We propose a residual based sparse approximate inverse (RSAI) preconditioning procedure, for the large sparse linear system Ax = b. Different from the SParse Approximate Inverse (SPAI) algorithm proposed by Grote and Huckle (SIAM Journal on Scientific Computing, 18 (1997), pp. 838-853.), RSAI uses only the dominant other than all the information on the current residual and augments sparsity patterns adaptively during loops. In order to control the sparsity of M, we develop two practical algorithms RSAI(f ix) and RSAI(tol). RSAI(f ix) retains the prescribed number of large nonzero entries and adjusts their positions in each column of M among all available ones, in which the number of large entries is increased by a fixed number at each loop. In contrast, the existing indices of M by SPAI are untouched in subsequent loops and a few most profitable indices are added to each column of M from the new candidates in the next loop. RSAI(tol) is a tolerance based dropping algorithm and retains all large entries by dynamically dropping small ones below some tolerances, and it better suits for the problem where the numbers of large entries in the columns of A −1 differ greatly. When the two preconditioners M have almost the same or comparable numbers of nonzero entries, the numerical experiments on real-world problems demonstrate that RSAI(f ix) is highly competitive with SPAI and can outperform the latter for some problems. We also make comparisons of RSAI(f ix), RSAI(tol), and power sparse approximate inverse(tol) proposed Jia and Zhu (Numerical Linear Algebra with Applications, 16 (2009), pp. 259-299.) and incomplete LU factorization type methods and draw some general conclusions. KEYWORDSadaptive, F-norm minimization, Krylov solver, preconditioning, PSAI, RSAI, SPAI, sparse approximate inverse, sparsity pattern 1 Numer Linear Algebra Appl. 2017;24:e2080.wileyonlinelibrary.com/journal/nla
Aiming to obtain the acoustic attenuation performance of exhaust muffler of diesel engine and the influence of main structural parameters on its acoustic attenuation characteristics, the finite element analysis method and acoustic theory were adopted to numerically investigate the acoustic attenuation performance under the boundary condition of acoustic adiabatic propagation and muffler wall. It suggested that the noise cancellation effect of muffler was poor at the middle and low frequency in range of 0-3000 Hz, and the transfer loss of muffler was basically 0 dB pass frequency at 1100 Hz. According to previous single-factor study experience, the structural factors, such as the expansion ratio, insertion length of outlet perforated pipe, the distance between the diaphragm and the front part of muffler, have influences on the acoustic performance of muffler at low frequency. Thus, they were taken as the starting point to study the influence of multiple interaction factors on the muffling performance by using orthogonal design method combined with the finite element analysis method. The influence degree of different structure parameters on the acoustic performance of muffler and the optimized structure parameters were obtained. Through the analysis on the acoustic characteristic of the optimized muffler, it indicated that the transmission loss of the improved muffler had significant increase in other frequency range except the range of 650-800 Hz and 2500-2700 Hz, especially at frequency of 1100 Hz compared with the original muffler. In the range of 0-3000 Hz, the mean of transmission loss of the improved muffler was about 9.8 dB larger than that of original muffler, which indicated that better noise cancellation effect was achieved. The improved muffler also provided a certain reference for the structural improvement of similar muffler.
The ever-expanding power system is developed into an interconnected pattern of power grids. Zone partitioning is an essential technique for the operation and management of such an interconnected power system. Owing to the transmission capacity limitation, transmission congestion may occur with a regional influence on power system. If transmission congestion is considered when the system is decomposed into several regions, the power consumption structure can be optimized and power system planning can be more reasonable. At the same time, power resources can be properly allocated and system safety can be improved. In this paper, we propose a power system zone partitioning method where the potential congested branches are identified and the spectral clustering algorithm is improved. We transform the zone partitioning problem into a graph segmentation problem by constructing an undirected weighted graph of power system where the similarities between buses are measured by the power transfer distribution factor (PTDF) corresponding to the potential congested branches. Zone partitioning results show that the locational marginal price (LMP) in the same zone is similar, which can represent regional price signals and provide regional auxiliary decisions.
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