LU Decomposition Method Research Articles

Numerical simulation of shale gas reservoir based on ILU-GMRES method

ABSTRACT The numerical simulation of shale gas reservoirs requires transforming mathematical models into large linear equation systems, and solving large linear equation systems requires a lot of time. A novel ILU-GMRES method for solving large linear equations is presented, which treats the sub matrix discretized at nodes as the basic elements of the coefficient matrix. By combining the incomplete LU decomposition method (ILU) with the generalized minimum residue method (GMRES), the discretized large linear equations are iteratively solved. A comparison was made between the new method and other methods for calculating the actual shale gas reservoir model. The results reveal that the new method has a faster calculation speed in solving linear equation systems. The calculation time of the novel ILU-GMRES method is 1029s, the calculation time of the generalized minimum residual method (GMRES) method is 2709s, the calculation time of the Gauss–Seidel method (GS) is 3786s, and the calculation time of the successive over relaxation method (SOR) method is 3386s, the calculation time of the preprocessing conjugate gradient method (PCG) is 3115s. The calculation speed of the novel method is almost 2.6 times faster than the GMRES method, the novel method is more effective in the numerical simulation of shale gas.

Meshfree algorithms for analysis and computational modeling of multidimensional hyperbolic wave models

PurposeThis article offered two meshfree algorithms, namely the local radial basis functions-finite difference (LRBF-FD) approximation and local radial basis functions-differential quadrature method (LRBF-DQM) to simulate the multidimensional hyperbolic wave models and work is an extension of Jiwari (2015).Design/methodology/approachIn the evolvement of the first algorithm, the time derivative is discretized by the forward FD scheme and the Crank-Nicolson scheme is used for the rest of the terms. After that, the LRBF-FD approximation is used for spatial discretization and quasi-linearization process for linearization of the problem. Finally, the obtained linear system is solved by the LU decomposition method. In the development of the second algorithm, semi-discretization in space is done via LRBF-DQM and then an explicit RK4 is used for fully discretization in time.FindingsFor simulation purposes, some 1D and 2D wave models are pondered to instigate the chastity and competence of the developed algorithms.Originality/valueThe developed algorithms are novel for the multidimensional hyperbolic wave models. Also, the stability analysis of the second algorithm is a new work for these types of model.

Pass-efficient randomized LU algorithms for computing low-rank matrix approximation

Abstract Low-rank matrix approximation is extremely useful in the analysis of data that arises in scientific computing, engineering applications, and data science. However, as data sizes grow, traditional low-rank matrix approximation methods, such as singular value decomposition (SVD) and column pivoting QR decomposition (CPQR), are either prohibitively expensive or cannot provide sufficiently accurate results. A solution is to use randomized low-rank matrix approximation methods such as randomized SVD, and randomized LU decomposition on extremely large data sets. In this paper, we focus on the randomized LU decomposition method. Then we propose a novel randomized LU algorithm, called SubspaceLU, for the fixed low-rank approximation problem. SubspaceLU is based on the sketch of the co-range of input matrices and allows for an arbitrary number of passes of the input matrix, v ≥ 2 {v\geq 2} . Numerical experiments on CPU show that our proposed SubspaceLU is generally faster than the existing randomized LU decomposition, while remaining accurate. Experiments on GPU shows that our proposed SubspaceLU can gain more speedup than the existing randomized LU decomposition. We also propose a version of SubspaceLU, called SubspaceLU_FP, for the fixed precision low-rank matrix approximation problem. SubspaceLU_FP is a post-processing step based on an efficient blocked adaptive rank determination Algorithm 5 proposed in this paper. We present numerical experiments that show that SubspaceLU_FP can achieve close results to SVD but faster in speed. We finally propose a single-pass algorithm based on LU factorization. Tests show that the accuracy of our single-pass algorithm is comparable with the existing single-pass algorithms.

