A Parallel Solving Algorithm on GPU for the Time-Domain Linear System with Diagonal Sparse Matrices

2019 ◽  
pp. 73-84
Author(s):  
Yifei Xia ◽  
Jiaquan Gao ◽  
Guixia He
Keyword(s):  
Linear System ◽  
Time Domain ◽  
Sparse Matrices ◽  
The Time Domain

scholarly journals Nonstationary First Threshold Crossing Reliability for Linear System Excited by Modulated Gaussian Process

2018 ◽  
Vol 2018 ◽  
pp. 1-17
Author(s):  
Rita Greco ◽  
Giuseppe Carlo Marano ◽  
Alessandra Fiore ◽  
Ivo Vanzi
Keyword(s):  
Linear System ◽  
Gaussian Process ◽  
Time Domain ◽  
Reliability Evaluation ◽  
Random Input ◽  
Common Assumption ◽  
Direct Extension ◽  
Threshold Crossing ◽  
The Time Domain

A widely used approach for the first crossing reliability evaluation of structures subject to nonstationary Gaussian random input is represented by the direct extension to the nonstationary case of the solution based on the qualified envelope, originally proposed for stationary cases. The most convenient way to approach this evaluation relies on working in the time domain, where a common assumption used is to adopt the modulation of stationary envelope process instead of the envelope of modulated stationary one, by utilizing the so-called “preenvelope” process. The described assumption is demonstrated in this work, also showing that such assumption can induce some errors in the envelope mean crossing rate.

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.

Solution of large-scale 3D controlled-source electromagnetic modeling problem using efficient iterative solvers

Geophysics ◽  
2021 ◽  
pp. 1-64
Author(s):  
Changkai Qiu ◽  
Changchun Yin ◽  
Yunhe Liu ◽  
Xiuyan Ren ◽  
Hui Chen ◽  
...  
Keyword(s):  
Linear System ◽  
Linear Equation ◽  
Frequency Domain ◽  
Time Domain ◽  
Large Scale ◽  
Equation System ◽  
Iterative Solvers ◽  
Controlled Source ◽  
Linear Equation System ◽  
The Time Domain

With geophysical surveys evolving from traditional 2D to 3D models, the large volume of data adds challenges to inversion, especially when aiming to resolve complex 3D structures. An iterative forward solver for a controlled-source electromagnetic method (CSEM) requires less memory than that for a direct solver; however, it is not easy to iteratively solve an ill-conditioned linear system of equations arising from finite-element discretization of Maxwell’s equations. To solve this problem, we have developed efficient and robust iterative solvers for frequency- and time-domain CSEM modeling problems. For the frequency-domain problem, we first transform the linear system into its equivalent real-number format, and then introduce an optimal block-diagonal preconditioner. Because the condition number of the preconditioned linear equation system has an upper bound of √2, we can achieve fast solution convergence when applying a flexible generalized minimum residual solver. Applying the block preconditioner further results in solving two smaller linear systems with the same coefficient matrix. For the time-domain problem, we first discretize the partial differential equation for the electric field in time using an unconditionally stable backward Euler scheme. We then solve the resulting linear equation system iteratively at each time step. After the spatial discretization in the frequency domain, or space-time discretization in the time domain, we exploit the conjugate-gradient solver with auxiliary-space preconditioners derived from the Hiptmair-Xu decomposition to solve these real linear systems. Finally, we check the efficiency and effectiveness of our iterative methods by simulating complex CSEM models. The most significant advantage of our approach is that the iterative solvers that we adopt have almost the same accuracy and robustness as direct solvers but require much less memory, rendering them more suitable for large-scale 3D CSEM forward modeling and inversion.

Apodisation in the time domain

1992 ◽  
Vol 2 (4) ◽  
pp. 615-620
Author(s):  
G. W. Series
Keyword(s):  
Time Domain ◽  
The Time Domain
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