Linear Iterative Method Research Articles

A Locally Weighted Linear Regression Look-Up Table-Based Iterative Reconstruction Method for Dual Spectral CT.

Compared with traditional computed tomography (CT), dual spectral CT (DSCT) exhibits superior material distinguishability and thus has broad prospects in industrial and medical fields. In iterative DSCT algorithms, accurately modeling forward-projection functions is crucial, but it is very difficult to analytically provide accurate functions. In this article, we propose a locally weighted linear regression look-up table-based (LWLR-LUT) iterative reconstruction method for DSCT. First, the proposed method uses LWLR to establish LUTs for the forward-projection functions through calibration phantoms, achieving good local information calibration. Second, the reconstructed images can be iteratively obtained through the established LUTs. The proposed method not only does not require knowledge of the X-ray spectra and the attenuation coefficients, but also implicitly accounts for some scattered radiation while fitting locally the forward-projection functions in the calibration space. Both numerical simulations and real data experiments demonstrate that the proposed method can achieve highly accurate polychromatic forward-projection functions and greatly improve the quality of the images reconstructed from scattering-free and scattering projections. The proposed method is simple and practical, and achieves good material decomposition effects for objects with different complex structures through simple calibration phantoms.

2D magnetotelluric inversion based on ResNet

In this study, a deep learning algorithm was applied to two-dimensional magnetotelluric (MT) data inversion. Compared with the traditional linear iterative inversion methods, the MT inversion method based on convolutional neural networks (CNN) does not rely on the selection of the initial model parameters and does not fall into the local optima. Although the CNN inversion models can provide a clear electrical interface division, their inversion results may remain prone to abrupt electrical interfaces as opposed to the actual underground electrical situation. To solve this issue, a neural network with a residual network architecture (ResNet-50) was constructed in this study. With the apparent resistivity and phase pseudo-section data as the inputs and with the resistivity parameters of the geoelectric model as the training labels, the modified ResNet-50 model was trained end-to-end for producing samples according to the corresponding production strategy of the study area. Through experiments, the training of the ResNet-50 with the dice loss function effectively solved the issue of over-segmentation of the electrical interface by the cross-entropy function, avoided its abrupt inversion, and overcame the computational inefficiency of the traditional iterative methods. The proposed algorithm was validated against MT data measured from a geothermal field prospect in Huanggang, Hubei Province, which showed that the deep learning method has opened up broad prospects in the field of MT data inversion.

Research on one-dimensional digital twin algorithm of plate heat exchanger

Based on the differential idea, this article establishes two segmented algorithms which are the linear equations method and iterative method for solving the one-dimensional digital twin of the temperature field and pressure field of plate heat exchanger (PHE). The segmented algorithms are to consider the change of the physical properties of the fluid along the flow direction, which makes the solution of the temperature field and pressure field more accurate than that without segmentation. It can still get other important parameters besides temperature and is easy to converge with a few iterative steps. When the boundary conditions are known, it can accurately solve the heat transfer, wall temperature, convective heat transfer coefficient, temperature field and pressure field of fluid. Under six groups of test conditions, the heat exchange, temperature and pressure calculated by the two algorithms are compared with the commercial software Flownex, when the number of segments is 100, the average maximum error of each segment is not more than −0.39%, −0.97%, and 0.18%, respectively. To balance accuracy and calculation speed, this article also explores the impact of the number of segments on accuracy. It is found that when the number of segments increases to 10, the impact on the calculation accuracy can be ignored. Compared with Flownex, these two algorithms have smaller errors, and the solution speed of solving linear equations method is faster than that of iteration. It provides application value for emerging digital twin technology, data center chilled water system and other projects and industries that need to accurately control the outlet temperature and wall temperature of PHE.

Double Dynamic Linearization-Based Higher Order Indirect Adaptive Iterative Learning Control.

In this article, a higher order indirect adaptive iterative learning control (HO-iAILC) scheme is developed for nonlinear nonaffine systems. The inner loop adopts a P -type controller whose set-point is updated iteratively by learning from the iterations. To this end, an ideal nonlinear learning control law is designed in the outer loop. It is then transferred to a linear parametric-learning controller with a corresponding parameter estimation law by introducing an iterative dynamic linearization (IDL) method. This IDL method is also used to gain an iterative linear data model of the nonlinear system. A parameter iterative updating algorithm is utilized for estimating the unknown parameters of the obtained linear data model. Finally, the HO-iAILC is presented that utilizes additional error information to improve the control performance and employs two iterative adaptive mechanisms to deal with uncertainties. The convergence of the proposed HO-iAILC scheme is proved by using two basic mathematical tools, namely: 1) contraction mapping and 2) mathematical induction. Simulation studies are conducted for the verification of the theoretical results.

