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.

The four dimensional variational data assimilation with multiple regularization parameters as a weak constraint(Tikh-4D-Var) and its preliminary application on typhoon initialization

Traditional variational data assimilation （VDA） with only one regularization parameter constraint cannot produce optimal error tuning for all observations. In this paper, a new data assimilation method of ＂four dimensional variational data assimilation （4D-Var） with multiple regularization parameters as a weak constraint （Tikh-4D-Var）＂ is proposed by imposing different reg- ularization parameters for different observations. Meanwhile, a new multiple regularization parameters selection method, which is suitable for actual high-dimensional data assimilation system, is proposed based on the posterior information of 4D-Var system. Compared with the traditional single regularization parameter selection method, computation of the proposed multiple regularization parameters selection method is smaller. Based on WRF3.3.1 4D-Vat data assimilation system, initiali- zation and simulation of typhoon Chaba （2010） with the new Tikh-4D-Var method are compared with its counterpart 4D-Var to demonstrate the effectiveness of the new method. Results show that the new Tikh-4D-Var method can accelerate the con vergence with less iterations. Moreover, compared with 4D-Var method, the typhoon track, intensity （including center surface pressure and maximum wind speed） and structure prediction are obviously improved with Tikh-4D-Var method for 72-h pre- diction. In addition, the accuracy of the observation error variances can be reflected by the multiple regularization parameters.

Adaptive Parameter Selection for Total Variation Image Deconvolution

In this paper,we propose a discrepancy rule-based method to automatically choose the regularization parameters for total variation image restoration problems. The regularization parameters are adjusted dynamically in each iteration.Numerical results are shown to illustrate the performance of the proposed method.

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 Preconditioned Fractional Tikhonov Regularization Method for Large Discrete Ill-posed Problems

The generalized Tikhonov regularization method is one of the most classical methods for the solution of linear systems of equations that arise from the discretization of linear ill-posed problems.However,the approximate solution obtained by the Tikhonov regularization method in general form may lack many details of the exact solution.Combining the fractional Tikhonov method with the preconditioned technique,and using the discrepancy principle for determining the regularization parameter,we present a preconditioned projected fractional Tikhonov regularization method for solving discrete ill-posed problems.Numerical experiments illustrate that the proposed algorithm has higher accuracy compared with the existing classical regularization methods.

Gas emission source term estimation with 1-step nonlinear partial swarm optimization-Tikhonov regularization hybrid method

Source term identification is very important for the contaminant gas emission event. Thus, it is necessary to study the source parameter estimation method with high computation efficiency, high estimation accuracy and reasonable confidence interval. Tikhonov regularization method is a potential good tool to identify the source parameters. However, it is invalid for nonlinear inverse problem like gas emission process. 2-step nonlinear and linear PSO （partial swarm optimization）-Tikhonov regularization method proposed previously have estimated the emission source parameters successfully. But there are still some problems in computation efficiency and confidence interval. Hence, a new 1-step nonlinear method combined Tikhonov regularizafion and PSO algorithm with nonlinear forward dispersion model was proposed. First, the method was tested with simulation and experiment cases. The test results showed that 1-step nonlinear hybrid method is able to estimate multiple source parameters with reasonable confidence interval. Then, the estimation performances of different methods were compared with different cases. The estimation values with 1-step nonlinear method were close to that with 2-step nonlinear and linear PSO-Tikhonov regularization method, 1-step nonlinear method even performs better than other two methods in some cases, especially for source strength and downwind distance estimation. Compared with 2-step nonlinear method, 1-step method has higher computation efficiency. On the other hand, the confidence intervals with the method proposed in this paper seem more reasonable than that with other two methods. Finally, single PSO algorithm was compared with 1-step nonlinear PSO-Tikhonov hybrid regularization method. The results showed that the skill scores of 1-step nonlinear hybrid method to estimate source parameters were close to that of single PSO method and even better in some cases. One more important property of 1-step nonlinear PSO-Tikhonov regularization method is its reasonable confidence interval, which is not obtained by single PSO algorithm. Therefore, 1-step nonlinear hybrid regularization method proposed in this paper is a potential good method to estimate contaminant gas emission source term.

Evolvement regularity of signal control parameters of urban road in cold region

In snow-icy road environment, the survey data indicate that the largest decrease in traffic flow running characters occurs when snow and ice begin to accumulate on the road surface. Saturation flow is decreased by 16% , speed is decreased by 30% , and start-up lost time is increased by 27%. Based on the signal control theory of HCM and Webster, the character values of traffic flow in different urban road environments were investigated, and the evolvement regularity of signal control parameters such as cycle, split, green time, offset, yellow time and red time in snow-icy road environment was analyzed. The impact factors and the changes in the scope of signal control parameters were achieved. Simulation results and practical application show that the signal control plan of road enviromnent without snow and ice will increase the vehicle delay, stop length and traffic congestion in snow-icy road environment. Thus, the traffic signal control system should address a suitable signal control plan based on different road environments.

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.

Solving Multicollinearity in Dam Regression Model Using TSVD

Targeting the multicollinearity problem in dam statistical model and error perturbations resulting from the monitoring process, we built a regularized regression model using Truncated Singular Value Decomposition （TSVD）. An earth-rock dam in China is presented and discussed as an example. The analysis consists of three steps： multicollinearity detection, regularization pa- rameter selection, and crack opening modeling and forecasting. Generalized Cross-Validation （GCV） function and L-curve criterion are both adopted in the regularization parameter selection. Partial Least-Squares Regression （PLSR） and stepwise regression are also included for comparison. The result indicates the TSVD can promisingly solve the multicollinearity problem of dam regression models. However, no general rules are available to make a decision when TSVD is superior to stepwise regression and PLSR due to the regularization parameter-choice problem. Both fitting accuracy and coefficients＇ reasonability should be considered when evaluating the mode/reliability.

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.