Iterative Conjugate Gradient Algorithm Research Articles

Separable Synchronous Gradient‐Based Iterative Algorithms for the Nonlinear ExpARX System

ABSTRACTThis article is aimed to study the parameter identification of the ExpARX system. To overcome the computational complexity associated with a large number of feature parameters, a parameter separation scheme based on the different features of the identification model is introduced. In terms of the phenomenon that the coupling parameters lead to the inability of algorithms, a separable synchronous interactive estimation method is introduced to eliminate the coupling parameters and perform parameter estimation in accordance with the hierarchical principle. For the purpose of achieving high‐accuracy performance and reducing complexity, a separable synchronous gradient iterative algorithm is derived by means of gradient search. In order to improve the identification accuracy, a separable synchronous multi‐innovation gradient iterative algorithm is proposed by introducing the multi‐innovation identification theory. In order to improve the convergence speed, a separable synchronous multi‐innovation conjugate gradient iterative algorithm is proposed by introducing the conjugate gradient theory. Finally, a simulation example and a real‐life example of piezoelectric ceramics are used to verify the effectiveness of the proposed algorithm.

Numerical Reconstruction of Heat Source on a Semi-Infinite Rod

Inverse heat conduction on an unbounded domain has always been the focus and difficulty in engineering and mechanical fields. In this paper, we consider the inverse problem of identifying the heat source by using the terminal measured temperature data. Being different from other ordinary heat conduction problems, our mathematical model is defined in a semi-infinite region, which may lead to many numerical methods that cannot be used directly. Based on the artificial boundary method, we first introduce the truncated boundary and derive the exact boundary conditions on it. Then we use the finite difference method to establish an approximate difference scheme for the forward problem. For the inverse problem, we design the Landweber and conjugate gradient iterative algorithm and do numerical experiments. Numerical results verify the correctness and effectiveness of our algorithm.

Background-oriented schlieren experimental and simulation study of the effect of sidewalls on optical turbulence anisotropy in a thermal convection turbulence simulator

The anisotropic optical structure characteristics within the Rayleigh-Bénard (RB) turbulence simulator have been investigated through experiment and simulation. We collected data using background-oriented schlieren technology and extracted displacements using optical flow algorithms. The wavefront phase can be recovered using the conjugate gradient iterative algorithm. We analyzed the optical structural characteristics within the turbulence simulator under different temperature differences. It is observed that with an increased temperature difference, the spatial correlation of the refractive index field decreases at an accelerated rate as the separation distance increases, and the degree of anisotropy characterized by the phase structure function increases. Large eddy simulation and split-step beam propagation method complement experimental results. Numerical simulations demonstrate consistency with experimental data. The results show that the von Karman power spectral density fits the simulation and experimental phase structure function very well relative to the Kolmogorov power spectral density, which demonstrates that the simulation system combining dynamics simulation and optical simulation can achieve power spectral characteristics that are consistent with reality. The experimental and simulation results are crucial for understanding the flow field's optical structural properties inside the turbulence simulator. They will help to construct a suitable turbulence simulator for optical engineering applications and establish a rational scheme for analyzing its internal optical structural properties.

Convolutional equivalent layer for magnetic data processing

We have developed a fast equivalent-layer method for processing large-scale magnetic data. We demonstrate that the sensitivity matrix associated with an equivalent layer of dipoles can be arranged to a block-Toeplitz Toeplitz-block (BTTB) structure for the case in which the observations and dipoles are aligned on a horizontal and regularly spaced grid. The product of a BTTB matrix and an arbitrary vector represents a discrete convolution and can be efficiently computed via 2D fast Fourier transform. In this case, the matrix-vector product uses only the elements forming the first column of the BTTB matrix and saves computational time and memory. Our convolutional equivalent-layer method uses this approach to compute the matrix-vector products in the iterative conjugate gradient algorithm with the purpose of estimating the physical-property distribution over the equivalent layer for large data sets. Synthetic tests of total-field anomaly data indicate a decrease in floating-point operations and computation runtime of our method compared to the classical approach of solving the least-squares normal equations via Cholesky decomposition. Faster results are obtained for millions of data, showing drastic decreases in computer memory usage and runtime and allowing for performances of magnetic data processing of large data sets on regular desktop computers. Our results also indicate that, compared to the classical Fourier approach, the magnetic data processing with our method requires a similar computation time but produces significantly smaller border effects without using any padding scheme and also is more robust to deal with data on irregularly spaced points or irregularly observation surfaces. A test with irregularly spaced field data over the Carajás Province, Brazil, confirms the efficiency of our method by estimating the physical-property distribution over the equivalent layer and computing the upward continuation.