Improving the performance of classical linear algebra iterative methods via hybrid parallelism

We propose fork-join and task-based hybrid implementations of four classical linear algebra iterative methods (Jacobi, Gauss–Seidel, conjugate gradient and biconjugate gradient stabilized) on CPUs as well as variations of them. This class of algorithms, that are ubiquitous in computational frameworks, are duly documented and the corresponding source code is made publicly available for reproducibility. Both weak and strong scalability benchmarks are conducted to statistically analyse their relative efficiencies.The weak scalability results assert the superiority of a task-based hybrid parallelisation over MPI-only and fork-join hybrid implementations. Indeed, the task-based model is able to achieve speedups of up to 25% larger than its MPI-only counterpart depending on the numerical method and the computational resources used. For strong scalability scenarios, hybrid methods based on tasks remain more efficient with moderate computational resources where data locality does not play an important role. Fork-join hybridisation often yields mixed results and hence does not seem to bring a competitive advantage over a much simpler MPI approach.

Application of linear iterative methods for the proximity effect correction in electron beam lithography

The Jacobi linear iterative method and weight Jacobi method (WJM) are introduced for solving the large-scale linear problem in the proximity effect correction (PEC) of electron beam lithography. Based on the discussion of PEC physics, a symmetrical and positive defined proximity interaction matrix is constructed to ensure the convergence of the methods. It shows that zeroing the center exposure fraction in the point spread function matrix is equal to the operation of splitting the proximity interaction matrix. Then, the Jacobi method is ready for the PEC calculation. The iterative can be performed in the Fourier domain due to the inherent parallelization of the Jacobi method. The convergent property of the Jacobi method is discussed and then testified by the PEC simulation. Compared with the classical Jacobi method, an improvement of 100% in convergence efficiency can be achieved by introducing the optimized relaxation parameter quasi-ωopt in the WJM. By combining the WJM and Gold nonlinear iteration method, a new method that shows an order of magnitude superior in accuracy to the WJM is proposed. Results indicate the methods introduced here could be used to calculate the PEC problem efficiently.

Multi-Objective Optimal DG Placement Approach Using Analytical Hierarchy Process

Distributed generation (DG) has been increasingly integrated into distribution systems in the last decade. The selection of the optimal location and capacity of these DGs is an important issue. This selection is based on minimizing losses, reducing emissions, reducing voltage deviation, and much more. This paper proposes a new optimization approach with a multi-objective function for maximizing DG capacities and minimizing both system losses and voltage deviation, which are subject to system constraints. The weight of each attribute of the multi-objective function is estimated using the analytical hierarchical process (AHP). The optimal DG size is calculated for all possible buses, and hence, the optimal bus can be located. Moreover, an algorithm is introduced to decrease the number of iterations when integrating more than one DG. The proposed method is an iterative linear programming method based on the familiar Newton-Raphson sensitivity matrix. The proposed approach has been tested on 69 and IEEE 33 bus systems. The simulation results are compared with previous literature results.

An efficient fourth-order accurate conservative scheme for Riesz space fractional Schrödinger equation with wave operator

In this study, we investigate a high-order accurate conservative finite difference scheme by utilizing a fourth-order fractional central finite difference method for the two-dimensional Riesz space-fractional nonlinear Schrödinger wave equation. The conservation laws of the discrete difference scheme are shown. Meanwhile, the exactness, uniqueness, and prior estimate of the numerical solution are rigorously established. Then, it is proved that the proposed scheme is unconditionally convergent in the discrete L2 and Hγ/2 norm, where γ is a fractional order. Furthermore, we demonstrate that when the fractional order γ and the spatial grid number J increase, the block-Toeplitz coefficient matrix generated by the spatial discretization becomes ill-conditioned. As a result, we adopt an effective linearized iteration method for the nonlinear system, allowing it to be solved efficiently by the Krylov subspace solver with an appropriate circulant preconditioner, in which the fast Fourier transform is applied to speed up the computational cost at each iterative step. Finally, numerical experiments are presented to validate the theoretical findings and the efficiency of the fast algorithm.

