Conjugate Gradient Iterative Method Research Articles

AN EFFICIENT ITERATIVE METHOD TO SIMULATE THE VARYING STATIC RECEPTANCE OF A THIN-WALLED WORKPIECE

This paper introduces an accelerated iterative Conjugate Gradient method (CG-method) for simulating the varying static receptance of a thin-walled workpiece. The proposed approach enhances computational efficiency by computing an optimal initial displacement guess for a given cutter location. This guess is based on the previously calculated displacement vector from the preceding cutter location, serving as the start iteration vector for the CG-method. The validation of efficiency gains through this method is demonstrated by a significant reduction in computation time for a numerical example. Moreover, to ensure the accuracy of the accelerated iterative CG-method, comparisons are made with the Cholesky method. The results confirm the precision of the proposed approach.

Logarithmic total variation regularization via preconditioned conjugate gradient method for sparse reconstruction of bioluminescence tomography

Background and objectiveBioluminescence Tomography (BLT) is a powerful optical molecular imaging technique that enables the noninvasive investigation of dynamic biological phenomena. It aims to reconstruct the three-dimensional spatial distribution of bioluminescent sources from optical measurements collected on the surface of the imaged object. However, BLT reconstruction is a challenging ill-posed problem due to the scattering effect of light and the limitations in detecting surface photons, which makes it difficult for existing methods to achieve satisfactory reconstruction results. In this study, we propose a novel method for sparse reconstruction of BLT based on a preconditioned conjugate gradient with logarithmic total variation regularization (PCG-logTV). MethodThis PCG-logTV method incorporates the sparsity of overlapping groups and enhances the sparse structure of these groups using logarithmic functions, which can preserve edge features and achieve more stable reconstruction results in BLT. To accelerate the convergence of the algorithm solution, we use the preconditioned conjugate gradient iteration method on the objective function and obtain the reconstruction results. We demonstrate the performance of our proposed method through numerical simulations and in vivo experiment. Results and conclusionsThe results show that the PCG-logTV method obtains the most accurate reconstruction results, and the minimum position error (LE) is 0.254mm, which is 26%, 31% and 34% of the FISTA (0.961), IVTCG (0.81) and L1-TV (0.739) methods, and the root mean square error (RMSE) and relative intensity error (RIE) are the smallest, indicating that it is closest to the real light source. In addition, compared with the other three methods, the PCG-logTV method also has the highest DICE similarity coefficient, which is 0.928, which means that this method can effectively reconstruct the three-dimensional spatial distribution of bioluminescent light sources, has higher resolution and robustness, and is beneficial to the preclinical and clinical studies of BLT.

Numerical method for quasi-static adhesive elastic contact subjected to tangential loading

This paper presents a three-dimensional (3D) numerical model for simulating the adhesive contact with linear isotropic elasticity considering the interaction between normal and tangential loads. The model is based on the Boussinesq-Cerruti integral equations for elasticity, the Maugis-Dugdale (MD) model for adhesion, and the friction/adhesion interaction theory in the McMeeking model; it uses the Coulomb's law of friction to identify the occurrence of local microslip. Efficient and accurate determination of the contact behavior, together with adhesive and stick/slip regions, uses the iterative conjugate gradient method (CGM) and the discrete convolution and fast Fourier transform (DC-FFT) algorithm. The model analyzes adhesive-contact pressure, shear traction, and subsurface stress fields. It is used to study the effects of increasing tangential loading on the variations of stick/slip zones and adhesive-energy dissipation, as well as the influence of sinusoidal roughness on surface adhesion behavior. Compared with most theoretical models, the proposed numerical model has no restrictions on surface geometry and roughness of contact bodies, and it should have a wider range of applications.

Fuzzy Adaptive Parameter in the Dai–Liao Optimization Method Based on Neutrosophy

The impact of neutrosophy has increased rapidly in many areas of science and technology in recent years. Furthermore, numerous applications of the neutrosophic theory have become more usual. We aim to use neutrosophy to enhance Dai–Liao conjugate gradient (CG) iterative method. In particular, we suggest and explore a new neutrosophic logic system intended to compute the essential parameter t required in Dai–Liao CG iterations. Theoretical examination and numerical experiments signify the effectiveness of the introduced method for controlling t. By incorporation of the neutrosophy in the Dai–Liao conjugate gradient principle, we established novel Dai–Liao CG iterations for solving large-scale unconstrained optimization problems. Global convergence is proved under standard assumptions and with the use of the inexact line search. Finally, computational evidence shows the computational effectiveness of the proposed fuzzy neutrosophic Dai–Liao CG method.

