Numerical estimation of choice of the regularization parameter for NMR T2 inversion

Nuclear Magnetic inversion is the basis of NMR Resonance （NMR） T2 logging interpretation. The regularization parameter selection of the penalty term directly influences the NMR T2 inversion result. We implemented both norm smoothing and curvature smoothing methods for NMR T2 inversion, and compared the inversion results with respect to the optimal regular- ization parameters （（Xopt） which were selected by the dis- crepancy principle （DP）, generalized cross-validation （GCV）, S-curve, L-curve, and the slope of L-curve methods, respectively. The numerical results indicate that the DP method can lead to an oscillating or oversmoothed solution which is caused by an inaccurately estimated noise level. The （Xopt selected by the L-curve method is occa- sionally small or large which causes an undersmoothed or oversmoothed T2 distribution. The inversion results from GCV, S-curve and the slope of L-curve methods show satisfying inversion results. The slope of the L-curve method with less computation is more suitable for NMR T2 inversion. The inverted T2 distribution from norm smoothing is better than that from curvature smoothing when the noise level is high.

Effect of regularization parameters on geophysical reconstruction

In this paper we discuss the edge-preserving regularization method in the reconstruction of physical parameters from geophysical data such as seismic and ground-penetrating radar data. In the regularization method a potential function of model parameters and its corresponding functions are introduced. This method is stable and able to preserve boundaries, and protect resolution. The effect of regularization depends to a great extent on the suitable choice of regularization parameters. The influence of the edge-preserving parameters on the reconstruction results is investigated and the relationship between the regularization parameters and the error of data is described.

Optimization of regularization parameter of inversion in particle sizing using light extinction method

In particle sizing by light extinction method, the regularization parameter plays an important role in applying regularization to find the solution to ill-posed inverse problems. We combine the generalized cross-validation （GCV） and L-curve criteria with the Twomey-NNLS algorithm in parameter optimization. Numerical simulation and experimental validation show that the resistance of the newly developed algorithms to measurement errors can be improved leading to stable inversion results for unimodal particle size distribution.

Retrieval of Atmospheric Temperature and Moisture Vertical Profiles from Satellite Advanced Infrared Sounder Radiances with a New Regularization Parameter Selecting Method

Considering the characteristics of nonlinear problems,a new method based on the L-curve method and including the concept of entropy was designed to select the regularization parameter in the one-dimensional variational analysis-based sounding retrieval method.In the first iteration,this method uses an empirical regularization parameter derived by minimizing the entropy of variables.During subsequent iterations,it uses the L-curve method to select the regularization parameter in the vicinity of the regularization parameter selected in the last iteration.The new method was employed to select the regularization parameter in retrieving atmospheric temperature and moisture profiles from Atmospheric Infrared Sounder radiance measurements selected from the first day of each month in 2008.The results show that compared with the original L-curve method,the new method yields 5.5%and 2.5%improvements on temperature and relative humidity profiles,respectively.Compared with the discrepancy principle method,the improvements on temperature and relative humidity profiles are 1.6%and 2.0%,respectively.

Regularization and Choice of the Parameter for the Third Kind Nonlinear Volterra-Stieltjes Integral Equation Solutions

The article is considering the third kind of nonlinear Volterra-Stieltjes integral equations with the solution by Lavrentyev regularizing operator. A uniqueness theorem was proved, and a regularization parameter was chosen. This can be used in further development of the theory of the integral equations in non-standard problems, classes in the numerical solution of third kind Volterra-Stieltjes integral equations, and when solving specific problems that lead to equations of the third kind.

Downward continuation of airborne geomagnetic data based on two iterative regularization methods in the frequency domain

Downward continuation is a key step in processing airborne geomagnetic data. However,downward continuation is a typically ill-posed problem because its computation is unstable; thus, regularization methods are needed to realize effective continuation. According to the Poisson integral plane approximate relationship between observation and continuation data, the computation formulae combined with the fast Fourier transform（FFT）algorithm are transformed to a frequency domain for accelerating the computational speed. The iterative Tikhonov regularization method and the iterative Landweber regularization method are used in this paper to overcome instability and improve the precision of the results. The availability of these two iterative regularization methods in the frequency domain is validated by simulated geomagnetic data, and the continuation results show good precision.

A New Extended BIC and Sequential Lasso Regression Analysis and Their Application in Classification

In this paper, firstly, we propose a new method for choosing regularization parameter λ for lasso regression, which differs from traditional method such as multifold cross-validation, our new method gives the maximum value of parameter λ directly. Secondly, by considering another prior form over model space in the Bayes approach, we propose a new extended Bayes information criterion family, and under some mild condition, our new EBIC (NEBIC) is shown to be consistent. Then we apply our new method to choose parameter for sequential lasso regression which selects features by sequentially solving partially penalized least squares problems where the features selected in earlier steps are not penalized in the subsequent steps. Then sequential lasso uses NEBIC as the stopping rule. Finally, we apply our algorithm to identify the nonzero entries of precision matrix for high-dimensional linear discrimination analysis. Simulation results demonstrate that our algorithm has a lower misclassification rate and less computation time than its competing methods under considerations.

An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.

