AbstractSteady-state modeling of water distribution networks (WDNs) is the calculation of flow rates in pipes and nodal pressures for a given set of boundary conditions (i.e., water levels, pump ...curves, nodal demands, and so forth). The solution procedure is based on simultaneous solving of the energy and mass conservation equations in the network. There is a Newton-based method entitled the global gradient algorithm (GGA) for solving these equations and it is used for many commercial software programs. The GGA solves a positive definite symmetric linear system for finding head pressures and also updates the discharge of pipes at each iteration. After a predefined number of iterations, the GGA converges to the final solution quickly and it is the most efficient algorithm for WDN modeling. In this paper, for improving the convergence rate, a new method called the multilinear technique is presented that solves the nonlinear system of equations. In this method, nonlinear terms of energy equations are linearized based on maximum and minimum allowable discharge in pipes. Therefore, the set of continuity and energy equations is converted into a linear system and by solving this linear system a good initial solution is obtained. Then, this new solution is used for the linearization process in the next iteration. The process continues until convergence with the final solution with reasonable accuracy. To demonstrate the robustness and effectiveness of the multilinear algorithm, several real and hypothetical WDNs from a small to a large scale are tested. Results in benchmark and real networks show that after two iterations the multilinear algorithm converges with acceptable precision. The simulation of 1,000 hypothetical networks shows that the computational efficiency of the multilinear method in terms of time is almost half of those obtained by the GGA.
In comprehensive fMRI studies of brain function, the data structures often contain higher-order ways such as trial, task condition, subject, and group in addition to the intrinsic dimensions of time ...and space. While multivariate bilinear methods such as principal component analysis (PCA) have been used successfully for extracting information about spatial and temporal features in data from a single fMRI run, the need to unfold higher-order data sets into bilinear arrays has led to decompositions that are nonunique and to the loss of multiway linkages and interactions present in the data. These additional dimensions or ways can be retained in multilinear models to produce structures that are unique and which admit interpretations that are neurophysiologically meaningful. Multiway analysis of fMRI data from multiple runs of a bilateral finger-tapping paradigm was performed using the parallel factor (PARAFAC) model. A trilinear model was fitted to a data cube of dimensions voxels by time by run. Similarly, a quadrilinear model was fitted to a higher-way structure of dimensions voxels by time by trial by run. The spatial and temporal response components were extracted and validated by comparison to results from traditional SVD/PCA analyses based on scenarios of unfolding into lower-order bilinear structures.