A fast implicit difference scheme for solving high-dimensional time-space fractional nonlinear Schrödinger equation

In this work, an efficient implicit difference scheme is developed for solving the high-dimensional time-space fractional nonlinear Schrödinger equation. The derived scheme is constructed by utilizing a fast evaluation of Caputo fractional derivative based on the - formula; meanwhile, the compact finite difference with matrix transfer technique is adopted for the spatial discretization. Moreover, a linearized iteration method based on the fast discrete sine transform technique is considered to solve the nonlinear system effectively. Because the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. The stability, truncation error and convergence analysis of the discrete scheme are discussed in detail. Furthermore, a fast iterative algorithm is provided. Finally, several numerical examples are given to verify the efficiency and accuracy of the derived scheme, and a comparison with similar work is presented.

A Single-Pixel High-Precision Imaging Technique Based on a Discrete Zernike Transform for High-Efficiency Image Reconstructions

Single-pixel imaging (SPI) has attracted increasing attention in recent years because of its advantages in imaging systems. However, a low reconstruction quality and a long reconstruction time have hindered the development of SPI. Hence, in this study, we propose a Zernike SPI (ZSPI) technique to reduce the number of illumination patterns and reconstruction time whilst retaining robustness. First, the ZSPI technique was theoretically demonstrated. Phase-shifting Zernike moment projections were used to illuminate the target and an inverse Zernike transform was used to reconstruct the desired image. In order to prove the feasibility, numerical simulations were carried out with different sample ratios (SRs) ranging from 0.1 to 0.3; an acceptable reconstruction appeared at approximately 0.1. This result indicated that ZSPI could obtain satisfactory reconstruction results at low SRs. Further simulation and physical experiments were compared with different reconstruction algorithms, including noniterative, linear iterative, and nonlinear iterative methods under speckle modulation patterns at a sample of 0.1 in terms of different targets. The results revealed that ZSPI had a higher imaging quality and required less imaging time, particularly for low-frequency targets. The method presented in this study has advantages for the high-efficiency imaging of low-frequency targets, which can provide a new solution for the SPI method.

Nonlinear response of beams with viscoelastic elements by an iterative linearization method

An iterative successive linearization approach based on energy optimization is presented to estimate the response statistics of a moderately largely vibrating beam endowed with a fractional derivative, subject to combined harmonic and random excitations. The presented method is developed by constructing two sets of optimized linear systems, so that the responses of the new system are equivalent to those of the original system in the sense of average and statistical characteristics. Specifically, the iterative deterministic and statistical linearization methods are utilized successively, based on optimizations of the average and mean square difference between the potential energies associated with the original systems and their linear counterparts. Numerical examples are given with Monte Carlo data to demonstrate the reliability and accuracy of the presented method.

Application of modified iterative method to simulate rainfall infiltration in unsaturated soils

A modified Gauss-Seidel iterative method (MP(m)-GS) with multistep preconditioner is developed to solve partial differential equations of rainfall infiltration. The finite difference method (FDM) is applied to numerical discretization. Furthermore, a system of linear equations is solved using an iterative scheme. For some unfavorable numerical conditions, such as large initial conditions, classical Picard iterative method usually has low accuracy and computational efficiency. Conventional linear iterative methods have a slower convergence rate, such as Gauss-Seidel (GS) and Jacobi iterative methods, particularly for small discrete space step size. Thus, MP(m)-GS is developed to simulate rainfall infiltration into unsaturated soils. The analytical solutions are employed to verify the proposed methods. The results indicate that the MP(m)-GS has higher computational efficiency and accuracy than the improved Picard methods. Compared with the conventional linear iterative methods GS and SOR, MP(m)-GS also demonstrates faster convergence rate and higher computation efficiency. The results show nice applications in modeling rainfall infiltration in unsaturated soils.

Data-driven adaptive tuning of iterative learning control

In this paper, we propose two data-driven adaptive tuning (DDAT) approaches of iterative learning control (ILC) for nonlinear non-affine systems. First, a compact-form iterative dynamic linearization (CFIDL) method is introduced to transfer the original nonlinear system into a linear data model. Then, we design an objective function for the tuning of the learning gains of a PD-type ILC law. By optimizing the designed cost function subjected to the linear data model, a CFIDL-based DDAT method is proposed, where only the real I/O data are used without requiring any mechanistic model information. Furthermore, the results are extended by introducing a partial-form iterative dynamic linearization (PFIDL) method for the purpose of utilizing more additional control information. Following the similar steps, a PFIDL-based DDAT method is developed for learning gain tuning of the PD-type ILC scheme. Both the proposed DDAT methods can help the PD-type ILC have a better robustness against to the uncertainties since they can use the real I/O data to iteratively tune the learning gains. The convergence of the DDAT-based PD-type ILC methods has been proved rigorously. The effectiveness of the two proposed DDAT-based ILC methods are further verified through simulations.

