Adaptive complex frequency with V-cycle GMRES for preconditioning 3D Helmholtz equation

Geophysics ◽  
2021 ◽  
pp. 1-40
Author(s):  
Wenhao Xu ◽  
Yang Zhong ◽  
Bangyu Wu ◽  
Jinghuai Gao ◽  
Qing Huo Liu
Keyword(s):  
Linear System ◽  
Helmholtz Equation ◽  
3D Models ◽  
Numerical Results ◽  
Spatial Sampling ◽  
Gmres Method ◽  
Complex Frequency ◽  
Iterative Solver ◽  
Helmholtz Equations ◽  
Sampling Density

Solving the Helmholtz equation has important applications in various areas, such as acoustics and electromagnetics. Using an iterative solver together with a proper preconditioner is key for solving large 3D Helmholtz equations. The performance of existing Helmholtz preconditioners usually deteriorates when the minimum spatial sampling density is small (approximately four points per wavelength [PPW]). To improve the efficiency of the Helmholtz preconditioner at a small minimum spatial sampling density, we have adopted a new preconditioner. In our scheme, the preconditioning matrix is constructed based on an adaptive complex frequency that varies with the minimum spatial sampling density in terms of the number of PPWs. Furthermore, the multigrid V-cycle with a GMRES smoother is adopted to effectively solve the corresponding preconditioning linear system. The adaptive complex frequency together with a GMRES smoother can work stably and efficiently at different minimum spatial sampling densities. Numerical results of three typical 3D models show that our scheme is more efficient than the multilevel GMRES method and shifted Laplacian with multigrid full V-cycle and a symmetric Gauss-Seidel smoother for preconditioning the 3D Helmholtz linear system, especially when the minimum spatial sampling density is large (approximately 120 PPW) or small (approximately 4 PPW).

An iterative solver for the 3D Helmholtz equation

2017 ◽  
Vol 345 ◽  
pp. 330-344 ◽  
Author(s):  
Mikhail Belonosov ◽  
Maxim Dmitriev ◽  
Victor Kostin ◽  
Dmitry Neklyudov ◽  
Vladimir Tcheverda
Keyword(s):  
Helmholtz Equation ◽  
Iterative Solver

The Linear System Solver of Programs Named CORTH and KYLIN2

2017 ◽  
Author(s):  
Pingzhou Ming ◽  
Junjie Pan ◽  
Xiaolan Tu ◽  
Dong Liu ◽  
Hongxing Yu
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Nuclear Power ◽  
Experimental Results ◽  
Gmres Method ◽  
Thermal Hydraulics ◽  
Lattice Calculation ◽  
Minimal Residual

Sub-channel thermal-hydraulics program named CORTH and assembly lattice calculation program named KYLIN2 have been developed in Nuclear Power Institute of China (NPIC). For the sake of promoting the computing efficiency of these programs and achieving the better description on fined parameters of reactor, the programs’ structure and details are interpreted. Then the characteristics of linear systems of these programs are analyzed. Based on the Generalized Minimal Residual (GMRES) method, different parallel schemes and implementations are considered. The experimental results show that calculation efficiencies of them are improved greatly compared with the serial situation.

Numerical simulation of squeezing Cu–Water nanofluid flow by a kernel-based method

2021 ◽  
pp. 2250005
Author(s):  
Mojtaba Fardi ◽  
Yasir Khan
Keyword(s):  
Numerical Simulation ◽  
Hilbert Space ◽  
Linear System ◽  
Numerical Simulations ◽  
Numerical Results ◽  
Kernel Matrix ◽  
Nanofluid Flow ◽  
Parallel Disks ◽  
Engineering Problems ◽  
Well Conditioning

The main aim of this paper is to propose a kernel-based method for solving the problem of squeezing Cu–Water nanofluid flow between parallel disks. Our method is based on Gaussian Hilbert–Schmidt SVD (HS-SVD), which gives an alternate basis for the data-dependent subspace of “native” Hilbert space without ever forming kernel matrix. The well-conditioning linear system is one of the critical advantages of using the alternate basis obtained from HS-SVD. Numerical simulations are performed to illustrate the efficiency and applicability of the proposed method in the sense of accuracy. Numerical results obtained by the proposed method are assessed by comparing available results in references. The results demonstrate that the proposed method can be recommended as a good option to study the squeezing nanofluid flow in engineering problems.