Semiexplicit time integration of a reduced magnetic vector potential magneto‐quasistatic field formulation

AbstractIn this article, a reduced vector potential formulation for transient magneto‐quasistatic field problems is presented. The finite element method for spatial discretization yields a system of differential algebraic equations (DAE). Applying a generalized Schur complement the DAE system decomposes into a system of ordinary differential equations (ODE) and an algebraic equation. In order to avoid computationally expensive linearization schemes to solve nonlinear eddy current problems the ODE system is integrated in time using a semiexplicit Euler scheme. The vector potential is split into a source vector potential and a reduced vector potential which describes the eddy current and magnetization effects. The reduced formulation is compared with the underlying nonreduced formulation regarding the overall performance. In addition, the proper orthogonal decomposition method is applied to the iterative preconditioned conjugate gradient method to minimize the number of iterations per time step.

Spectral Methods in Nonlinear Optics Equations for Non-Uniform Grids Using an Accelerated NFFT Scheme

In this work, we propose the use of non-homogeneous grids in 1D and 2D for the study of various nonlinear physical equations using spectral methods. As is well known, the use of spectral methods allow a faster resolution of the problem via the application of the ubiquitous Fast Fourier Transform (FFT) algorithm. We will center our investigation on the search of fast and accurate schemes to solve the spectral operators in the Fourier space. In particular, we will use the Conjugate Gradient (CG) iterative method, with a preconditioning matrix to accelerate the inversion process of the non-uniform Fast Fourier Transform (NFFT). As it will be shown, the results obtained are in good agreement with the expected values.

Using incomplete Cholesky factorization to increase the time step in molecular dynamics simulations

Incomplete Cholesky (IC) factorization is widely used as a preconditioner for accelerating the convergence of the conjugate gradient iterative method. The IC factorization has been used in continuum mechanics and other applications that require solutions of elliptic partial differential equations. In this study, we propose an efficient use of the IC factorization to increase the time step and accelerate molecular dynamics simulations. Previously, we proposed the semi-implicit Hessian correction (SimHec) scheme (Washio et al., 2021) for overdamped Langevin dynamics of polymer simulations. SimHec constructs an approximation of the Hessian matrix by superposing the corrected elemental Hessian matrices associated with the interactions within a limited bandwidth along the polymer chain. The resulting narrow-bandwidth system is efficiently solved by an overlapped skyline solver in parallel. In this study, we integrate the non-local interactions in the polymer chain into the corrected Hessian matrix, and approximate the components of the Hessian matrix outside the narrow bandwidth using the IC factorization. This strategy allows us to use time steps that are 150–1000 times larger than those used in the explicit scheme without a severe increase in the computational load.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method.
 Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Identification of stationary source in the anomalous diffusion equation

In this paper, we consider the initial-boundary value problem of determining the stationary right-hand side function in the anomalous diffusion equation with a Caputo fractional derivative with respect to time. The value of the solution of the problem at the final time moment is set as the overdetermination condition. In order to carry out the numerical solution, the iterative conjugate gradient method is used, while at each iteration a direct problem is solved by the finite-difference method using a purely implicit difference scheme. The computational experiment results for the model problem are presented to confirm the efficiency of this new method.

Solution of the Cauchy problem for the wave equation using iterative regularization

We propose a regularization method based on the iterative conjugate gradient method for the solution of a Cauchy problem for the wave equation in one dimension. This linear but ill-posed Cauchy problem consists of finding the displacement and flux on a hostile and inaccessible part of the medium boundary from Cauchy data measurements of the same quantities on the remaining friendly and accessible part of the boundary. This inverse boundary value problem is recast as a least-squares minimization problem that is solved by using the conjugate gradient method whose iterations are stopped according to the discrepancy principle for obtaining stable reconstructions. The objective functional associated is proved Fréchet differentiable and a formula for its gradient is derived. The well-posed direct, adjoint and sensitivity problems present in the conjugate gradient method are solved by using a finite-difference method. Two numerical examples to illustrate the accuracy and stability of the proposed numerical procedure are thoroughly presented and discussed.

