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Design and Analysis of Power Supply Networks
Published in Louis Scheffer, Luciano Lavagno, Grant Martin, EDA for IC Implementation, Circuit Design, and Process Technology, 2018
David Blaauw, Sanjay Pant, Rajat Chaudhry, Rajendran Panda
Several direct [18] and iterative [19] approaches are available to solve the linear system of equations as in Equation (20.2). Direct techniques rely on factorizing the LHS matrix once and then using the LU factors repeatedly in a simple backward and forward substitution procedure [18] to solve the system at every time-step. Iterative methods, on the other hand, rely on efficient convergence techniques to steer the iterations from an initial guess to the final solution. In this section, we analyze the relative merits and limitations of these methods as applied to solving large power networks. The size and structure of the conductance matrix of the power grid is important in determining the type of linear solution technique that should be used. Typically, the power grid contains millions of nodes, but the conductance matrix is sparse (typically, less than five entries per row/column). This matrix is also symmetric positive-definite, but for a purely resistive network, it may be ill-conditioned.
Numerical Methods and Computational Tools
Published in Raj P. Chhabra, CRC Handbook of Thermal Engineering Second Edition, 2017
Atul Sharma, Salil S. Kulkarni, K. Hrisheekesh, Amit Agrawal, Shyamprasad Karagadde, Amitabh Bhattacharya, Rajneesh Bhardwaj
Iterative solvers. In these solvers, one starts with an initial guess x0 to the solution of Equation 5.4.21 and generates a sequence of approximations xi, which converges to the exact solution of Equation 5.4.21. The main advantage of these method is that they do not need any decompositions and primarily involve matrix vector multiplications. For symmetric positive definite matrices, one can use the conjugate gradient method (see Hestenes and Stiefel, 1952), whereas for a general matrix one can use the Generalized minimal residual method (GMRES) method (see Saad and Schultz, 1986). The rate of convergence of an iterative method is heavily influenced by the choice of the preconditioners. For symmetric positive definite matrices, some of the preconditioners that can be used include incomplete Cholesky and diagonal scaling. For a general matrix, the preconditioners that are commonly used are based on the incomplete LU factorizations. More details of the iterative methods are found in Saad (2003). The detailed survey on the types of preconditioners for large systems is given in Benzi (2002).
Background and Introduction
Published in Ramin S. Esfandiari, Numerical Methods for Engineers and Scientists Using MATLAB®, 2017
An iterative method is a process that starts with an initial guess and computes successive approximations of the solution of a problem until a reasonably accurate approximation is obtained. As we will demonstrate throughout the book, iterative methods are used to find roots of algebraic equations, solutions of systems of algebraic equations, solutions of differential equations, and much more. An important issue in an iterative scheme is the manner in which it is terminated. There are two ways to stop a procedure: (1) when a terminating condition is satisfied, or (2) when the maximum number of iterations is exceeded. In principle, the terminating condition should check to see whether an approximation calculated in a step is within a prescribed tolerance of the true value. In practice, however, the true value is not available. As a result, one practical form of a terminating condition is whether the difference between two successively generated quantities by the iterative method is within a prescribed tolerance. The ideal scenario is when an algorithm meets the terminating condition, and at a reasonably fast rate. If it does not, then the total number of iterations performed should not exceed a prescribed maximum number of iterations.
Performance analysis of parameter estimator on non-linear iterative methods for ultra-wideband positioning
Published in International Journal of Image and Data Fusion, 2023
Chuanyang Wang, Bing He, Liangliang Shi, Weiduo Huang, Liuxu Shan
The iterative method is a procedure that uses a rough initial guess to generate a sequence of improving approximate solutions for non-linear equations (W et al. 2004). The traditional method solves the non-linear distance equations by linearising them, and then, the solution can be found iteratively (H 1976). However, the iterative approaches require and rely heavily on the initial starting values to ensure global minimum and faster convergence to the correct solution (Awange and Fukuda 2007). The numerical characteristics of a number of iterative descent algorithms for solving non-linear LS problems have been discussed, particularly in the Gauss–Newton method (Teunissen 1990). The estimators are usually inherently biased by linearising the non-linear expression, and the bias comes from non-zero higher-order terms (Yan et al. 2008).
Distribution free approach on electrical energy supply chain using geometric shipment policy
Published in International Journal of Sustainable Engineering, 2021
Different researchers developed different types of model under the consideration of production systems with safety stocks and equal sized shipments but no one developed an integrated distribution system with unequal sized shipments for an electricity power system with stochastic electricity demand, transmission cost and distribution cost. In particular two types of lead time concept are considered which are lead time depends on the setup and transportation time and other is the lead time depends on transportation time only. Using the two types of lead time, most of the researchers considered a backorder quantity follows a normal distribution, uniform distribution, etc. But in this paper, we have considered a shortage cost follows a free distribution concept. An iterative solution technique is used to find an optimal solution numerically. An iterative method is a mathematical procedure that uses an initial guess to generate a sequence of improving approximate solutions for a class of problems, in which the approximation is derived from the previous ones. There is a big research gap in this direction, which is fulfilled by this research. In this paper, section 2 represents a problem definition, notations and assumptions, section 3 describes a mathematical model, section 4 describes the solution methodology of the model, section 5 represents the numerical example of the model and section 6 explains the sensitivity analysis and conclusion of the model.
Evaluating the performance of an Inexact Newton method with a preconditioner for dynamic building system simulation
Published in Journal of Building Performance Simulation, 2022
Zhelun Chen, Jin Wen, Anthony J. Kearsley, Amanda Pertzborn
We investigate the application of a NK method from the solver package NITSOL (Pernice and Walker 1998) that solves Equation (1) by solving Equation (2) iteratively using a Krylov subspace method. This method is an Inexact Newton method with Backtracking (INB) algorithm, where linear systems are solved using Generalized Minimal RESidual (GMRES) (Saad and Schultz 1986), a Krylov subspace method. The implementation allows for the application of a preconditioner to accelerate and improve convergence of the iterative method. Details about the INB framework, the GMRES algorithm, and the preconditioning technique are summarized in the following subsections.