numerical examples and the applicability of the proposed method at different errors has been analyzed via the TL with different height.

Split Bregman iteration based reconstruction algorithm for electrical capacitance tomography

The electrical capacitance tomography (ECT) technique uses the measured capacitance data to reconstruct the permittivity distribution in a specific measurement area, in which the performances of reconstruction algorithms play a crucial role in the reliability of measurement results. According to the Tikhonov regularization technique, a new cost function with the total least squares technique and the ℓ1-norm based regularizer is presented, in which measurement noises, model deviations and the influence of the outliers in the measurement data are simultaneously considered. The split Bregman technique and the fast-iterative shrinkage-thresholding method are combined into a new iterative scheme to solve the proposed cost function efficiently. Numerical experiment results show that the proposed algorithm achieves the boost in the precision of reconstruction, and under the noise-free condition the image errors for the imaging targets simulated in this paper, that is, 8.4%, 12.4%, 13.5% and 6.4%, are smaller than the linear backprojection (LBP) algorithm, the Tikhonov regularization (TR) algorithm, the truncated singular value decomposition (TSVD) algorithm, the Landweber algorithm and the algebraic reconstruction technique (ART).

Estimation of thermal impulse response of a multi-layer building wall through in-situ experimental measurements in a dynamic regime with applications

The dynamic thermal characteristics of the components of a building have a primary influence on the energy performance of the building’s envelope under real environmental conditions. In this study, a novel approach for estimation of the thermal impulse response (TIR) functions and determination of the dynamic thermal characteristics of a multilayer façade wall with unknown thermal properties, structure, and dimensions is proposed. Unlike existing approaches, such as those presented by Luo et al. (2010) and Fernandes et al. (2015), which are based on the use of known physical parameters and dimensions of the considered structure for determination of the transfer function, the proposed framework is based solely on data from in-situ experimental measurements of surface temperatures and thermal fluxes through the inner and outer wall surfaces in a dynamic regime.Consequently, the estimated TIR functions and dynamic thermal characteristics reflect the actual physical conditions of the considered building wall. The building wall is modelled as a two-input, two-output linear time-invariant (LTI) dynamic system where the surface temperatures and fluxes from both sides are used as system inputs and outputs, respectively. The input and output quantities are related by the convolution integrals and TIR functions. The TIR functions are obtained using the measured data and the least square estimator. As the corresponding system of linear equations is ill-posed, the Tikhonov regularization technique with a single parameter is implemented to overcome the numerical difficulties. The optimal regularization parameter is obtained using the L-curve approach. The estimated TIR functions are validated by comparison with the analytical solutions. The dynamic thermal characteristics of the considered building wall with unknown parameters are determined using the Fourier transform (FT) of the estimated TIR functions. The practical applications of the estimated TIR functions related to the energy performance of buildings (EPB) and energy efficiency, along with additional validation, are demonstrated by the evaluation of the dynamic thermal characteristics, cumulative heat losses, heat accumulation, conductive part of thermal transmittance (U-value), and surface heat fluxes, using only the estimated TIR functions and a control set of the experimental data.

Robust sparse linear regression for tokamak plasma boundary estimation using variational Bayes

Precise control of the shape of plasma in a tokamak requires reliable reconstruction of the plasma boundary. The problem of boundary estimation can be reduced to a simple linear regression with a potentially infinite amount of regressors. This regression problem poses some difficulties for classical methods. The selection of regressors significantly influences the reconstructed boundary. Also, the underlying model may not be valid during certain phases of the plasma discharge. Formal model structure estimation technique based on the automatic relevance principle yields a version of sparse least squares estimator. In this contribution, we extend the previous method by relaxing the assumption of Gaussian noise and using Student’s t-distribution instead. Such a model is less sensitive to potential outliers in the measurement. We show on simulations and real data that the proposed modification improves estimation of the plasma boundary in some stages of a plasma discharge. Performance of the resulting algorithm is evaluated with respect to a more detailed and computationally costly model which is considered to be the “ground truth” The results are also compared to those of Lasso and Tikhonov regularization techniques.