Early results on deep unfolded conjugate gradient‐based large‐scale MIMO detection

Deep learning (DL) is attracting considerable attention in the design of communication systems. This paper derives a deep unfolded conjugate gradient (CG) architecture for large-scale multiple-input multiple-output detection. The proposed technique combines the advantages of a model-driven approach in readily incorporating domain knowledge and deep learning in effective parameters learning. The parameters are trained via backpropagation over a data flow graph inspired from the iterative conjugate gradient method. We derive the closed-form expressions for the gradients for parameters training and discuss early results on the performance in a statistically identical and independent distributed channel where the training overhead is considerably low. It is worth noting that the loss function is based on the residual error that is not an explicit function of the desired signal, which makes the proposed algorithm blind. As an initial framework, we will point to the inherent issues and future directions.

Numerical modeling of non-stationary heat problems in a two-phase medium

The simulation of thermal fields and melting permafrost is an important step for engineering construction in many regions of Russia. Water-ice phase transitions in a 3D complicated computation domain must be taken into account. The enthalpy statement with an implicit finite volume method was used to create a good discretization and a local mesh refinement to focus an area near the oil well. System of linear algebraic equations is solved using the iterative conjugate residual or conjugate gradient method in Krylov subspaces with the incomplete factorization algorithm in the Eisenstat modification as a preconditioner. A special high-parallel version of matrix generation and solving a system of the linear algebraic equations code have been developed and then efficiency has been estimated. The performance results of the developed code and temperature fields for different wells configuration (1 well and 4 wells placed as squared) during 5 years have been presented. The results validation was based on the comparison with previous papers and other algorithms. The results of simulations are close those to presented by other authors.

An augmented wavelet reconstructor for atmospheric tomography

Atmospheric tomography, i.e., the reconstruction of the turbulence profile in the atmosphere, is a challenging task for adaptive optics (AO) systems for the next generation of extremely large telescopes. Within the AO community, the solver of first choice is the so-called Matrix Vector Multiplication (MVM) method, which directly applies the (regularized) generalized inverse of the system operator to the data. For small telescopes this approach is feasible, however, for larger systems such as the European Extremely Large Telescope (ELT), the atmospheric tomography problem is considerably more complex, and the computational efficiency becomes an issue. Iterative methods such as the Finite Element Wavelet Hybrid Algorithm (FEWHA) are a promising alternative. FEWHA is a wavelet-based reconstructor that uses the well-known iterative preconditioned conjugate gradient (PCG) method as a solver. The number of floating point operations and the memory usage are decreased significantly by using a matrix-free representation of the forward operator. A crucial indicator for the real-time performance are the number of PCG iterations. In this paper, we propose an augmented version of FEWHA, where the number of iterations is decreased by 50% using an augmented Krylov subspace method. We demonstrate that a parallel implementation of augmented FEWHA allows the fulfilment of the real-time requirements of the ELT.

3D electromagnetic tomography using a single layer sensor array

Electromagnetic tomography (EMT) is a non-invasive technique to gain the induced voltages and to reconstruct the distribution of electromagnetic properties. A volume imaging using a single layer sensor array, namely, Single Layer 3D-EMT, has been developed in this paper. To solve the forward problem, three kinds of sensor arrays are established with different number of coils,—8, 12 and 16, to conduct simulations. Four typical conductivity distributions are used to verify the 3D-EMT imaging with single layer sensor array. The calculation of sensitivity maps in 3D views is presented, and the sensitivity maps of different layers, from z = 0 to z = 20, are analyzed. The conjugate gradient iterative method has been adopted in image reconstruction. It can be seen from the simulation results that the 3D reconstructed images using a single layer sensor array can display the distribution of the measured objects in position, shape, and size. By calculating the relative error, the quality of the reconstructed images is evaluated andthe result shows that the 12-coil sensor array has the best image quality among three sensor arrays. In addition, the single layer 3D experimental system has been established to evaluate the effectiveness of 3D imaging using a single layer sensor array. In order to save the hardware resources, the paper presents the author's research work about the possibility of Single Layer 3D-EMT imaging in simulations and experiments. This can also provide a guide to explore the potential application with electromagnetic testing technique.

Q-Compensated Denoising of Seismic Data

It is widely known that strong noise can decrease the quality of seismic data. However, the anelastic attenuation could be more important to account for the weak amplitude and low quality of seismic data. Here, we develop an inversion framework to simultaneously compensate for the attenuation of seismic data and remove noise, thereby enhancing the quality of seismic data. Instead of directly applying a compensation operator to the input seismic data, we formulate an inverse problem that connects the sparse reflectivity model and the raw seismic data via the convolution and attenuation functions. The random noise is assumed to be the unpredicted part of the forward modeling process. We use the L2-norm regularization for the data misfit and impose a sparsity constraint onto the reflectivity series, e.g., using the L1-norm constraint. We use an iterative preconditioned conjugate gradient method to solve the L1-norm constrained least-squares optimization problem and obtain the reflectivity series. The denoised and compensated data are obtained by applying the convolution operator to the reflectivity. We use several synthetic and field seismic data to illustrate the effectiveness of the presented method.