Wavelets computational modeling of nonlinear coupled reaction–diffusion models arising in chemical processes

This work submits the computational modeling of nonlinear coupled reaction–diffusion (RD) models via uniform scale‐2 Haar wavelets (S2HWs) based numerical algorithm. The nonlinear RD models have many applications in mathematical chemistry, chemical and biological processes, etc. In the design of the algorithm, first of all, semi‐discretization is done in time, and after that, the quasi‐linearization process is used for linearization. Finally, the full discretization is done with the help of S2HWs, and the enlisted linear system is solved by LU decomposition method. Also, the convergence analysis of the algorithm has discussed. In the simulation part, different types of patterns such as Turing, pulse‐splitting, self‐replicating pulse patterns, etc. of different models are captured.

Steeply dipping structural target oriented viscoacoustic prismatic reverse time migration in frequency domain and its application

AbstractSteeply dipping structural imaging is a great challenge due to its poor illumination. Conventional migration methods are unable to produce an accurate image of complex steeply dipping structures. The prismatic wave can improve the illumination of steeply dipping structures and is often used to improve the imaging results of such structures. Traditional elastic wave theory assumes that seismic waves do not attenuate when propagating through subsurface media. However, during seismic wave propagation, the wave energy decays exponentially due to the absorption and attenuation of the ground layer. Subsurface attenuation leads to amplitude loss and phase distortion of seismic waves, resulting in blurring of migration amplitudes when this attenuation is not taken into account during imaging. To address this issue, a frequency‐domain Q‐compensated prismatic reverse time migration method is proposed, which derives Q‐compensated prismatic wavefield propagation operators. In the proposed frequency‐domain Q‐compensated prismatic reverse time migration, Q attenuation is fully compensated along three propagation paths and two propagation types of prismatic waves. The optimized four‐order mixed 25‐point difference format and LU decomposition method are used to solve the Q‐compensated prismatic wavefield propagation equations with high computational efficiency. Numerical and field data examples demonstrate that the proposed frequency‐domain Q‐compensated prismatic reverse time migration method can compensate for deep attenuation energy and improve the imaging resolution of steeply dipping structures.

A two-phase power flow algorithm of traction power supply system based on traction substation-network decoupling

Power flow analysis of traction power supply system (TPSS) is an effective approach to assess supply capacity and power quality, serving the whole process of system design, operation and optimization. The coupling effect between adjacent power supply arms poses challenges for the accurate and efficient solution, especially with the penetration of power converters and energy equipment. This paper proposes a two-phase power flow decoupling algorithm. TPSS is divided into traction substation and two decoupled power supply arms by introducing current ratio coefficients and unit matrices. Specifically, the equivalent voltage source model of traction substation and the modified chain network model of arms are solved by using forward–backward sweep method and LU decomposition method, respectively. Then the TPSS power flow can be obtained by alternative iteration of two solving phases. Simulations verify the proposed algorithm exhibits great compatibility at various operation modes. Additionally, the power supply capacity is mapped to the algorithm convergence. Compared to conventional methods, the algorithm achieves superior performance on accuracy and efficiency.

Log(v) 3LPF: A Linear Power Flow Formulation for Unbalanced Three-Phase Distribution Systems

In this work, we introduce Log(v) 3LPF, a linear power flow solver for unbalanced three-phase distribution systems. Log(v) 3LPF uses a logarithmic transform of the voltage phasor to linearize the AC power flow equations around the balanced case. We incorporate the modeling of ZIP loads, transformers, capacitor banks, switches and their corresponding controls and express the network equations in matrix-vector form. With scalability in mind, special attention is given to the computation of the inverse of the system admittance matrix, Ybus. We use the Sherman-Morrison-Woodbury identity for an efficient computation of the inverse of a rank-k corrected matrix and compare the performance of this method with traditional LU decomposition methods using Big- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}$</tex-math></inline-formula> notation. We showcase the solver for a variety of network sizes, ranging from tens to thousands of nodes, and compare the Log(v) 3LPF with commercial-grade software, such as OpenDSS.

