Inoltro questo avviso di seminario che sembra di argomento interessante per gli iscritti alla lista.
-federico
-------- Forwarded Message -------- Subject: [Dipartimento.di] Mauriana Pesaresi Seminar Date: Wed, 11 May 2016 10:53:01 +0200 From: Francesca Pratesi pratesi@di.unipi.it Organization: University of Pisa To: dottorandi@di.unipi.it, dipartimento@di.unipi.it
Dear all, I would like to remind you that the next Mauriana Pesaresi Seminar will be held tomorrow, 12 May. Below further information on the seminars.
Title2: Impact of Equations Ordering on Inexact and Quasi-Inverse Newton Methods.
Speaker2: Giulio Masetti Date2: 12 May 2016, 13:00 Place2: East Seminar Room
Abstract2: The Newton-Raphson method has a prominent role in applied science and engineering. Our research group at SEDC lab., ISTI/CNR, is involved in dependability-related assessment of Smart Electrical Grids and the Newton-Raphson method is used to solve numerically the system of power-flow equations that characterize the electrical network.
In general, given a system of non-linear equations written in the form F(u)=0, where F is a continuously differentiable function that maps a n-dimensional vector u to a n-dimensional vector F(u), one of the solutions can be found via the Newton-Raphson method. At each iteration of this method, a linear system that involves the Jacobian matrix J_{F}(u) of F has to be solved and the equation ordering does not affect it. At increasing the size of the problem, solving with a direct method the linear system is often impractical (w.r.t. time and space complexity, and accuracy), even if the Jacobian is a sparse matrix. Therefore, in these cases iterative methods have been studied, where, the equation ordering becomes a critical aspect to analyse carefully.
Studies related to equations ordering are available on techniques employed in medium-size linear systems, such as the Gauss-Seidel method. In this method, at each iteration, permutations of Jacobian rows and columns, that do not depend on the initial equation ordering, are exploited to gain in CPU time.
Jacobian-free techniques, developed to solve very large linear problems, offer interesting opportunities for investigations on the impact of initial equations ordering over the number of iterations. One of the most effective algorithms in this category is the Inexact-Newton-Krylov: it gets rid of the Jacobian matrix and approximates the matrix-vector product J_{F}(u)v with the finite difference [F(u+sv)-F(u)]/s. Another variant of Newton- Raphson is the Quasi-Inverse-Newton: it approximates the inverse of the Jacobian J_{F}^{-1}(u) with a diagonal matrix and solves directly the linear system. A seminal article, written by L. Dutto and published on the International Journal for Numerical Methods in Engineering in 1993, numerically presents the influence of initial ordering over the preconditioner of Inexact- Newton-Krylov GMRES, and then on the number of iterations. However, it does not address the impact of ordering on the core of the method, that is the adopted Jacobian-free techniques. To go further such existing results, our study focuses on Jacobian- free methods and explores the influence of the initial ordering over them.
In this talk, we discuss our approach to prove that the initial equations ordering affects Inexact-Newton-Krylov and Quasi- Inverse-Newton, also presenting a preliminary analysis to quantify the resulting impact. A trivial bivariate function will be used as case study and numerical experiments will be presented. Interestingly, the equations ordering does not only affect the efficiency in reaching a solution, but also the solution itself.
As next step, we are working on exploring more realistic scenarios in the power grid domain. Traditionally, the characteristics of the domain at hand has influenced the application of mathematical structures. For example, when using Gauss-Seidel in the power-flow context, there exists a particular initial ordering such that the Jacobian is a four-block matrix and each block corresponds to well defined electrical quantities. Using Jacobian-free methods the application-oriented initial equations ordering could not be the best choice with respect to iterations number and reachable solutions. We believe this is an important feature, that the user of these methods need to be aware of.