Incomplete Cholesky Conjugate Gradient Research Articles

On cost-efficient parallel iterative solvers for 3D frequency-domain seismic multisource viscoelastic anisotropic wave modeling

Solving large sparse linear systems in 3D frequency-domain seismic wave modeling, especially in viscoelastic anisotropic media, poses significant challenges due to the increasing number of discrete moduli and nonzero elements in the linear system matrix. The computational load surpasses that of acoustic or viscoacoustic media, making it even more challenging when dealing with multisource problems. Popular scientific tools for solving a linear system, such as multifrontal massively parallel sparse direct solver (MUMPS), structured matrix package (STRUMPACK), and portable extensible toolkit for scientific computation (PETSc), can be used, but their applicability to our specific problem has not been comprehensively evaluated. Our study aims to tackle the challenges in solving large, sparse, complex-valued symmetric linear systems with multiple right-hand side vectors for 3D frequency-domain seismic wave modeling. We have adopted the preconditioned conjugate gradient iterative algorithms as the foundation for our research, introducing two highly cost-effective parallel iterative solvers: the parallel symmetric successive overrelaxation conjugate gradient (P-SSORCG) and the parallel incomplete Cholesky conjugate gradient (P-ICCG). These novel solvers are subjected to a comprehensive comparative analysis against well-established scientific tools, such as MUMPS, STRUMPACK, and PETSc, in the context of 3D frequency-domain seismic wave modeling. We show their promising performances in a practical 3D SEG/EAGE overthrust model and demonstrate that the grouped P-SSORCG offers an efficient alternative to parallel direct solvers, particularly in situations wherein computational resources are limited.

Coupled Incomplete Cholesky and Jacobi Preconditioned Conjugate Gradient on the New Generation of Sunway Many-Core Architecture

Coupled Incomplete Cholesky and Jacobi Preconditioned Conjugate Gradient on the New Generation of Sunway Many-Core Architecture

GPU implementation of an incomplete Cholesky conjugate gradient solver for a FEM-generated system using full kernel consolidation

GPU implementation of an incomplete Cholesky conjugate gradient solver for a FEM-generated system using full kernel consolidation

Analysis of SAR reduction to human head with plasma photonic crystals shield using ICCG-SFDTD method

A novel and effective biological electromagnetic method using plasma photonic crystals shield to protect the human head from electromagnetic radiation of an ultra-high-field magnetic resonance imaging (MRI) frequency is explored. An improved incomplete Cholesky conjugate gradient (ICCG)-based symplectic finite-difference time-domain (ICCG-SFDTD) method, which maintains the property of the Hamilton system and ensures energy conservation numerically with high precision, is applied to a bioelectromagnetic simulation of human head models without shield, with plasma shield and with plasma photonic crystals shield. Results show that plasma photonic crystals shield is superior to plasma shield with the same thickness in reducing the specific absorption rate (SAR) and electric field levels of the human head. In addition, the relationship of SAR and electric field in the human head models with the parameters of plasma photonic crystals, such as the plasma frequency, plasma collision frequency and dielectric constant of the medium is studied in order to provide a guidance to design plasma photonic crystals protective layer for biological tissue.

Wireline Logs Constraint Borehole-to-Surface Resistivity Inversion Method and Water Injection Monitoring Analysis

Detection of the oil–water encroachment front is of great importance to water injection of oil reservoirs. In borehole-to-surface electrical imaging (BSEI), a high-power direct current is applied into a borehole through a well case and the electric potential on the surface, which is affected by the subsurface electrical change, is measured. However, further accurate interpretation of BSEI data is difficult due to the weak surface response, deep target layer, long inversion time, and uncertainty in unconstrained inversion. Therefore, a new method to enhance the surface response of the anomalous body and a new three-dimensional inversion approach based on the damped least-squares method are proposed. Simulation of the water injection and fracturing process was modeled in three dimensions using the finite difference method and the incomplete Cholesky conjugate gradient method. The inversion approach was applied by using the log data to construct a layered resistivity model and constrain the inversion. The forward modeling results suggest that the electric potential gradient can enhance the response of electrical variations in the target layer and help estimate the water injection direction, depending on the distance of electrical anomalies and the current source. In actual water injection monitoring, the BSEI inversion results suggest that layered resistivity model constrained three-dimensional inversion can improve the precision and accuracy of the resistivity inversion results and outline the water injection channeled to the adjacent wells.