A new methodology for the estimation of wheel–rail contact forces at a high-frequency range

Online measurement of wheel–rail contact forces is nowadays in demand for evaluating safety and manoeuvring the condition in real time and in real operation. In this study, the wheel–rail contact forces are estimated using a novel indirect identification method based on the measured radial strain on the wheel web. Further, the strain response of the rolling wheel is derived using an analytical solution of the disk under a rotating load, and a scheme was prepared for the identification of the rolling wheel parameters and its corresponding characteristic matrix. An appropriate angular strain configuration is employed to eliminate the effect of wheel rotation. The Tikhonov regularization technique is employed to solve the ill-posed least square problem and to attenuate the effect of noisy measurement and numerical uncertainty during the estimation of the forces. A finite element model of the rotating load is then constructed to investigate the effectiveness and accuracy of the proposed methodology. The effects of the rotating speed, loading and measurement noise on the estimated normal force are studied. It is found that neglecting the effect of the rotating speed causes a notable error particularly in the high-speed range.

Reconstruction method for temperature field measurement using ultrasonic tomography

The gas temperature field measurement is crucial for the monitoring of the objects of interest. Owing to the distinct advantages, including non-invasive sensing, high safety, etc., the ultrasonic tomography (UT) method is considered to be a promising visualization measurement method, in which acquiring high-quality images is vital for real-world applications. In this paper, a new UT measurement system that integrates the Data Acquisition Card and a Single Chip Microcomputer is proposed for the temperature distribution measurement. A hardware system is designed to alleviate the inertia delay phenomenon. A reconstruction model that simultaneously emphasizes the inaccurate properties on the reconstruction model and the measurement data is proposed. By means of introducing the radial basis function method and the framework of the Tikhonov regularization technique, a two-stage reconstruction method is proposed to alleviate the ill-posed nature in the UT inverse problem. A new objective functional is proposed to convert the original reconstruction task into an optimization problem, and the simulated annealing algorithm is deployed to effectively solve the proposed objective functional. Experimental results validate the feasibility and effectiveness of the proposed reconstruction method.

Structural damage detection based onl1regularization using natural frequencies and mode shapes

Conventional vibration-based damage detection methods employ the Tikhonov regularization in model updating to deal with the problems of underdeterminacy and measurement noise. However, the Tikhonov regularization technique tends to provide over smooth solutions that the identified damage is distributed to many structural elements. This result does not match the sparsity property of the actual damage scenario, in which structural damage typically occurs at a small number of locations only in comparison with the total elements of the entire structure. In this study, an l1 regularization-based model updating technique is developed by utilizing the sparsity of the structural damage. Both natural frequencies and mode shapes are employed during the model updating. A strategy of selecting the regularization parameter for the l1 regularization problem is also developed. A numerical and an experimental examples are utilized to demonstrate the effectiveness of the proposed damage detection method. The results showed that the proposed l1 regularization-based method is able to locate and quantify the sparse damage correctly over a large number of elements. The effects of the mode number on the damage detection results are also investigated. The advantage of the present l1 regularization over the traditional l2 regularization method in damage detection is also demonstrated.

Investigation of gas–solid bubbling fluidized beds using ECT with a modified Tikhonov regularization technique

Electrical capacitance tomography (ECT) provides a non‐intrusive means to visualize cross‐sectional material distribution of gas–solid bubbling fluidized beds. Successful application of ECT strongly depends on the image reconstruction algorithm used. For on‐line measurements of bubbling fluidized beds, employing an algorithm that can produce high‐quality images without extensive computation is necessary. Using the conventional Tikhonov regularization algorithm, image quality in the central area is basically satisfied but suffers from artifacts in the near‐wall region. To solve this problem, a similar division operation learned from linear back projection was introduced to modify the conventional Tikhonov algorithm. Both numerical simulations and experiments were performed to evaluate the modified technique. The results indicate that the artifacts can be effectively removed and the reconstructed image quality is similar to Landweber method with dozens of iterations. Furthermore, the modified Tikhonov technique shows high accuracy when obtaining important hydrodynamic parameters in gas–solid bubbling fluidized beds. © 2017 American Institute of Chemical Engineers AIChE J, 63: 29–41, 2018

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.