Three dimensional radial echo planar imaging for functional MRI.

To demonstrate a novel 3D radial echo planar imaging (3D REPI) sequence for flexible, rapid, and motion-robust sampling in fMRI. The 3D REPI method expands on the recently described golden angle rotated EPI trajectory using radial batched internal navigator echoes (TURBINE) approach by exploiting the unused perpendicular direction in the EPI readout to form fast analogues of rotated stack of stars or spirals trajectories that cover all 3 dimensions of k-space. An iterative conjugate gradient algorithm with SENSE reconstruction and time-segmented non-uniform fast Fourier transform (FFT) was used for parallel imaging acceleration and to account for the effects of B0 inhomogeneity. The golden angle rotation allowed for sliding window reconstruction schemes to be applied in brain BOLD fMRI experiments. Combined whole brain visual and motor fMRI experiments were successfully carried out on a clinical 3T scanner at 2 mm isotropic and 1 × 1 × 2 mm3 resolutions using the 3D REPI design. Improved sampling characteristics and image quality were observed for twisted trajectories at the expense of prolonged readout times and off-resonance effects. The ability to correct for rigid motion correction was also demonstrated. 3D REPI presents a flexible approach for segmented volumetric fMRI with motion correction and high in-plane spatial resolutions. Improved BOLD fMRI brain activation maps were obtained using a sliding window reconstruction.

Deep Unfolded Extended Conjugate Gradient Method for Massive MIMO Processing with Application to Reciprocity Calibration

In this paper, we consider deep unfolding the standard iterative conjugate gradient (CG) algorithm to solve a linear system of equations. Instead of being adjusted with known rules, the parameters are learned via backpropagation to yield the optimal results. However, the proposed unfolded CG (UCG) is extended wherein a scalar parameter is substituted by a matrix-parameter to augment the degrees of freedom per layer. Once the training is completed, the UCG has revealed to require far a smaller number of layers than the number of iterations needed using the standard iterative CG. It is also shown to be very robust to noise and outperforms the standard CG in low signal to noise ratio (SNR) region. A key merit of the proposed approach is the fact that no explicit training data is dedicated to the learning phase as the optimization process relies on the residual error which is not explicitly expressed as a function of the desired data. As an example, the proposed UCG is applied to solve the reciprocity calibration problem encountered in massive MIMO (Multiple-Input Multiple-Output) systems.

Fast Parallel FDFD Algorithm for Solving Electromagnetic Scattering Problems

The finite difference frequency domain (FDFD) method is very suitable for working out narrowband problems and resonance problems. However, the FDFD method needs to solve a large complex sparse matrix equation. With the increase of computing scale, the dimension of matrix will increase rapidly, which is difficult to simulate. For improving the computational efficiency of solving the large complex sparse matrix equation and extend the application scope of the FDFD method, a fast parallel FDFD method on the basis of message passing interface (MPI) shared memory technology is proposed in this paper, which is used to solve the electromagnetic scattering problems of electrically large targets. Based on the conjugate gradient iterative algorithm, the large complex sparse matrix is reasonably distributed to each process according to the unequal row allocation scheme, so as to guarantee the load balancing of each process. In addition, the intermediate vectors utilized in total processes are stored in the shared memory of MPI, which reduces the communication time and the consumption of computer memory. The proposed parallel FDFD method is employed to solve the bistatic RCS of the PEC sphere, composite Von warhead and an automobile, compared with the serial FDFD method, the parallel FDFD method greatly improves the computational efficiency when the memory is not increased much.

