Implementation of Modified Conjugate Gradient Algorithm in Electromagnetic Tomography Lab System

A generalized data windowing scheme for adaptive conjugate gradient algorithms

Self-scaled conjugate gradient training algorithms

An adaptive array code acquisition for direct-sequence/ code-division multiple access (DS/CDMA) systems was recently proposed to enhance the performance of the conventional correlator-based method. The scheme consists of an adaptive spatial and an adaptive temporal filter, and can simultaneously perform beamforming and code-delay estimation. Unfortunately, the scheme uses a least-mean-square (LMS) adaptive algorithm, and its convergence is slow. Although the recursive-least-squares (RLS) algorithm can be applied, the computational complexity will greatly increase. In this paper, we solve the dilemma with a low-complexity conjugate gradient (LCG) algorithm, which can be considered as a special case of a modified conjugate gradient (MCG) algorithm. Unlike the original conjugate gradient (CG) algorithm developed for adaptive applications, the proposed method, exploiting the special structure inherent in the input correlation matrix, requires a low computational-complexity. It can be shown that the computational complexity of the proposed method is on the same order of the LMS algorithm. However, the convergence rate is improved significantly. Simulation results show that the performance of adaptive array code acquisition with the proposed CG algorithm is comparable to that with the original CG algorithm.

Transient analysis of large-scale power delivery network (PDN) is a critical task to ensure the functional correctness and desired performance of today’s integrated circuits (ICs), especially if significant transient noises are induced by clock and/or power gating due to the utilization of extensive power management. In this paper, we propose an efficient algorithm for PDN transient analysis based on sparse approximation. The key idea is to exploit the fact that the transient response caused by clock/power gating is often localized and the voltages at many other “inactive” nodes are almost unchanged, thereby rendering a unique sparse structure. By taking advantage of the underlying sparsity of the solution structure, a modified conjugate gradient algorithm is developed and tuned to efficiently solve the PDN analysis problem with low computational cost. Our numerical experiments based on standard benchmarks demonstrate that the proposed transient analysis with sparse approximation offers up to $2.2\times $ runtime speedup over other traditional methods, while simultaneously achieving similar accuracy.

A modified conjugate gradient algorithm is presented, which possesses the sufficient descent condition without any line search. The global convergence with the weak Wolfe-Powell (WWP) line search rule is established for nonconvex function under suitable conditions. Numerical results show that the given method is interesting.

Image reconstruction algorithm of electromagnetic tomography based on fractional Kalman filter

The gradient methods for the solution of large sparse sets of linear finite element equations in subsurface flow and land subsidence modeling may be extraordinarily accelerated by a suitable matrix modification. While the accelerated steepest descent turns out to converge fairly fast, the modified conjugate gradient (MCG) scheme exhibits a much faster convergence. For the range of the problem size N explored in the present paper (800≤N≤2200) the iterations required by MCG to provide very accurate results prove to be of the order of \(\sqrt {\text{N}} .\)

Although numerical calculations of heat transfer and fluid flow can provide detailed insights into welding processes and welded materials, these calculations are complex and unsuitable in situations where rapid calculations are needed. A recourse is to train and validate a neural network, using results from a well tested heat and fluid flow model to significantly expedite calculations and ensure that the computed results conform to the basic laws of conservation of mass, momentum and energy. Seven feedforward neural networks were developed for gas metal arc (GMA) fillet welding, one each for predicting penetration, leg length, throat, weld pool length, cooling time between 800°C and 500°C, maximum velocity and peak temperature in the weld pool. Each model considered 22 inputs that included all the welding variables, such as current, voltage, welding speed, wire radius, wire feed rate, arc efficiency, arc radius, power distribution, and material properties such as thermal conductivity, specific heat and temperature coefficient of surface tension. The weights in the neural network models were calculated using the conjugate gradient (CG) method and by a hybrid optimisation scheme involving the CG method and a genetic algorithm (GA). The neural network produced by the hybrid optimisation model produced better results than the networks based on the CG method with various sets of randomised initial weights. The CG method alone was unable to find the best optimal weights for achieving low errors. The hybrid optimisation scheme helped in finding optimal weights through a global search, as evidenced by good agreement between all the outputs from the neural networks and the corresponding results from the heat and fluid flow model.