Research on application of LU decomposition and CG method in nonlinear iteration of transmission line finite element method

PurposeThe purpose of this study is to develop a reliable finite element algorithm based on the transmission line method (TLM) to solve the nonlinear problem in electromagnetic field calculation.Design/methodology/approachIn this paper, the TLM has been researched and applied to solve nonlinear iteration in FEM. LU decomposition method and the Jacobi preconditioned conjugate gradient method have been investigated to solve the equations in transmission line finite element method (FEM-TLM). The algorithms have been developed in C++ language. The algorithm is applied to analyze the magnetic field of a long straight current-carrying wire and a single-phase transformer.FindingsFEM-TLM is more effective than traditional FEM in nonlinear iteration. The results of FEM-TLM have been compared and analyzed under different calculation scales. It is found that the LU decomposition method is more suitable for FEM-TLM because there is no need to repeatedly assemble the global coefficient matrix in the iterative solution process and it is not affected by the disturbance of the right-hand vector.Originality/valueAn effective algorithm is provided for solving nonlinear problems in the electromagnetic field, which can save a lot of computing costs. The efficiency of LU decomposition and CG method in FEM-TLM nonlinear iteration is investigated, which also makes a preliminary exploration for the research of FEM-TLM parallel algorithms.

Iterative Method of Thomas Algorithm on The Case Study of Energy Equation

Implicit method is one of the finite difference method and is widely used for discretization some of ordinary or partial differential equations, such like: advection equation, heat transfer equation, burger equation, and many others. Implicit method is unconditionally stable and has been proved with the approximation of Von-Neumann stability criterion. Actually, implicit method is always identical to block matrices (tri-diagonal matrices or penta-diagonal matrices). These matrices can be solved numerically by Thomas algorithm including Gauss elimination using pivot or not, backward or forward substitution. Furthermore, it can be also solved using LU decomposition method with the elimination of lower triangle matrices first and then the elimination of upper triangle matrices. In this research, Thomas algorithm is used to solve numerically for the problem of convective flow on boundary layer, especially for energy equation with the variation of Prandtl number ( ).

Recursive versus nonrecursive Richardson algorithms: systematic overview, unified frameworks and application to electric grid power quality monitoring

Sufficiently accurate, fast and computationally efficient solution of the system of linear equations is required in many estimation problems. Richardson iteration is one of the main solvers for linear equations, which provides optimization possibilities for time critical and accuracy critical applications. Convergence rate improvement and reduction of the computational complexity of the Richardson iteration are the most important problems in the area. The introduction of Newton–Schulz iterations is the efficient way for convergence rate improvement and the paper starts with systematic overview of the high-order Newton–Schulz matrix inversion algorithms. In addition, the unified framework for recursive computationally efficient convergence accelerators and error models for a number of combinations of Richardson and Newton–Schulz iterations is developed. A new nonrecursive parameter estimation concept is introduced and compared in this paper with recursive estimation. Recursive and nonrecursive Richardson algorithms together with the standard LU decomposition method were applied to the electric grid power quality monitoring problem. The algorithms were tested for the detection of the sag and swell signatures in the voltage and current signals on real data in three-phase power system. Nonrecursive Richardson algorithms which save close to half of the computational time compared to LU decomposition method were recommended for power quality monitoring applications.