Secure Joint Transmitter-Receiver Spatial Media-Based Modulation

Joint transmitter-receiver spatial modulation (JSM) is a promising multiple-input multiple-output communication technique that achieves transmit diversity, receive diversity, and multiplexing gain simultaneously. However, the conventional JSM scheme has security risks due to the broadcast nature of the wireless channels. In addition, the spectral efficiency of JSM is not satisfactory since the antenna elements are not fully utilized and a single modulated symbol is transmitted only. In this paper, we propose a secure joint transmitter-receiver spatial media-based modulation (JSMBM) to improve the security and spectral efficiency. Specifically, multiple radio frequency (RF) mirrors are placed close by the transmitting antennas and the transmit index bits can be mapped to the mirror activation patterns. As a result, the spectral efficiency scales linearly with the number of RF mirrors, instead of logarithmical scaling with the number of transmitting antennas. Moreover, the proposed JSMBM scheme enables the transmission of multiple modulated symbols such that the spectral efficiency is further enhanced. In order to protect the proposed JSMBM scheme against passive eavesdropping, we use friendly jamming and probabilistic shaping in a collaborative manner to improve the secrecy performance. The mechanism of probabilistic shaping is to adjust the probability of transmitting each symbol such that the favorable symbols are transmitted with higher probability. We propose a simplified iterative conjugate gradient algorithm to optimize the input probability mass function, which maximizes the secrecy rate. Simulation results validate the proposed algorithm and show that the proposed secure JSMBM scheme achieves higher secrecy rate than the benchmark schemes, especially when the eavesdropper has more receive antennas than the legitimate receiver.

A new multiple-point grade estimation method by implicit volterra series

Ore grade estimation plays an important role in recoverable resource calculations and mining projects. Classic geostatistical estimation methods are based on two-point statistics and cannot make use of multiple-point and higher-order statistics of the data. Very few studies have been conducted on determination of the possible effects of these features on the results of ore grade estimation. In view of this, the present study introduced a new multiple-point interpolation method based on the implicit Volterra series. In this regard, least square support vector machines (LS-SVM) were used to implicitly estimate the Volterra series coefficients. Regularized risk minimization in the LS-SVM can increase generalization of the Volterra series and reduce sensitivity of these series to data noise. Also, conjugate gradient iterative algorithm was used to solve the linear equations of the implicit Volterra series estimation problem. This sets the stage to solve large-scale problems for Volterra series. With the use of Volterra series, multiple-point multiplicative interactions of spatial data can be used in the ore grade estimation process. In this study, multiple point spatial templates were used to extract multiplicative features of the input data. Therefore, an algorithm was presented to search the 3D space by these templates. The applicability of the regularized implicit Volterra series in capturing the higher-order features of the spatial data is successfully evaluated in a synthetic example. Also, the generalization ability of the introduced method is successfully examined in a synthetic 2D example and one real 3D mining example at the Sarcheshmeh porphyry copper deposit. The results revealed that the introduced multiple-point method could present better results than the ordinary kriging method.

Acceleration of 3D potential field data inversion using a BB iterative algorithm

As the big-data age arrives, the efficiency of three dimensional inversions of potential field data can be paid enough attention by scholars. A new inversion method is considered to deal with the large potential field data with a fast rate of convergence. Hence, in this paper, a fast inversion method is researched. And the gravity data is used as an example of potential field data to test the efficiency of our inversion method. To achieve this aim, the study region will be divided into huge amounts of rectangular prisms with unknown constant physical properties of rock. The traditional smooth inversion method is the main principle, and a new Barzilai-Borwein iterative algorithm is applied to ensure the rapid rate of convergence of the inversion method. To compare the rate of convergence of the BB (Barzilai-Borwein) iterative algorithm, the iterative gradient descent algorithm and the iterative conjugate gradient algorithm are used as the compared algorithms. To test high efficiency of the new fast developed inversion method, the large synthetic gravity data are performed. The contrast analysis results can easily reflect the high efficiency of our new inversion method. The great practical value of our inversion method is expounded by a real gravity data. Therefore, the new inversion method may have a great influence on the potential field data inversion.

Research on the Method of Electrical Impedance Tomography Based on Conjugate Gradient Iterative Algorithm

Research on the Method of Electrical Impedance Tomography Based on Conjugate Gradient Iterative Algorithm

Reduced optimal models via cross Gramian for continuous linear time‐invariant systems

In this study, the authors focus on the problem of model order reduction (MOR) suitable for continuous linear time-invariant (LTI) systems. Specifically, for single-input single-output (SISO) LTI systems, a couple of MOR algorithms via the cross Gramian are presented. The authors first give a cross Gramian-based optimal MOR iterative algorithm to reduce the SISO system. Further, the authors explore the optimisation problem on the Stiefel manifold. Based on the geometric notions on this manifold, a conjugate gradient iterative algorithm (CGIAHM) is proposed. The conjugate gradient direction used to search for the minimiser is derived by applying the notion of vector transport. It is worth mentioning that the conjugate gradient used in the CGIAHM algorithm is a decent direction for the cost function due to the ingenious construction of this algorithm. In addition, the authors’ algorithms are extended to solve the optimal MOR problem for general LTI systems. Finally, two numerical examples demonstrate the effectiveness of their algorithms.