A modified conjugate gradient method based on the Tikhonov system for computerized tomography (CT)

The modified conjugate gradient (CG) method, called the conjugate guided gradient (CGG) method, is a robust iterative inversion method producing a parsimonious model estimation. The CG method for solving least-squares (LS) (i.e., ℓ2-norm minimization) problems is modified to solve for different norms or different minimization criteria by guiding the gradient vector appropriately during iteration steps. Guiding is achieved by iteratively reweighting either the residual vector or the gradient vector during iteration steps like the iteratively reweighted least-squares IRLS method does. Robustness is achieved by weighting the residual vector and parsimonious model estimation is obtained by weighting the gradient vector. Unlike the IRLS method, however, the CGG method doesn't change the corresponding forward operator of the problem and is implemented in a linear inversion template. Therefore, the CGG method requires less computation than the IRLS method. Since the solution in the CGG method is found in a least-squares sense along the gradient direction guided by the weights, this solution can be interpreted as the LS solution located in the guided gradient direction. Guiding the gradient gives us more flexibility in the choice of weighting parameters than the IRLS method. I applied the CGG method to velocity-stack inversion, and the results show that the CGG method gives a far more robust and parsimonious model estimation than the standard ℓ2-norm solution, with results comparable to the ℓ1-norm IRLS solution.

A new CMA adaptive beamforming algorithm based on modified conjugate gradient method is proposed. This algorithm focuses on a new technique for data block modification method with conjugate gradient, and overcomes the shortcoming of traditional CMA algorithms, which has slow convergence speed or vast complexity. Computer simulations show that the algorithm has better convergence performance and better SINR capability than other kinds of CMA algorithms.

This study introduces an improved three-term conjugate gradient (CG) algorithm which fulfills the descent conditions and exhibits good global convergence properties. The development of this new algorithm is based on the findings of recently introduced generalized RMIL CG technique. The algorithm has been modified so that the search direction would always meet the sufficient descent condition no matter the line-search techniques employed. Under some given assumptions, the algorithm’s global convergence result for the general nonconvex functions is established. The numerical efficacy of the suggested three-term CG algorithm is assessed through comparisons with four other CG algorithms. The assessment is performed using an array of unconstrained optimization benchmark functions. The acquired findings indicate that all the considered algorithms exhibit similar performance, particularly the variants of the RMIL method. One algorithm is more robust and slightly faster than the rest. However, it is essential to highlight that the newly developed CG formula outperforms all the other algorithms, including a recently presented three-term PRP algorithm. The study further expanded the proposed method to address an image restoration problem. The computational experiments yielded promising results, proving that the proposed CG algorithm surpasses others. It produces higher-quality output images, requires less CPU time (sec), and achieves higher PSNR values.

The three-dimensional (3D) reconstruction of Electromagnetic Tomography (EMT) is an important task for many applications, such as the non-destructive testing of inner defects in rail systems. Additionally, image reconstruction algorithms utilizing deep learning methods have been verified to be useful in recent years. Therefore, the interpretability of deep learning is a question that is relevant to its application in other areas. This paper proposes an innovative rotational convolution pattern, Conv-P, for convolutional neural network (CNN) image reconstruction in a 3D EMT system. This pattern is based on the projection relationships inherent in tomographic imaging, where each convolution is performed on adjacent projections along the excitation rotation direction. The advantage of this pattern is that it can generate the convolution process by utilizing the 3D structural information from real sensors. To verify the effectiveness of this convolution pattern, we constructed a 3D dual-layer 16-coil EMT model and tested its image reconstruction performance. The results demonstrate that, compared with two common convolution patterns, Conv-P achieves a 4.7% and 4.1% increase in the Image Correlation Coefficient (CC), a 19.8% and 13.1% reduction in the Relative Image Error (IE), a 0.67% and 1.59% increase in the Peak Signal-to-Noise Ratio (PSNR), and a 3.24% and 0.74% increase in the Structural Similarity Index Measure (SSIM).

At present, the conjugate gradient (CG) method of Hager and Zhang (Hager and Zhang, SIAM Journal on Optimization, 16(2005)) is regarded as one of the most effective CG methods for optimization problems. In order to further study the CG method, we develop the Hager and Zhang's CG method and present two modified CG formulas, where the given formulas possess the value information of not only the gradient but also the function. Moreover, the sufficient descent condition will be holden without any line search. The global convergence is established for nonconvex function under suitable conditions. Numerical results show that the proposed methods are competitive to the normal conjugate gradient method.

Conjugate gradient (CG) method is a line search algorithm mostly known for its wide application in solving unconstrained optimization problems. Its low memory requirements and global convergence properties makes it one of the most preferred method in real life application such as in engineering and business. In this paper, we present a new CG method based on AMR* and CD method for solving unconstrained optimization functions. The resulting algorithm is proven to have both the sufficient descent and global convergence properties under inexact line search. Numerical tests are conducted to assess the effectiveness of the new method in comparison to some previous CG methods. The results obtained indicate that our method is indeed superior.