scholarly journals A PRECONDITIONED METHOD FOR THE SOLUTION OF THE ROBBINS PROBLEM FOR THE HELMHOLTZ EQUATION

2010 ◽  
Vol 52 (1) ◽  
pp. 87-100 ◽  
Author(s):  
JIANG LE ◽  
HUANG JIN ◽  
XIAO-GUANG LV ◽  
QING-SONG CHENG
Keyword(s):  
Linear System ◽  
Helmholtz Equation ◽  
Analytical Formula ◽  
System Of Equations ◽  
Two Dimensional ◽  
Sparse System ◽  
Sine Transform ◽  
Minimum Residual ◽  
Preconditioned Matrix ◽  
Preconditioned Iterative

AbstractA preconditioned iterative method for the two-dimensional Helmholtz equation with Robbins boundary conditions is discussed. Using a finite-difference method to discretize the Helmholtz equation leads to a sparse system of equations which is too large to solve directly. The approach taken in this paper is to precondition this linear system with a sine transform based preconditioner and then solve it using the generalized minimum residual method (GMRES). An analytical formula for the eigenvalues of the preconditioned matrix is derived and it is shown that the eigenvalues are clustered around 1 except for some outliers. Numerical results are reported to demonstrate the effectiveness of the proposed method.

scholarly journals Practical Model Proposed for the Structural Analysis of Segmental Tunnels

2020 ◽  
Vol 10 (23) ◽  
pp. 8514
Author(s):  
Jatziri Y. Moreno-Martínez ◽  
Arturo Galván ◽  
Fernando Peña ◽  
Franco Carpio
Keyword(s):  
Finite Element ◽  
Soft Soil ◽  
Limit State ◽  
Structural Response ◽  
3D Models ◽  
Complex Model ◽  
Numerical Results ◽  
Point Of View ◽  
Simplified Model ◽  
Ultimate Limit State

The construction of tunnels has become increasingly common in city infrastructure; tunnels are used to connect different places in a region (for transportation and/or drainage). In this study, the structural response of a typical segmental tunnel built in soft soil was studied using a simplified model which considers the coupling between segmental rings. From an engineering point of view, there is a need to use simple and reliable finite element models. Therefore, a 1D model based on the Finite Element Method (FEM) composed of beam elements to model the segments and elastic-linear springs and non-linear springs to model the mechanical behavior of the joints was performed. To validate the modeling strategy, the numerical results were compared to (lab-based) experimental results, under an Ultimate Limit State, obtained from the literature, and a comparison between numerical results considering a 3D numerical complex model which included the nonlinearity of concrete, reinforcing steel and the joints was performed. With this simplified model, we obtained a prediction of approximately 95% of the ultimate loading capacity compared to the results developed in the experimental and 3D models. This proposed model will help engineers in practice to create “rational” structural designs of segmental tunnel linings when a “low” interaction between rings is expected.

An optimized Schwarz method with relaxation for the Helmholtz equation: the negative impact of overlap

2019 ◽  
Vol 53 (1) ◽  
pp. 249-268
Author(s):  
Yongxiang Liu ◽  
Xuejun Xu
Keyword(s):  
Convergence Rate ◽  
Helmholtz Equation ◽  
Domain Decomposition ◽  
Negative Impact ◽  
Mesh Size ◽  
Careful Analysis ◽  
Numerical Results ◽  
Wave Number ◽  
Convergence Behavior ◽  
Schwarz Method

In this paper we study how the overlapping size influences the convergence rate of an optimized Schwarz domain decomposition (DD) method with relaxation in the two subdomain case for the Helmholtz equation. Through choosing suitable parameters, we find that the convergence rate is independent of the wave number k and mesh size h, but sensitively depends on the overlapping size. Furthermore, by careful analysis, we obtain that the convergence behavior deteriorates with the increase of the overlapping size. Numerical results which confirm our theory are given.