New Conjugate Gradient Method Addressing Large Scale Unconstrained Optimization Problem

An iterative conjugate gradient (CG) method is prominently known for dealing with unconstrained optimization problem. A new CG method which is modified by Wei Yao Liu (WYL) method is tested by standard test functions. Moreover, the step size is calculated using exact line search. Theoretical proofs on convergence analysis are shown. As a result, this new CG is comparable to the other methods in finding the optimal points by measuring the total iterations required as well as the computing time. Numerical results showed the execution between three CG methods in details.

Numerical algorithms for solving discrete Lyapunov tensor equation

In this paper, we present several numerical algorithms to solve the discrete Lyapunov tensor equation, which is the generalized form of discrete Lyapunov matrix equation. Based on the structure of the tensor equation, we firstly propose a simple iterative (SI) method to consider the numerical solution of the discrete Lyapunov tensor equation. Then we extend the gradient based iterative (GBI) method and the residual norm steepest descent (RNSD) iterative method to study the tensor equation, respectively. Furthermore, we develop a residual norm conjugate gradient (RNCG) iterative method to accelerate the convergence speed of the RNSD method. Convergence analysis shows that the proposed methods converge to an exact solution for any initial value. Finally, some numerical examples are provided to illustrate the efficiency and validity of these methods proposed.

Negative Reactance Impacts Power Flow Convergence Using Conjugate Gradient Method

It is usually considered in power systems that the B matrices in fast decoupled power flow (B’ and B”) are symmetric and positive-definite. The fast decoupled power flow (FDPF) based on the conjugate gradient (CG) iterative method was well developed, because the CG iterative method has a good convergence property and a lower memory requirement which can be easily implemented in the parallel computation. However, a rare yet important phenomenon hasn’t been well addressed in that “negative reactance” may exist in the practical power system models, which could affect the definiteness of B matrices in FDPF and the CG convergence performance. In this brief, the eigenvalues of B matrices with negative reactance are investigated and the convergence of CG for FDPF is discussed. Several large-scale practical systems are tested, which could provide some insights for the study of the FDPF using the CG method with negative reactances.

An efficient Legendre–Galerkin spectral element method for the steady flows in rectangular cavities

An efficient Legendre–Galerkin spectral element method for the steady flows in rectangular cavities is proposed in this paper. Firstly, we eliminate the singularity of biharmonic equation in rectangular cavity at the corner by the singularity substraction technique. Then we construct some appropriate interior basis functions and interface basis functions which maintain -continuity. Consequently, the discrete variational formulation is reduced to a linear system with block diagonal and well-conditioned coefficient matrix, which can be efficiently solved by the conjugate gradient iteration method. Finally, several numerical examples are given to show the effectiveness of our numerical method. The present method is used to solve the creeping flows in rectangular cavities, the numerical results are compared well with the benchmark steady solutions provided by the finite difference method.

A Discrete Dipole Approximation Solver Based on the COCG-FFT Algorithm and Its Application to Microwave Breast Imaging.

We introduce the discrete dipole approximation (DDA) for efficiently calculating the two-dimensional electric field distribution for our microwave tomographic breast imaging system. For iterative inverse problems such as microwave tomography, the forward field computation is the time limiting step. In this paper, the two-dimensional algorithm is derived and formulated such that the iterative conjugate orthogonal conjugate gradient (COCG) method can be used for efficiently solving the forward problem. We have also optimized the matrix-vector multiplication step by formulating the problem such that the nondiagonal portion of the matrix used to compute the dipole moments is block-Toeplitz. The computation costs for multiplying the block matrices times a vector can be dramatically accelerated by expanding each Toeplitz matrix to a circulant matrix for which the convolution theorem is applied for fast computation utilizing the fast Fourier transform (FFT). The results demonstrate that this formulation is accurate and efficient. In this work, the computation times for the direct solvers, the iterative solver (COCG), and the iterative solver using the fast Fourier transform (COCG-FFT) are compared with the best performance achieved using the iterative solver (COCG-FFT) in C++. Utilizing this formulation provides a computationally efficient building block for developing a low cost and fast breast imaging system to serve under-resourced populations.