A Regularized Super Resolution Algorithm for Generalized Gaussian Noise

In this paper,an iterative regularized super resolution (SR) algorithm considering non-Gaussian noise is proposed.Based on the assumption of a generalized Gaussian distribution for the contaminating noise,an lp norm is adopted to measure the data fidelity term in the cost function.In the meantime,a regularization functional defined in terms of the desired high resolution (HR) image is employed,which allows for the simultaneous determination of its value and the partly reconstructed image at each iteration step.The convergence is thoroughly studied.Simulation results show the effectiveness of the proposed algorithm as well as its superiority to conventional SR methods.

Two-dimensional NMR inversion based on fast norm smoothing method

Two-dimensional(2D)nuclear magnetic resonance(NMR)inversion operates with massive echo train data and is an ill-posed problem.It is very important to select a suitable inversion method for the 2D NMR data processing.In this study,we propose a fast,robust,and effective method for 2D NMR inversion that improves the computational efficiency of the inversion process by avoiding estimation of some unneeded regularization parameters.Firstly,a method that combines window averaging(WA)and singular value decomposition(SVD)is used to compress the echo train data and obtain the singular values of the kernel matrix.Subsequently,an optimum regularization parameter in a fast manner using the signal-to-noise ratio(SNR)of the echo train data and the maximum singular value of the kernel matrix are determined.Finally,we use the Butler-Reeds-Dawson(BRD)method and the selected optimum regularization parameter to invert the compressed data to achieve a fast 2D NMR inversion.The numerical simulation results indicate that the proposed method not only achieves satisfactory 2D NMR spectra rapidly from the echo train data of different SNRs but also is insensitive to the number of the final compressed data points.

AUTOMATIC AUGMENTED GALERKIN ALGORITHMS FOR FREDHOLM INTEGRAL EQUATIONS OF THE FIRST KIND

In recent papers, Babolian & Delves [2] and Belward[3] described a Chebyshev series method for the solution of first kind integral equations. The expansion coefficients of the solution are determined as the solution of a mathematical programming problem.The method involves two regularization parameters, Cf and r, but values assigned to these parameters are heuristic in nature. Essah & Delves[7] described an algorithm for setting these parameters automatically, but it has some difficulties. In this paper we describe three iterative algorithms for computing these parameters for singular and non-singular first kind integral equations. We give also error estimates which are cheap to compute. Finally, we give a number of numerical examples showing that these algorithms work well in practice.

BLOCK-SYMMETRIC AND BLOCK-LOWER-TRIANGULAR PRECONDITIONERS FOR PDE-CONSTRAINED OPTIMIZATION PROBLEMS＊

Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained optimization problems is usually a major computational task. Because of the complexion for directly seeking the solution of PDE-constrained op- timization problem, we transform it into a system of linear equations of the saddle-point form by using the Galerkin finite-element discretization. For the discretized linear system, in this paper we construct a block-symmetric and a block-lower-triangular preconditioner, for solving the PDE-constrained optimization problem. Both preconditioners exploit the structure of the coefficient matrix. The explicit expressions for the eigenvalues and eigen- vectors of the corresponding preconditioned matrices are derived. Numerical implementa- tions show that these block preconditioners can lead to satisfactory experimental results for the preconditioned GMRES methods when the regularization parameter is suitably small.

Localization and characterization of intermittent pollutant source in buildings with ventilation systems:Development and validation of an inverse model

Terrorist attacks through building ventilation systems are becoming an increasing concern.In case pollutants are intentionally released in a building with mechanical ventilation systems,it is critical to localize the source and characterize its releasing curve.Previous inverse modeling studies have adopted the adjoint probability method to identify the source location and used the Tikhonov regularization method to determine the source releasing profile,but the selection of the prediction model and determination of the regularization parameter remain challenging.These limitations can affect the identification accuracy and prolong the computational time required.To address the difficulties in solving the inverse problems,this work proposed a Markov-chain-oriented inverse approach to identify the temporal release rate and location of a pollutant source in buildings with ventilation systems and validated it in an experimental chamber.In the modified Markov chain,the source term was discrete by each time step,and the pollutant distribution was directly calculated with no iterations.The forward Markov chain was reversed to characterize the intermittently releasing profile by introducing the Tikhonov regularization method,while the regularized parameter was determined by an automatic iterative discrepancy method.The source location was further estimated by adopting the Bayes inference.With chamber experiments,the effectiveness of the proposed inverse model was validated,and the impact of the sensor performance,quantity and placement,as well as pollutant releasing curves on the identification accuracy of the source intensity was explicitly discussed.Results showed that the inverse model can identify the intermittent releasing rate efficiently and promptly,and the identification error for pollutant releasing curves with complex waveforms is about 20%.

Infrared image enhancement based on adaptive weighted guided filter

The physical principle of infrared imaging leads to the low contrast of the whole image,the blurring of contour and edge details,and it is also sensitive to noise.To improve the quality of infrared image and visual effect,an adaptive weighted guided filter(AWGF) for infrared image enhancement algorithm was proposed.The core idea of AWGF algorithm is to propose an adaptive strategy to update the weights of guided filter(GF) parameters,which not only improves the accuracy of regularization parameter estimation in GF theory,but also achieves the purpose of removing infrared image noise and improving its detail contrast.A large number of real infrared images were used to verify AWGF algorithm,and good experimental results were obtained.Compared with other guided filtering algorithms,the halo phenomenon at the edge of infrared images processed by the AWGF algorithm is significantly avoided,and the evaluation parameter values of information entropy(IE),average gradient(AG),and moment of inertia(MI)are relatively high.This shows that the quality of infrared image processed by the AWGF algorithm is better.