Souza de Cursi Eduardo - Uncertainty Quantification and Stochastic Modeling with Matlab

Eduardo Souza de Cursi Rubens Sampaio First published 2015 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Press Ltd 27-37 St Georges Road London SW19 4EU UK www.iste.co.uk Elsevier Ltd The Boulevard, Langford Lane Kidlington, Oxford, OX5 1GB UK www.elsevier.com Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

For information on all Elsevier publications visit our website at http://store.elsevier.com/ ISTE Press Ltd 2015 The rights of Eduardo Souza de Cursi and Rubens Sampaio to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. MATLAB is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This books use or discussion of MATLAB software or related products does not constituteendorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use ofthe MATLAB software. British Library Cataloguing in Publication Data A CIP record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress ISBN 978-1-78548-005-8 Printed and bound in the UK and US

Until recently, the deterministic view of Nature prevailed in Physics.

Starting from the Enlightenment, there was the belief that Mechanics was completely described by Newtons laws. If the initial conditions of a mechanical systems were known, then its state could be completely determined in the future as well as in the past. This was precisely the viewpoint of Lagrange and it ruled from the 18th Century to around the middle of the 20th Century. However this panglossian view of Nature was disputed, especially with the advent of Thermodynamics that challenged the belief in reversibility and started the study of complex random systems. In addition, the discovery of simple systems with chaotic behavior, that despite being deterministic, showed a very complex behavior, that the mechanical problems are far from simple. Quantum Theory, however, showed that the processes in Nature are not deterministic, but stochastic.

This change in paradigm is not yet accepted, as is well described in the book of Thomas Kuhn that discusses sociological aspects of sciences. The main ideas of probability were slow to develop. Perhaps, because chance events were interpreted as the wish of the gods and hence were not believed to be random. The first theoretical works on probability were connected with games of chance and since the set of possibilities are finite, the studies are combinatorial, but the big breakthrough being when the continuous case is studied. Stochastic modeling is, of course, harder than deterministic modeling and the implementation of the model is more costly. Let us look at this in a simple example.

In the simplest deterministic continuous model, the parameters of the model are constants. The equivalent of this in the stochastic case is to transform the constants in random variables. For each realization of the random variable, its value is also a constant, but the constant may change with the realization following a certain distribution of probability. So, the most simple object from a stochastic model random variables is formed of functions, hence objects of infinite dimension. Also, in a coherent deterministic model, we expect a solution if we fix suitable conditions. In a stochastic model this is not the case.

For each realization of the stochastic data, the parameters are chosen and a deterministic problem is solved. Then, with the set of solutions obtained, statistics are made and the main result of the problem is the distribution of probability of the results. Thus, the main element to be determined is the distribution of the possible solutions. The values obtained from a single realization are of no importance, the distribution of the values is the important result. The understanding of this fact came very slowly for the manufacturing industry, for example, and only after long years of application of quality control in the manufacturing came the realization that they were dealing with a process stochastic in nature. Since then, stochastic modeling has been essential in Engineering.

Today, reliability represents a new way of design and it takes into account the inevitable uncertainties of the processes. The solution of a stochastic model can be decomposed into three steps: generation of the stochastic data following their distribution of probability, solution of a deterministic problem for each sample generated, and finally computation of statistics with the results (for example, to construct a histogram) until they show a certain persistence (do not change an error criterion accordingly). Histograms represent the approximation of the solution of a stochastic problem. When we want an approximation of the solution of a stochastic problem, a histogram is the best approximation. If they are hard to find, or costly to compute, we make do with mean and dispersion, or a certain number of moments. In some situations, such as the ones described in , we compute only a particular probability since computation of moments is too expensive.

This book presents the main ideas of Stochastic Modeling and Uncertainty Quantification using Functional Analysis as the main tool. More specifically, Hilbert Spaces and orthogonal projections are the basic objects leading to the methods exposed. As presented in the text, some ideas often considered as complex, such as Conditional Expectations, may be developed in a systematic way by considering their definition as orthogonal projections in convenient Hilbert spaces. present methods for optimization under uncertainties. In order to help the reader interested in practical applications, listings of Matlab programs implementing the main methods are included in the text. In each chapter, the reader will find these listings which complete the methodological presentation by a practical implementation.