Transmit Precoded Spatial Modulation: Maximizing the Minimum Euclidean Distance Versus Minimizing the Bit Error Ratio

In this paper, we investigate a pair of transmit precoding (TPC) algorithms conceived for spatial modulation (SM) systems communicating over flat-fading multiple-input multiple-output (MIMO) channels. In order to retain all the benefits of conventional SM, we design the TPC matrix to be diagonal and introduce two design criteria for optimizing the elements of the TPC matrix. Specifically, we first investigate a TPC design based on maximizing the minimum Euclidean distance $d_{\bf min }$ (max- $d_{\bf min }$ ) between the SM signal points at the receiver side. A closed-form solution of the optimal max- $d_{\bf min }$ -based TPC matrix is derived. Then, another TPC design algorithm is proposed for directly minimizing the bit error ratio (BER) upper bound of SM, which is capable of jointly optimizing the overall Euclidean distance between all received signal points. In the minimum BER (min-BER)-based TPC algorithm, the theoretical gradient of the BER with respect to the diagonal TPC matrix is derived and a simplified iterative conjugate gradient (SCG) algorithm is invoked for TPC optimization. Our simulation results demonstrate that the proposed max- $d_{\bf min }$ -based TPC algorithm is optimal in terms of the minimum distance. However, increasing $d_{\bf min }$ does not achieve a further BER improvement. We also confirm that the min-BER-based TPC outperforms the max- $d_{\bf min }$ -based TPC schemes in terms of the achievable BER performance.

Phantom Validation of Tc-99m Absolute Quantification in a SPECT/CT Commercial Device.

Aim. Similar to PET, absolute quantitative imaging is becoming available in commercial SPECT/CT devices. This study's goal was to assess quantitative accuracy of activity recovery as a function of image reconstruction parameters and count statistics in a variety of phantoms. Materials and Methods. We performed quantitative 99mTc-SPECT/CT acquisitions (Siemens Symbia Intevo, Erlangen, Germany) of a uniform cylindrical, NEMA/IEC, and an anthropomorphic abdominal phantom. Background activity concentrations tested ranged: 2–80 kBq/mL. SPECT acquisitions used 120 projections (20 s/projection). Reconstructions were performed with the proprietary iterative conjugate gradient algorithm. NEMA phantom reconstructions were obtained as a function of the iteration number (range: 4–48). Recovery coefficients, hot contrast, relative lung error (NEMA phantom), and image noise were assessed. Results. In all cases, absolute activity and activity concentration were measured within 10% of the expected value. Recovery coefficients and hot contrast in hot inserts did not vary appreciably with count statistics. RC converged at 16 iterations for insert size > 22 mm. Relative lung errors were comparable to PET levels indicating the efficient integration of attenuation and scatter corrections with adequate detector modeling. Conclusions. The tested device provided accurate activity recovery within 10% of correct values; these performances are comparable to current generation PET/CT systems.

Improving the Reconstruction of Vector Fields Using Mixed Finite Element Methods and Optimal Preconditioning

In this article, we study numerically a diagnostic model, based on mass conservation, to recover solenoidal vector fields from experimental data. Based on a reformulation of the mathematical model as a saddle‐point problem, we introduce an iterative preconditioned conjugate gradient algorithm, applied to an associated operator equation of elliptic type, to solve the problem. To obtain a stable algorithm, we use a second‐order mixed finite element approximation for discretization. We show, using synthetic vector fields, that this new approach, yields very accurate solutions at a low computational cost compared to traditional methods with the same order of approximation. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1137–1154, 2016

Image‐based estimation method for field inhomogeneity in brain echo‐planar images with geometric distortion using k‐space textures

ABSTRACTEcho‐planar imaging (EPI) can suffer from geometrical distortion due to magnetic field inhomogeneity. To correct the geometric distortions in EPI, a magnetic field map is used. Our purpose was to develop a novel image‐based method for estimating the field inhomogeneity map from the distorted EPI image and T1‐weighted image of the brain using k‐space textures. Based on magnetic resonance imaging physics, our method synthesizes the distorted image to match the measured EPI image through the generating process of EPI image by updating the estimated field inhomogeneity map. The estimation process was performed to minimize the cost function, which was defined by the synthesized EPI image and the measured EPI image with geometric distortion, using an iterative conjugate gradient algorithm. The proposed method was applied to simulation and human data. To evaluate the performance of the proposed method quantitatively, we used the normalized root mean square error (NRMSE) between the ground truth and the results estimated by our proposed method. In simulation data, the values of the NRMSE between the ground truth and the estimated field inhomogeneity map were <0.08. In both simulation and human data, the estimated EPI images were very similar to input EPI images, and the NRMSE values between them were <0.09. The results of the simulated and human data demonstrated that our method produced a reasonable estimation of the field inhomogeneity map. The estimated map could be used for distortion correction in EPI images. © 2015 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 45B: 142–152, 2015