Structural damage detection by a new iterative regularization method and an improved sensitivity function

A new sensitivity-based damage detection method is proposed to identify and estimate the location and severity of structural damage using incomplete noisy modal data. For these purposes, an improved sensitivity function of modal strain energy (MSE) based on Lagrange optimization problem is derived to adapt the initial sensitivity formulation of MSE to damage detection problem with the aid of new mathematical approaches. In the presence of incomplete noisy modal data, the sensitivity matrix is sparse, rectangular, and ill-conditioned, which leads to an ill-posed damage equation. To overcome this issue, a new regularization method named as Regularized Least Squares Minimal Residual (RLSMR) is proposed to solve the ill-posed damage equation. This method relies on Krylov subspace and exploits bidiagonalization and iterative algorithms to solve linear mathematical systems. For the majority of Krylov subspace methods, conventional direct methods for the determination of an optimal regularization parameter may not be proper. To cope with this limitation, a hybrid technique is introduced that depends on the residual of RLSMR method, the number of iterations, and the bidiagonalization algorithm. The accuracy and performance of the improved and proposed methods are numerically examined by a planner truss by incorporating incomplete noisy modal parameters and finite element modeling errors. A comparative study on the initial and improved sensitivity functions is conduced to investigate damage detectability of these sensitivity formulations. Furthermore, the accuracy and robustness of RLSMR method in detecting damage are compared with the well-known Tikhonov regularization method. Results show that the improved sensitivity of MSE is an efficient tool for using in the damage detection problem due to a high sensitivity to damage and reliable damage detectability in comparison with the initial sensitivity function. Additionally, it is observed that the RLSMR method with the aid of the hybrid technique successfully solves the ill-posed damage equation and provides better damage detection results compared with the Tikhonov regularization technique.

A stable numerical scheme for a time fractional inverse parabolic equation

In this paper, we consider a time fractional inverse heat conduction problem of finding the temperature distribution and the heat flux on the boundary , when the time fractional derivative is interpreted in the sense of Caputo. We prove that this problem is an ill-posed problem. For finding a stable solution, the Tikhonov regularization technique is applied. A finite difference scheme is considered by using the given temperature at a point and the given heat flux on the boundary . Stability analysis by using the Von Neumann (or Fourier) method and convergence are discussed. Numerical examples show that the proposed method is stable and works well.

An inversion method for cometary atmospheres

Remote observation of cometary atmospheres produces a measurement of the cometary emissions integrated along the line of sight. This integration is the so-called Abel transform of the local emission rate. The observation is generally interpreted under the hypothesis of spherical symmetry of the coma. Under that hypothesis, the Abel transform can be inverted. We derive a numerical inversion method adapted to cometary atmospheres using both analytical results and least squares fitting techniques. This method, derived under the usual hypothesis of spherical symmetry, allows us to retrieve the radial distribution of the emission rate of any unabsorbed emission, which is the fundamental, physically meaningful quantity governing the observation. A Tikhonov regularization technique is also applied to reduce the possibly deleterious effects of the noise present in the observation and to warrant that the problem remains well posed. Standard error propagation techniques are included in order to estimate the uncertainties affecting the retrieved emission rate. Several theoretical tests of the inversion techniques are carried out to show its validity and robustness. In particular, we show that the Abel inversion of real data is only weakly sensitive to an offset applied to the input flux, which implies that the method, applied to the study of a cometary atmosphere, is only weakly dependent on uncertainties on the sky background which has to be subtracted from the raw observations of the coma. We apply the method to observations of three different comets observed using the TRAPPIST telescope: 103P/ Hartley 2, F6/ Lemmon and A1/ Siding Spring. We show that the method retrieves realistic emission rates, and that characteristic lengths and production rates can be derived from the emission rate for both CN and C2 molecules. We show that the retrieved characteristic lengths can differ from those obtained from a direct least squares fitting over the observed flux of radiation, and that discrepancies can be reconciled for by correcting this flux by an offset (to which the inverse Abel transform is nearly not sensitive). The A1/Siding Spring observations were obtained very shortly after the comet produced an outburst, and we show that the emission rate derived from the observed flux of CN emission at 387nm and from the C2 emission at 514.1nm both present an easily-identifiable shoulder that corresponds to the separation between pre- and post-outburst gas. As a general result, we show that diagnosing properties and features of the coma using the emission rate is easier than directly using the observed flux, because the Abel transform produces a smoothing that blurs the signatures left by features present in the coma. We also determine the parameters of a Haser model fitting the inverted data and fitting the line-of-sight integrated observation, for which we provide the exact analytical expression of the line-of-sight integration of the Haser model.