The More Fractal the Architecture the More Intensive the Color of Flower: A Superpixel-Wise Analysis towards High-Throughput Phenotyping

A breeder can select a visually appealing phenotype, whether for ornamentation or landscaping. However, the organic vision is not accurate and objective, making it challenging to bring a reliable phenotyping intervention into implementation. Therefore, the objective of this study was to develop an innovative solution to predict the intensity of the flower’s color upon the external shape of the crop. We merged the single linear iterative clustering (SLIC) algorithm and box-counting method (BCM) into a framework to extract useful imagery data for biophysical modeling. Then, we validated our approach by fitting Gompertz function to data on intensity of flower’s color and fractal dimension (SD) of the architecture of white-flower, yellow-flower, and red-flower varieties of Portulaca umbraticola. The SLIC algorithm segmented the images into uniform superpixels, enabling the BCM to precisely capture the SD of the architecture. The SD ranged from 1.938315 to 1.941630, which corresponded to pixel-wise intensities of 220.85 and 47.15. Thus, the more compact the architecture the more intensive the color of the flower. The sigmoid Gompertz function predicted such a relationship at radj2 > 0.80. This study can provide further knowledge to progress the field’s prominence in developing breakthrough strategies toward improving the control of visual quality and breeding of ornamentals.

An Iterative Linear Method with Variable Shear Stress Magnitudes for Estimating the Stress Tensor from Earthquake Focal Mechanism Data: Method and Examples

ABSTRACT Earthquake focal mechanism data provide information about the stress state at the origin of earthquakes. The inversion methods that are commonly used to infer the stress tensor from focal mechanisms have varying complexity but always rely on a number of assumptions. We present an iterative method built upon a classic linear stress tensor inversion that allows for relaxing the assumption on shear stress magnitudes while preserving the computational simplicity of the linear problem. Every iteration of our method computes the least-squares solution of the problem, which makes the method fast enough to estimate the inverted parameter errors with nonparametric resampling methods such as bootstrapping. Following previous studies, this method removes the fault plane ambiguity in focal mechanism data by selecting the nodal plane that best satisfies the Mohr–Coulomb failure criterion. We first test the performance and robustness to noise of the proposed method on synthetic data sets and then apply it to data from the Southern California and Geysers geothermal field data sets. We focus the study on investigating the consequences of relaxing the assumption of constant shear stress magnitudes. Our variable shear method successfully generalizes its constant shear counterpart: it is able to perform similarly when the constant shear assumption is a good approximation and provides more accurate results when it is not. We provide the Python package iterative linear stress inversion to implement the proposed method.

A marker-less monocular vision point positioning method for industrial manual operation environments

Vision-assisted technologies in industrial manual operation such as augmented reality (AR) are increasingly popular. They require high positioning accuracy and robustness to operate normally. However, narrow spaces, moving hands, or tools may occlude or obscure local visual features of operation environments and thus affect the positioning accuracy and robustness of operating position. The resultant misguidance may even cause misoperation of operators. This paper proposes a marker-less monocular vision point positioning method for vision-assisted manual operation in industrial environments. The proposed method can accurately and robustly locate the target point of operation using constraint minimization method even if the target area has no corresponding visual features in the case of occlusion and improper illumination. The proposed method has three phases: intersection generation, intersection optimization, and target point solving. In the intersection generation phase, a certain number of intersections of epipolar lines are generated as candidate target points using fundamental matrices. Here, the solving constraint is converted from point-to-line to point-to-points. In the intersection optimization phase, the intersections are optimized to two different sets through the iterative linear fitting and geometric mean absolute error methods. Here, the solving constraint is further converted from point-to-points to point-to-point sets. In the target point solving phase, the target point is solved as a constrained minimization problem based on the distribution constraint of the two intersection sets. Here, the solving constraint is finally converted from point-to-point sets to point-to-point, and the unique optimal solution is obtained as the target point. The experimental results show that this method has a better accuracy and robustness than the traditional homography matrix method for the practical industrial operation scenes.