Gas Flow in Occluded Respiratory Tree: A New Matrix-Based Approach

AbstractStudies suggest that both the size of airways and the number of bifurcations of the respiratory tree provide the best structural design to accomplish its function. However, constrictions and occlusions due to inflammation and pulmonary edema of the airways can inhibit normal air flowing through the respiratory tree, affecting gas exchange. It results in heterogeneity in gas exchange (and pulmonary perfusion) with adverse risk factors. In this study, we propose a methodology based on the airway tree admittance (reciprocal of impedance) to study this problem. This methodology is distinct from the traditional quantification, based on overall impedance using lump parameter models, and applies to a matrix formed by admittances of each airway of the entire conducting part of the bronchial tree. The generated system admittance matrix is highly sparse in nature, and thus to solve the same system, a modified block-based LU decomposition method is proposed to improve the space–time tradeoff. Our approach enables the determination of the local ventilation pattern and reduces the misevaluation, mainly in the cases that characterize the early-stage obstructive disorders. The key finding of the present study is to show that how the position and intensity of local obstruction in an airway can affect the overall as well as regional ventilation which can lead to impaired gas exchange.

Local RBF‐FD‐Based Mesh‐free Scheme for Singularly Perturbed Convection‐Diffusion‐Reaction Models with Variable Coefficients

This work analyze singularly perturbed convection‐diffusion‐reaction (CDR) models with two parameters and variable coefficients by developing a mesh‐free scheme based on local radial basis function‐finite difference (LRBF‐FD) approximation. In the evolvement of the scheme, time derivative is discretized by forward finite difference. After that, LRBF‐FD approximation is used for spatial discretization, and we obtained a system of linear equations. Then, the obtained linear system is solved by LU decomposition method in MATLAB. For numerical simulation, four singularly perturbed models are pondered to check the efficiency and chastity of the proposed scheme.

Cryptography using Automata Theory

Encryption and decryption are the two most crucial components of cryptography. Data protection is the main objective of both systems. We utilised encryption to transform plain text into ciphertext. Decryption, which works the other way around from encryption, is the process of converting encrypted text back into plain text. By using a finite state machine and the LU decomposition method, the created encryption solution ensures data secrecy for safe communication. In our suggested approach, we additionally employ lower and upper triangular matrices, which are obtained by decomposing a square matrix. During encryption, the key will be a lower triangular matrix modulated by a prime number, and during decryption, an upper triangular matrix modulated by a prime number. The tactic is beneficial. This tactic is helpful in sectors such as finance and military services where confidential material must be delivered.

Factorisation Path Based Refactorisation for High-Performance LU Decomposition in Real-Time Power System Simulation

The integration of renewable energy sources into modern power systems requires simulations with smaller step sizes, larger network models and the incorporation of complex nonlinear component models. These features make it more difficult to meet computation time requirements in real-time simulations and have motivated the development of high-performance LU decomposition methods. Since nonlinear component models cause numerical variations in the system matrix between simulation steps, this paper places a particular focus on the recomputation of LU decomposition, i.e., on the refactorisation step. The main contribution is the adoption of a factorisation path algorithm for partial refactorisation, which takes into account that only a subset of matrix entries change their values. The approach is integrated into the modern LU decomposition method NICSLU and benchmarked against the methods SuperLU and KLU. A performance analysis was carried out considering benchmark as well as real power systems. The results show the significant speedup of refactorisation computation times in use cases involving system matrices of different sizes, a variety of sparsity patterns and different ratios of numerically varying matrix entries. Consequently, the presented high-performance LU decomposition method can assist in meeting computation time requirements in real-time simulations of modern power systems.

AN IMPLICIT BLOCK JACOBI APPROACH FOR HIGH-ORDER FLUX RECONSTRUCTION METHOD

Recently, the flux reconstruction (FR) method has attracted more and more attentions for its simplicity and generality. However, it is still computationally expensive and time consuming when simulating the complex flow problems by FR method. There is a huge demand for developing appropriate efficient implicit solvers and parallel computing techniques for FR. This paper proposes an implicit high-order flux reconstruction solver on GPU platform based on the block Jacobi iteration method. As it is inefficient to solve the large global linear system resulting from spatial and implicit temporal discretization of FR directly. A block Jacobi approach is used to change the characteristics of the lift-hand matrix of the global linear system and this avoids the dependence of neighboring elements. Therefore, only the diagonal blocks of global matrix need to be stored and calculated. Then, the problem of solving the huge global linear system is transformed into solving a series of local linear equations simultaneously. Finally, these small local linear equations would be solved by the LU decomposition method in parallel on GPU platforms. Two typical cases, including subsonic flows over a bump and a NACA0012 airfoil, were simulated and compared with the multi-grid explicit Runge-Kutta scheme. The numerical results demonstrated that the present implicit method can reduce the iterations significantly. Meanwhile, the implicit solver has shown at least 10x speedup over the multi-grid Runge-Kutta scheme in all cases.