scholarly journals The Simplified Tikhonov Regularization Method for Identifying the Unknown Source for the Modified Helmholtz Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Fan Yang ◽  
HengZhen Guo ◽  
XiaoXiao Li
Keyword(s):  
Exact Solution ◽  
Helmholtz Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Numerical Results ◽  
Tikhonov Regularization Method ◽  
Convergence Estimate ◽  
Modified Helmholtz Equation ◽  
Unknown Source ◽  
Ill Posed

This paper discusses the problem of determining an unknown source which depends only on one variable for the modified Helmholtz equation. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the simplified Tikhonov regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.

scholarly journals A new iterative solver for the time-harmonic wave equation

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. E57-E63 ◽  
Author(s):  
C. D. Riyanti ◽  
Y. A. Erlangga ◽  
R.-E. Plessix ◽  
W. A. Mulder ◽  
C. Vuik ◽  
...  
Keyword(s):  
Wave Equation ◽  
Linear System ◽  
Time Domain ◽  
Multigrid Method ◽  
Harmonic Wave ◽  
Computational Time ◽  
Full Time ◽  
Iterative Solver ◽  
Time Harmonic ◽  
The Time Domain

The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can be solved efficiently by a direct method. In three dimensions, direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large. Iterative methods are an alternative. These methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence. The iterative solution of the time-harmonic wave equation has long been a notoriously difficult problem in numerical analysis. Recently, a new preconditioner based on a strongly damped wave equation has heralded a breakthrough. The solution of the linear system associated with the preconditioner is approximated by another iterative method, the multigrid method. The multigrid method fails for the original wave equation but performs well on the damped version. The performance of the new iterative solver is investigated on a number of 2D test problems. The results suggest that the number of required iterations increases linearly with frequency, even for a strongly heterogeneous model where earlier iterative schemes fail to converge. Complexity analysis shows that the new iterative solver is still slower than a time-domain solver to generate a full time series. We compare the time-domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively. The new iterative solver can be applied straightforwardly to 3D problems.

scholarly journals Modified Preconditioned GAOR Methods for Systems of Linear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xue-Feng Zhang ◽  
Qun-Fa Cui ◽  
Shi-Liang Wu
Keyword(s):  
Linear System ◽  
Convergence Rate ◽  
Least Squares ◽  
Linear Equations ◽  
Generalized Least Squares ◽  
Numerical Results ◽  
Least Squares Problem ◽  
Comparison Results ◽  
Systems Of Linear Equations ◽  
Better Than

Three kinds of preconditioners are proposed to accelerate the generalized AOR (GAOR) method for the linear system from the generalized least squares problem. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned generalized AOR (PGAOR) methods is better than that of the original GAOR methods. Finally, some numerical results are reported to confirm the validity of the proposed methods.

Use of Shifted Laplacian Operators for Solving Indefinite Helmholtz Equations

2015 ◽  
Vol 8 (1) ◽  
pp. 136-148 ◽  
Author(s):  
Ira Livshits
Keyword(s):  
Hybrid Approach ◽  
Numerical Results ◽  
Laplacian Operator ◽  
Two Dimensional ◽  
Basic Tool ◽  
Helmholtz Equations ◽  
Helmholtz Operator ◽  
Relaxation Properties ◽  
Laplacian Operators ◽  
Complex Damping

AbstractA shifted Laplacian operator is obtained from the Helmholtz operator by adding a complex damping. It serves as a basic tool in the most successful multigrid approach for solving highly indefinite Helmholtz equations — a Shifted Laplacian preconditioner for Krylov-type methods. Such preconditioning significantly accelerates Krylov iterations, much more so than the multigrid based on original Helmholtz equations. In this paper, we compare approximation and relaxation properties of the Helmholtz operator with and without the complex shift, and, based on our observations, propose a new hybrid approach that combines the two. Our analytical conclusions are supported by two-dimensional numerical results.

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