SPI-CGAN: Single-pixel imaging based on conditional generative adversarial network

Imaging quality at extremely sampling ratios is a key research topic in the field of single-pixel imaging (SPI). Although conventional methods can be used to reconstruct the object images at low sampling ratios, the reconstructed image is still visually unclear. To solve this problem, an SPI model based on a conditional generative adversarial network (SPI-CGAN) is proposed to achieve an end-to-end reconstruction of the object images and improve the image quality of the reconstruction at extremely low sampling ratios. To improve the stability of the training model, the objective function is composed of multiple loss functions. The validity of the model is verified through simulation data, which do not require a long time to collect. The optimized SPI-CGAN can reconstruct a sharp image edge at extremely low sampling ratios. Compared with a linear iterative method and a nonlinear iterative method, the proposed method performs better in terms of the quantitative indicators used.

Seeding Crop Detection Framework Using Prototypical Network Method in UAV Images

With the development of unmanned aerial vehicle (UAV), obtaining high-resolution aerial images has become easier. Identifying and locating specific crops from aerial images is a valuable task. The location and quantity of crops are important for agricultural insurance businesses. In this paper, the problem of locating chili seedling crops in large-field UAV images is processed. Two problems are encountered in the location process: a small number of samples and objects in UAV images are similar on a small scale, which increases the location difficulty. A detection framework based on a prototypical network to detect crops in UAV aerial images is proposed. In particular, a method of subcategory slicing is applied to solve the problem, in which objects in aerial images have similarities at a smaller scale. The detection framework is divided into two parts: training and detection. In the training process, crop images are sliced into subcategories, and then these subcategory patch images and background category images are used to train the prototype network. In the detection process, a simple linear iterative clustering superpixel segmentation method is used to generate candidate regions in the UAV image. The location method uses a prototypical network to recognize nine patch images extracted simultaneously. To train and evaluate the proposed method, we construct an evaluation dataset by collecting the images of chilies in a seedling stage by an UAV. We achieve a location accuracy of 96.46%. This study proposes a seedling crop detection framework based on few-shot learning that does not require the use of labeled boxes. It reduces the workload of manual annotation and meets the location needs of seedling crops.

Two-dimensional deep learning inversion of magnetotelluric sounding data

Abstract Conventional linear iterative methods for magnetotelluric sounding (MT) suffer from considerable limitations related to difficulties in selecting the initial model and local optima. On the other hand, conventional intelligent nonlinear methods exhibit slow convergence and low accuracy. In this study, we propose an inversion strategy based on the deep learning (DL) deep belief network (DBN) to realise the instantaneous inversion of MT data. A scaled momentum learning rate is introduced to improve the convergence performance of the restricted Boltzmann machine during the DBN pre-training stage, and a novel activation function (DSoft) is introduced to enhance the global optimisation capability during the DBN fine-tuning stage. To address the difficulty in designing the sample data when prior information is lacking, we employ the k-means++ algorithm to cluster the MT field data and use the clustering results as the prior information to guide the construction of the sample dataset. Then, based on the proposed DBN, we ensure end-to-end mapping directly from the apparent resistivity to the resistivity model. We implement two groups of experiments to demonstrate the validity of both improvements on the DBN. We consider six types of geoelectric model from the test set to demonstrate the feasibility and effectiveness of the proposed DBN method for MT 2D inversion, which we further compare with the well-known least-square regularisation method for several extended geoelectric models and field data. The qualitative and quantitative analyses show that the DL inversion method is promising as it can accurately delineate the subsurface structures and perform rapid inversion.

Refinement of Extended Accelerated Over-Relaxation Method for Solution of Linear Systems

Given any linear stationary iterative methods in the form z^(i+1)=Jz^(i)+f, where J is the iteration matrix, a significant improvements of the iteration matrix will decrease the spectral radius and enhances the rate of convergence of the particular method while solving system of linear equations in the form Az=b. This motivates us to refine the Extended Accelerated Over-Relaxation (EAOR) method called Refinement of Extended Accelerated Over-Relaxation (REAOR) so as to accelerate the convergence rate of the method. In this paper, a refinement of Extended Accelerated Over-Relaxation method that would minimize the spectral radius, when compared to EAOR method, is proposed. The method is a 3-parameter generalization of the refinement of Accelerated Over-Relaxation (RAOR) method, refinement of Successive Over-Relaxation (RSOR) method, refinement of Gauss-Seidel (RGS) method and refinement of Jacobi (RJ) method. We investigated the convergence of the method for weak irreducible diagonally dominant matrix, matrix or matrix and presented some numerical examples to check the performance of the method. The results indicate the superiority of the method over some existing methods.