Hierarchical block multi-color ordering: a new parallel ordering method for vectorization and parallelization of the sparse triangular solver in the ICCG method

In this paper, we propose a new parallel ordering method to vectorize and parallelize the sparse triangular solver, which is called hierarchical block multi-color ordering. In this method, the parallel forward and backward substitutions can be vectorized while preserving the advantages of block multi-color ordering, that is, fast convergence and fewer thread synchronizations. To evaluate the proposed method in a parallel ICCG (Incomplete Cholesky Conjugate Gradient) solver, numerical tests were conducted using seven test matrices on three types of computational nodes. The numerical results indicate that the proposed method outperforms the conventional block and nodal multi-color ordering methods in 18 out of 21 test cases, which confirms the effectiveness of the method.

Bucket-based multigrid preconditioner for solving pressure Poisson equation using a particle method

The execution time for incompressible particle methods is dominated by solving large sparse linear systems. In this study, a novel simple bucket-based multigrid (BMG) preconditioner is presented to retrieve linear scaling. In the algorithm, the domain is decomposed into cubic boxes, to enable recursive aggregation of the closest neighboring particles. Under a moving particle semi-implicit (MPS) discretization scheme, parametric, verification, and performance studies were conducted for basic problems. From the parametric study, the BMG preconditioner was accompanied with a Krylov subspace accelerated multigrid cycle strategy and incorporated into a generalized conjugate residual method. Regardless of dimension and degree of particle distribution irregularity, the method significantly outperformed a conjugate gradient (CG), an incomplete Cholesky CG, and a conventional plain aggregation-based algebraic multigrid preconditioned solver. For a dam-break problem with 1.3 M and 2.3 M particles in 2D and 3D, the proposed method, as a linear solver for the MPS method, was 27 and 5.4 times faster than the CG method.

An evaluation of reordering algorithms to reduce the computational cost of the incomplete Cholesky-conjugate gradient method

This paper is concerned with applying bandwidth and profile reduction reordering algorithms prior to computing an incomplete Cholesky factorization and using this as a preconditioner for the conjugate gradient method. Hundreds of reordering algorithms have been proposed to solve the problems of bandwidth and profile reductions since the mid-1960s. In previous publications, a large range of heuristics for bandwidth and/or profile reductions was reviewed. Based on this experience, 13 heuristics were selected as the most promising methods. These are evaluated in this paper along with a variant of the breadth-first search procedure that is proposed. Numerical results confirm the effectiveness of this modified reordering algorithm for linear systems derived from specific application areas. Moreover, the most promising heuristics for several application areas are identified when reducing the computational cost of the incomplete Cholesky-conjugate gradient method.

Three-dimensional forward modeling and inversion of borehole-to-surface electrical imaging with different power sources

Borehole-to-surface electrical imaging (BSEI) uses a line source and a point source to generate a stable electric field in the ground. In order to study the surface potential of anomalies, three-dimensional forward modeling of point and line sources was conducted by using the finite-difference method and the incomplete Cholesky conjugate gradient (ICCG) method. Then, the damping least square method was used in the 3D inversion of the formation resistivity data. Several geological models were considered in the forward modeling and inversion. The forward modeling results suggest that the potentials generated by the two sources have different surface signatures. The inversion data suggest that the low-resistivity anomaly is outlined better than the high-resistivity anomaly. Moreover, when the point source is under the anomaly, the resistivity anomaly boundaries are better outlined than when using a line source.

A research for plasma electromagnetic character using JEC-CN-FDTD algorithm based on ICCG method

For the first time, the current density convolution finite-difference time-domain (JEC-FDTD) method is extended to dispersive media using Crank–Nicolson difference scheme, and derive the one-dimensional JEC-CN-FDTD iterative equation of plasma. The incomplete Cholesky conjugate gradient (ICCG) is proposed to solve the large sparse matrix equation in the Crank–Nicolson finite-difference time-domain (CN-FDTD) method as the ICCG method improves speed of convergence, enhances stability and reduces memory consumption. The high accuracy and efficiency of the JEC-CN-FDTD method are confirmed by computing the characteristic parameters of electromagnetic wave through a collision plasma slab in one dimension, such as the reflection electric field, transmission electric field, magnitudes and phases of reflection coefficient and transmission coefficient. The results prove that the method performs stably, is of accuracy, and has certain advantages.

A Study on Plasma Photonic Crystals: Electromagnetic Characteristics Using ICCG-based JEC-CN-FDTD Algorithm

Abstract We extended the current density convolution finite-difference time-domain (JEC-FDTD) method to plasma photonic crystals using the Crank–Nicolson difference scheme and derived the one-dimensional JEC-Crank–Nicolson (CN)-FDTD iterative equation of plasma photonic crystals. The method eliminated the Courant–Friedrich–Levy (CFL) stability constraint and became completely unconditional stable form. The incomplete Cholesky conjugate gradient (ICCG) algorithm is proposed to solve the equation with a large sparse matrix in the CN-FDTD method as the ICCG method improves the speed of convergence, enhances stability, and reduces memory consumption. The JEC-CN-FDTD method is applied to study the characteristics of time domain and frequency domain in the plasma photonic crystal objects. The high accuracy and efficiency of the JEC-CN-FDTD method are confirmed by computing the characteristic parameters of plasma photonic crystals under different conditions such as the electric field distribution of electromagnetic wave, reflection coefficients, and transmission coefficients. Simulation study showed that the algorithm performed stably and could reduce memory consumption and facilitate computer programming.