Instance-specific canonical correlation analysis

Canonical Correlation Analysis (CCA) is one of the most popular statistical methods to capture the correlations between two variables. However, it has limitations as a linear and global algorithm. Although some variants have been proposed to overcome the limitations, neither of them achieves locality and nonlinearity at the same time. In this paper, we propose a novel algorithm called Instance-Specific Canonical Correlation Analysis (ISCCA), which approximates the nonlinear data by computing the instance-specific projections along the smooth curve of the manifold. First, we propose a least squares solution for CCA which will set the stage for the proposed method. Second, based on the framework of least squares regression, CCA is extended to the instance-specific case which obtains a set of locally linear smooth but globally nonlinear transformations. Third, ISCCA can be extended to semi-supervised setting by exploiting the unlabeled data to further improve the performance. The optimization problem is proved to be convex and could be solved efficiently by alternating optimization. And the globally optimal solutions could be achieved with theoretical guarantee. Moreover, for large scale applications, iterative conjugate gradient algorithm can be used to speed up the computation procedure. Experimental results demonstrate the effectiveness of our proposed method.

Three-dimensional structural topology optimization of aerial vehicles under aerodynamic loads

A previously developed density distribution-based structural topology optimization algorithm coupled with a Computational Fluid Dynamics (CFD) solver for aerodynamic force predictions is extended to solve large-scale problems to reveal inner structural details of a wing wholly rather than some specific regions. Resorting to an iterative conjugate gradient algorithm for the solution of the structural equilibrium equations needed at each step of the topology optimizations allowed the solution of larger size problems, which could not be handled previously with a direct equation solver. Both the topology optimization and CFD codes are parallelized to obtain faster solutions. Because of the complexity of the computed aerodynamic loads, a case study involving optimization of the inner structure of the wing of an unmanned aerial vehicle (UAV) led to topologies, which could not be obtained by intuition alone. Post-processing features specifically tailored for visualizing computed topologies proved to be good design tools in the hands of designers for identifying complex structural components.

An Enhanced NAS-RIF Algorithm for Blind Image Restoration Based on Total Variation Regularization

In order to improve the quality of blind image restoration, we propose an algorithm which combines Non-negativity and Support constraint Recursive Inverse Filtering (NAS-RIF) and adaptive total variation regularization. In the proposed algorithm, the total variation regularization constraint term is added in the NAS-RIF algorithm cost function. The majorization-minimization approach and conjugate gradient iterative algorithm are adopted to improve the convergence speed. We do the simulation experiments for the blurred classic test image which is added additive random noise. Experimental results show that the restoration effect of our algorithm is better than the spatially adaptive Tikhonov regularization method and the NAS-RIF spatially adaptive regularization algorithm, while the value of improvement of signal to noise ratio (ISNR) has improved.

High‐resolution multishot spiral diffusion tensor imaging with inherent correction of motion‐induced phase errors

To develop and compare three novel reconstruction methods designed to inherently correct for motion-induced phase errors in multishot spiral diffusion tensor imaging without requiring a variable-density spiral trajectory or a navigator echo. The first method simply averages magnitude images reconstructed with sensitivity encoding from each shot, whereas the second and third methods rely on sensitivity encoding to estimate the motion-induced phase error for each shot and subsequently use either a direct phase subtraction or an iterative conjugate gradient algorithm, respectively, to correct for the resulting artifacts. Numerical simulations and in vivo experiments on healthy volunteers were performed to assess the performance of these methods. The first two methods suffer from a low signal-to-noise ratio or from residual artifacts in the reconstructed diffusion-weighted images and fractional anisotropy maps. In contrast, the third method provides high-quality, high-resolution diffusion tensor imaging results, revealing fine anatomical details such as a radial diffusion anisotropy in cortical gray matter. The proposed sensitivity encoding + conjugate gradient method can inherently and effectively correct for phase errors, signal loss, and aliasing artifacts caused by both rigid and nonrigid motion in multishot spiral diffusion tensor imaging, without increasing the scan time or reducing the signal-to-noise ratio.