B-spline fitting of sunglasses lens contour based on adaptive sampling

To solve the problems of low accuracy and low efficiency of B-spline fitting in sunglasses lens contour cutting, a sampling method based on the characteristic function of a weighted combination of arc length and curvature square root was proposed, and then perform B-spline fitting according to the sampling points to improve the accuracy and efficiency. First, read and identify the sunglasses lens data in the DXF file, and disperse it to obtain a series of coordinate points; combine the defined feature function for uniform sampling, and adjust the parameter weight in the feature function to obtain a more reasonable sampling point for the sunglasses lens contour; Then use the LU decomposition and interpolation method to obtain the B-spline curve through all the sampling points; the maximum absolute error (MAE) and the root-mean-square error (RMSE) are used as the accuracy criteria, by increasing the number of sampling points iteratively until the accuracy is satisfied to construct a reasonable fitting curve.

Multi-Objective Management of Heterogeneous Coastal Aquifers.(Dept.C)

A nonlinear optimization problem based on both single and multiple objective management functions for sustainable utilization of heterogeneous coastal aquifers was formulated and solved. The sharp interface approach was assumed within this work. Finite Element Method (FEM) was applied on Sirack's formulation to simulate the hydraulic response of the steady slate heterogeneous aquifer under different objectives. The LU-decomposition method was exploited to inverse the conductance matrix, that enhanced the computation time. The constraint technique was used to simplify, without any approximation, the multi objective management problem to single objective management problem. Genetic Algorithm (GA) was used to solve the nonlinear optimization problem. A Fortran program was written and verified against a problem that has an analytical solution. A hypothetical example was solved for one and three objective functions. These func tions are maximum magnitude of extracted freshwater, minimum drop in water table level, and minimum increment in volume of intruded saltwater.

LU Decomposition Scheme for Solving m-Polar Fuzzy System of Linear Equations

This paper presents a new scheme for solving m-polar fuzzy system of linear equations (m-PFSLEs) by using LU decomposition method. We assume the coefficient matrix of the system is symmetric positive definite, and we discuss this point in detail with some numerical examples. Furthermore, we investigate the inconsistent m×nm-polar fuzzy matrix equation (m-PFME) and find the least square solution (LSS) of this system by using generalized inverse matrix theory. Moreover, we discuss the strong solution of m-polar fuzzy LSS of the inconsistent m-PFME. In the end, we present a numerical example to illustrate our approach.

Solving interval linear programming problem using generalized interval lu decomposition method

In this paper, we propose a method for solving interval linear programming problem by using generalized interval LU decomposition method based on Kaucher arithmetic for generalized interval numbers. Numerical example is also been provided.

A fast method for solving a block tridiagonal quasi-Toeplitz linear system

This paper addresses the problem of solving block tridiagonal quasi-Toeplitz linear systems. Inspired by [10], we propose a more general algorithm for such systems. The algorithm is based on a block decomposition for block tridiagonal quasi-Toeplitz matrices and the Sherman–Morrison–Woodbury inversion formula. We also compare the proposed approach to the standard block LU decomposition method and the Gauss algorithm. A theoretical error analysis is also presented. All algorithms have been implemented in Matlab. Numerical experiments performed on a wide variety of test problems show the effectiveness of our algorithm in terms of efficiency, stability and robustness.