Research on the transmission coefficient of plasma photonic crystals based on the ICCG–SFDTD method

Compared with the traditional finite-difference time-domain (FDTD) method, the symplectic finite-difference time-domain (SFDTD) method has the characteristics of high precision and low dispersion. However, because the higher-order difference is necessary for the calculation, a large sparse matrix is generated. It causes that the computational time is relatively long and the memory is more. To solve this problem, the incomplete Cholesky conjugate gradient (ICCG) method for solving the large sparse matrix needs to be taken into the SFDTD differential equations. The ICCG method can accelerate the iterations of the numerical calculation and reduce the memory with fast and stable convergence speed. The new ICCG–SFDTD method, which has both the advantages of the ICCG method and SFDTD method, is proposed. In this paper, the ICCG–SFDTD method is used for research on the characteristic parameters of the plasma photonic crystals (PPCs) under different conditions, such as the reflection electric field and the transmission coefficient, to verify the feasibility and accuracy of this method. The results prove that the ICCG–SFDTD method is accurate and has some advantages.

Comparison of parallel solvers for Moving Particle Semi-Implicit method

Purpose – The purpose of this paper is to find the best solver for parallelizing particle methods based on solving Pressure Poisson Equation (PPE) by taking Moving Particle Semi-Implicit (MPS) method as an example because the solution for PPE is usually the most time-consuming part difficult to parallelize. Design/methodology/approach – To find the best solver, the authors compare six Krylov solvers, namely, Conjugate Gradient method (CG), Scaled Conjugate Gradient method (SCG), Bi-Conjugate Gradient Stabilized (BiCGStab) method, Conjugate Gradient Squared (CGS) method with Symmetric Lanczos Algorithm (SLA) method and Incomplete Cholesky Conjugate Gradient method (ICCG) in terms of convergence, time consumption, parallel efficiency and memory consumption for the semi-implicit particle method. The MPS method is parallelized by the hybrid Open Multi-Processing (OpenMP)/Message Passing Interface (MPI) model. The dam-break flow and channel flow simulations are used to evaluate the performance of different solvers. Findings – It is found that CG converges stably, runs fastest in the serial way, uses the least memory and has highest OpenMP parallel efficiency, but its MPI parallel efficiency is lower than SLA because SLA requires less synchronization than CG. Originality/value – With all these criteria considered and weighed, the recommended parallel solver for the MPS method is CG.

A Straightforward Convergence Method for ICCG Simulation of Multiloop and Time-Stepping FE Model of Synchronous Generators with Simultaneous AC and Rectified DC Connections

Now electric machines integrate with power electronics to form inseparable systems in lots of applications for high performance. For such systems, two kinds of nonlinearities, the magnetic nonlinearity of iron core and the circuit nonlinearity caused by power electronics devices, coexist at the same time, which makes simulation time-consuming. In this paper, the multiloop model combined with FE model of AC-DC synchronous generators, as one example of electric machine with power electronics system, is set up. FE method is applied for magnetic nonlinearity and variable-step variable-topology simulation method is applied for circuit nonlinearity. In order to improve the simulation speed, the incomplete Cholesky conjugate gradient (ICCG) method is used to solve the state equation. However, when power electronics device switches off, the convergence difficulty occurs. So a straightforward approach to achieve convergence of simulation is proposed. At last, the simulation results are compared with the experiments.

Computer Simulation of Fracture Process in Compressive Failure of Cross-Ply Laminate of Composite Materials

This study investigates the computer simulation of fracture process in compressive failure of cross-ply laminate of composite materials. The purpose of this study is to establish the numerical analysis method to understand the fracture mechanism and predict the mechanical response of composite materials through the computer simulation. The stacking sequence of the simulated laminate is [0/90] s. The reinforcement fibers in 0-degree plies are modeled by circle cross-section beam elements to represent the three-dimensional effect in bending of fibers. Cohesive elements are inserted in the connection of beam elements to simulate the bending breaking of fibers. For the purpose of parallel computing, the domain decomposition method is applied, and for pre-conditioned conjugate gradient algorithm, incomplete Cholesky conjugate gradient method is applied. The simulated results show that in the initial state of the loading, the stress concentration occurs around the initial misalignment of fiber in 0-degree plies, and it also occurs around the area where fibers come close in 90-degree plies. At average applied strain 1.20 %, the fiber breaking damage initiates in 0-degree ply, and after this point the damage develops in the material. The simulated damage is close to the microscope picture of the actual composite materials obtained in the experiment. The current simulation is considered to correspond with the actual material deformation.