Substructural interface force identification with limited vibration measurements

This paper presents a dynamic substructural interface force identification approach with limited vibration measurements from the target substructure. The substructure is considered to be subjected to both the external excitation forces at some specific degrees-of-freedom (DOFs) within the substructure and the interface forces at interface DOFs. The interface force time histories are represented as orthogonal polynomials and their discrete Fourier transform is expressed in the matrix multiplication form which transforms the time domain interface forces into their Fourier transforms in the frequency domain. A frequency-time domain method is used to formulate the identification equation based on the relationship between two sets of response vectors in a substructure. The coefficients of orthogonal functions are identified with the Tikhonov regularization technique. Numerical studies on a seven-storey plane frame structure are conducted to investigate the accuracy and efficiency of the proposed substructure interface force identification approach.

Fundamental kernel-based method for backward space–time fractional diffusion problem

Based on kernel-based approximation technique, we devise in this paper an efficient and accurate numerical scheme for solving a backward space–time fractional diffusion problem (BSTFDP). The kernels used in the approximation are the fundamental solutions of the space–time fractional diffusion equation expressed in terms of inverse Fourier transform of Mittag-Leffler functions. The use of Inverse fast Fourier transform (IFFT) technique enables an accurate and efficient evaluation of the fundamental solutions and gives a robust numerical algorithm for the solution of the BSTFDP. Since the BSTFDP is intrinsic ill-posed, we apply the standard Tikhonov regularization technique to obtain a stable solution to the highly ill-conditioned resultant system of linear equations. For choosing optimal regularization parameter, we combine the regularization technique with the generalized cross validation (GCV) method for an optimal placement of the source points in the use of fundamental solutions. Meanwhile, the proposed algorithm also speeds up the previous method given in Dou and Hon (2014). Several numerical examples are constructed to verify the accuracy and efficiency of the proposed method.

Reconstruction of inaccessible boundary value in a sideways parabolic problem with variable coefficients—Forward collocation with finite integration method

We investigate a sideways problem of reconstructing an inaccessible boundary value for parabolic equation with variable coefficients. Formulating the sideways problem into a sequence of well-posed direct problems (DP) and a system of Ordinary Differential Equations (ODE), we combine the recently developed finite integration method (FIM) with radial basis functions (RBF) to iteratively obtain the solution of each DP by solving an ill-posed linear system. The use of numerical integration instead of finite quotient formula in FIM completely avoids the well known roundoff-discretization errors problem in finite difference method and the use of RBF as forward collocation method (FCM) gives a truly meshless computational scheme. For tackling the ill-posedness of the sideways problem, we adapt the traditional Tikhonov regularization technique to obtain stable solution to the system of ODEs. Convergence analysis is then derived and error estimate shows that the error tends to zero when perturbation δ→0. We can then obtain highly accurate and stable solution under some assumptions. Numerical results validate the feasibility and effectiveness of the proposed numerical algorithms.

Inverse Problems in Population Balances. Determination of Aggregation Kernel by Weighted Residuals

The phenomenology of particulate systems are embedded into the population balance model through breakage, aggregation, or growth kernels. Aggregation kernel is a difficult kernel to predict from first principle because of its nonlinear nature. In this work we demonstrate a new methodology to obtain the aggregation kernel through inverse problem approach. This new approach is based on the method of weighted residuals and does not rely on specific traits of the system like self-similarity. The residual approach reduces the inverse problem into solution of a set of linear equations. However, the system of equations is badly conditioned and therefore requires regularization for accurate solution. In this work Tikhonov regularization technique has been used. The new method has been demonstrated successfully for constant, sum, and product kernel.