An explicit method to calculate implicit spatial finite differences

The finite-difference solution of the second-order acoustic wave equation is a fundamental algorithm in seismic exploration for seismic forward modeling, imaging, and inversion. Unlike the standard explicit finite difference (EFD) methods that usually suffer from the so-called "saturation effect", the implicit FD methods can obtain much higher accuracy with relatively short operator length. Unfortunately, these implicit methods are not widely used because band matrices need to be solved implicitly, which is not suitable for most high-performance computer architectures. We introduce an explicit method to overcome this limitation by applying explicit causal and anti-causal integrations. We can prove that the explicit solution is equivalent to the traditional implicit LU decomposition method in analytical and numerical ways. In addition, we also compare the accuracy of the new methods with the traditional EFD methods up to 32nd order, and numerical results indicate that the new method is more accurate. In terms of the computational cost, the newly proposed method is standard 8th order EFD plus two causal and anti-causal integrations, which can be applied recursively, and no extra memory is needed. In summary, compared to the standard EFD methods, the new method has a spectral-like accuracy; compared to the traditional LU-decomposition implicit methods, the new method is explicit. It is more suitable for high-performance computing without losing any accuracy.

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.

Research on TFe Content of Hematite Based on LU-TELM-SOA and Selection of Band

Iron ore is an important raw material for the steel industry, so it is of great economic significance to determine the grade of the iron ore quickly and accurately. And the TFe content is the main indicator that determines the grade of the iron ore and whether the iron ore can be smelted directly. Unlike manual methods and methods for chemical analysis, the paper uses the selection of band for the near-infrared spectrum based on the pruning method and the two-hidden-layer extreme learning machine based on LU decomposition and seagull optimization algorithm (LU-TELM-SOA) to identify the TFe content. First of all, the paper proposes the selection of band based on the pruning method to retain the sensitive band of the near-infrared spectrum. Aiming at the problems of poor stability and low accuracy of a single LU-TELM (the two-hidden-layer extreme learning machine based on LU decomposition) model, the paper proposes LU-TELM-SOA. The experimental results show that LU-TELM-SOA has the advantages of high accuracy and strong stability.

New Models for Solving Time-Varying LU Decomposition by Using ZNN Method and ZeaD Formulas

In this paper, by employing the Zhang neural network (ZNN) method, an effective continuous-time LU decomposition (CTLUD) model is firstly proposed, analyzed, and investigated for solving the time-varying LU decomposition problem. Then, for the convenience of digital hardware realization, this paper proposes three discrete-time models by using Euler, 4-instant Zhang et al. discretization (ZeaD), and 8-instant ZeaD formulas to discretize the proposed CTLUD model, respectively. Furthermore, the proposed models are used to perform the LU decomposition of three time-varying matrices with different dimensions. Results indicate that the proposed models are effective for solving the time-varying LU decomposition problem, and the 8-instant ZeaD LU decomposition model has the highest precision among the three discrete-time models.

ANALYSIS OF NUMERICAL SOLUTION OF POISSON EQUATION BY ILU (0)-CG METHOD

The problem of generalization of the method is the main question that arises when studying the quality of iterative methods. The efficiency of solving systems using iterative methods directly depends on the assumptions about the system of equations to be solved. Prerequisites are used to provide a more efficient solution. Many types of prerequisites are currently known, for example, prerequisites based on the approximation of the system matrix: ILU, IQR, and ILQ; Prerequisites based on the approximation of the inverse matrix: a polynomial, rarely filled approximation of the inverse matrix (for example, AINV), an approximation in the factorized form of the inverse matrix (for example, FSAI, SPAI, etc.). This article analyzes the CG and CG methods with the preconditioner ILU (0) by the example of solving the two-dimensional Poisson equation. The CG method is usually used to solve any system of linear equations. ILU (0) was selected as a prerequisite for the article. The incomplete LU decomposition (ILU (0)) is an efficient precursor and is easily implemented. This suggests a system that can be solved to speed up the accumulation of CG and other iterative methods, that is, to reduce the number of iterations. The ILU (0) preconditioner is very easy to detect using the LU decomposition. Since the linear matrix was rarely filled, the CSR format was used to store the matrix in memory. ILU (0) + CG, i.e. the algorithm with a precondition, was assembled 5-8 times faster than the CG algorithm. Data on the number of iterations of convergence of the method without a preconditioner and with the ILU(0) preconditioner were obtained and analyzed.