2.5-D and 3-D DC resistivity modelling using an extrapolation cascadic multigrid method

SUMMARY Multigrid methods are well known for their high efficiency in solving elliptic boundary value problems. In this study, an improved extrapolation cascadic multigrid (EXCMG) method is presented to solve large sparse systems of linear equations, which are discretized from both 2.5-Dand3-DDCresistivitymodellingusingthefiniteelementmethods.Toincreasetheaccuracy,thesingularitygeneratedbythesourcetermisremovedbyreformulatingthesolutionwith thesecondarypotential.Inaddition,asetofnewandefficientFouriercoefficientispresentedto transform the solutions in the 2.5-D Fourier domain to the 3-D Cartesian domain. To show the efficiency and the ease-to-implement of EXCMG, we first implement the EXCMG methods to a two-layered model of both 2-D and 3-D and compare the results with the analytical solutions. It has been shown that the maximum relative error in apparent resistivity is no more than 0.4 per cent provided an appropriate grid size is chosen. Then the comparisons of EXCMG with two other iterative solvers [symmetric successive over-relaxation conjugate gradient (SSORCG) and incomplete Cholesky conjugate gradient (ICCG)] show that converging at a rate independent of the grid size, the EXCMG method is much more efficient than SSORCG and ICCG solvers. Moreover, the EXCMG method has been shown its potential for being generalized to large-scale 3-D problems, due to the fact that it becomes more efficient as the size of the problem increases.

Automatic Determination of Acceleration Factor Based on Residual and Functional in Shifted ICCG Method for 3-D Electromagnetic Field Analyses

This paper describes new strategy for determining an optimal acceleration factor in the shifted incomplete Cholesky conjugate gradient (ICCG) method. Although a useful method focused on the maximum diagonal entry in the incomplete Cholesky (IC) preconditioner was already proposed, it cannot always provide the optimal acceleration factor. In this paper, we propose new automatic determination methods based on the minimization of the norms of the remainder matrix, the residual, and the functional in the shifted ICCG method and discuss its effectiveness in the 3-D electromagnetic field analyses using the edge-based finite element method. It is concluded that the proposed methods based on the minimization of the norm of the residual or the functional can determine an appropriate acceleration factor within the acceptable computational cost and improve the convergence property of the shifted ICCG method.

Improvement of the Preconditioned MRTR Method With Eisenstat's Technique in Real Symmetric Sparse Matrices

The Incomplete Cholesky Conjugate Gradient (ICCG) method is widely used to solve indefinite algebraic equations obtained using an edge-based finite element method. However, when a linear solver based on the minimum residual is used, there is a possibility of reducing the elapsed time for a linear system. This paper shows the performance of the preconditioned minimized residual method based on the three-term recurrence formula of the CG-type (MRTR) method by comparing the MRTR method with the ICCG method for real symmetric sparse matrices. Furthermore, we intend to reduce computational costs by using Eisenstat's technique, and achieve more speed-up by applying a preconditioned residual to the convergence criterion.

Parallel Performance of Multithreaded ICCG Solver Based on Algebraic Block Multicolor Ordering in Finite Element Electromagnetic Field Analyses

We present a parallel multithreaded incomplete Cholesky-conjugate gradient (ICCG) solver for a linear system derived from a finite element electromagnetic field analysis. Algebraic block multicolor ordering is introduced to parallelize the solver with a high cache hit ratio and convergence comparable to the sequential solver. We develop the parallel ICCG solver based on reordering with modification for electromagnetic field analyses involving external circuits. The numerical results from practical models show that a 2.6- to 3.8-fold speedup compared with the sequential solver is attained using eight cores.

Surface-borehole electric potential survey in layered media and its applications

The method in which a source is set on the surface and electric potential is received in the borehole is called surface-borehole electric potential technique. Technique of surface-borehole electric potential was employed to study electric response of layered formation. The electric potential was obtained by solving Poisson equation with finite difference method. In the course of calculation, forward modeling with finite difference method was realized by adopting bandwidth non-zero storage technique and the incomplete Cholesky conjugate gradient method. The results show that method of surface-borehole can acquire anomalous electric potential corresponding to geo-electric layers. In addition, application of appropriate mathematical operator can improve the resolution. Moreover, overburden low resistivity layers have severe influence on measuring results of surface-borehole electric potential. However, bottom low resistivity layers play a positive role in the measurement.