Decomposition algorithm

The velocity Verlet algorithm may be derived by considering a standard approximate decomposition of the Liouville operator which preserves reversibility and is symplectic (which implies that volume in phase space is conserved). This approach [47] has had several beneficial consequences. 

The force decomposition algorithm maps all possible interactions to processors and does not require inter-processor communication during the force calculation phase of MD simulation. However, to obtain the net force on each particle for the update phase would need global communication. In this section, we will present parallel algorithms based on force decomposition. 

Furthermore, one may need to employ data transformation. For example, sometimes it might be a good idea to use the logarithms of variables instead of the variables themselves. Alternatively, one may take the square roots, or, in contrast, raise variables to the nth power. However, genuine data transformation techniques involve far more sophisticated algorithms. As examples, we shall later consider Fast Fourier Transform (FFT), Wavelet Transform and Singular Value Decomposition (SVD). 

The underlying theory of r-RESPA is somewhat involved, but the final result and const quent implementation is actually rather straightforward, being very closely related to th velocity Verlet integration scheme. For our four-way decomposition the algorithm woul... 

The right-hand side in Equation (6.18) is known and hence its solution yields the error 5x in the original solution. The procedure can be iterated to improve the solution step-by-step. Note that implementation of this algorithm in the context of finite element computations may be very expensive. A significant advantage of the LU decomposition technique now becomes clear, because using this technique [A] can be decomposed only once and stored. Therefore in the solution of Equation (6.18) only the right-hand side needs to be calculated. 

The decomposition algorithm nece.ssitates performing the factorization of the operator of the special structure... 

An alternative approach to the discretion of the translation parameter i( involves uniform sampling of the measured signal at all scales, i.e., u = kr, with k e Z. The resulting decomposition algorithm is of complexity OiN log N), and the associated reconstruction requires the computation of N log N coefficients, i.e., it contains redundant information. 

Let us now see how the theory of the wavelet-based decomposition and reconstruction of discrete-time functions can be converted into an efficient numerical algorithm for the multiscale analysis of signals. From Eq. (6b) it is easy to see that, given a discrete-time signal, FqU) we have... 

Multiscale process identification and control. Most of the insightful analytical results in systems identification and control have been derived in the frequency domain. The design and implementation, though, of identification and control algorithms occurs in the time domain, where little of the analytical results in truly operational. The time-frequency decomposition of process models would seem to offer a natural bridge, which would allow the use of analytical results in the time-domain deployment of multiscale, model-based estimation and control. 

The paper-and-pencir method of eigenvector decomposition can only be performed on small matrices, such as illustrated above. For matrices with larger dimensions one needs a computer for which efficient algorithms have been designed (Section 31.4). 

Note that the algebraic signs of the columns in U and V are arbitrary as they have been computed independently. In the above illustration, we have chosen the signs such as to be in agreement with the theoretical result. This problem does not occur in practical situations, when appropriate algorithms are used for singular vector decomposition. 

Fig. 31.13. Schematic example of three common algorithms for singular value and eigenvalue decomposition.

The power algorithm [21] is the simplest iterative method for the calculation of latent vectors and latent values from a square symmetric matrix. In contrast to NIPALS, which produces an orthogonal decomposition of a rectangular data table X, the power algorithm decomposes a square symmetric matrix of cross-products X which we denote by C. Note that Cp is called the column-variance-covariance matrix when the data in X are column-centered. 

A comparison of the performance of the three algorithms for eigenvalue decomposition has been made on a PC (IBM AT) equipped with a mathematical coprocessor [38]. The results which are displayed in Fig. 31.14 show that the Householder-QR algorithm outperforms Jacobi s by a factor of about 4 and is superior to the power method by a factor of about 20. The time for diagonalization of a square symmetric value required by Householder-QR increases with the power 2.6 of the dimension of the matrix. 

Fig. 31.14. Performance of three computer algorithms for eigenvalue decomposition as a function of the dimension of the input matrix. The horizontal and vertical scales are scaled logarithmically. Execution time is proportional to a power 2.6 of the dimension.

Having a closer look at the pyramid algorithm in Fig. 40.43, we observe that it sequentially analyses the approximation coefficients. When we do analyze the detail coefficients in the same way as the approximations, a second branch of decompositions is opened. This generalization of the discrete wavelet transform is called the wavelet packet transform (WPT). Further explanation of the wavelet packet transform and its comparison with the DWT can be found in [19] and [21]. The final results of the DWT applied on the 16 data points are presented in Fig. 40.44. The difference with the FT is very well demonstrated in Fig. 40.45 where we see that wavelet describes the locally fast fluctuations in the signal and wavelet a the slow fluctuations. An obvious application of WT is to denoise spectra. By replacing specific WT coefficients by zero, we can selectively remove... 

Among the methods presented above, we have chosen the best implemented in computer codes (algorithms for 2D and 3D morphology) available at http // www.ichf.edu.pl/mfialkowski/morph.html. The 3D program uses the tetragonal simplex decomposition (Section III.B) with a consequent triangulation scheme... 

The staircase matrix structure of the 2S-MILP (see Figure 9.9) is exploited by 2S-MILP-specific decomposition based algorithms [9,10], The constraint matrix of the 2S-MILP consists of 12 subproblems Wm that are tied together by the first-stage variables x and the corresponding matrix column [ATj. .. Th]r. The main steps of decomposition based algorithms for 2S-MILPs are ... 

Scenario Decomposition Based Branch-and-Bound Algorithm... 

Before a new stage decomposition based hybrid evolutionary algorithm is proposed in Section 9.4, we briefly review the algorithm for general 2S-MILPs of Caroe and Schultz [11] which is regarded as the state-of-the-art exact algorithm for 2S-MILPs... 

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