From Linear To Non Linear Polytechnique-Books Pdf

From Linear to Non Linear polytechnique
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Taylor Francis Group,6000 Broken Sound Parkway NW Suite 300. Boca Raton FL 33487 2742,2016 by Taylor Francis Group LLC. CRC Press is an imprint of Taylor Francis Group an Informa business. No claim to original U S Government works,Printed on acid free paper. Version Date 20160509, International Standard Book Number 13 978 1 4987 4622 9 Hardback. This book contains information obtained from authentic and highly regarded sources Reasonable. efforts have been made to publish reliable data and information but the author and publisher cannot. assume responsibility for the validity of all materials or the consequences of their use The authors and. publishers have attempted to trace the copyright holders of all material reproduced in this publication. and apologize to copyright holders if permission to publish in this form has not been obtained If any. copyright material has not been acknowledged please write and let us know so we may rectify in any. future reprint, Except as permitted under U S Copyright Law no part of this book may be reprinted reproduced.
transmitted or utilized in any form by any electronic mechanical or other means now known or. hereafter invented including photocopying microfilming and recording or in any information stor. age or retrieval system without written permission from the publishers. For permission to photocopy or use material electronically from this work please access www copy. right com http www copyright com or contact the Copyright Clearance Center Inc CCC 222. Rosewood Drive Danvers MA 01923 978 750 8400 CCC is a not for profit organization that pro. vides licenses and registration for a variety of users For organizations that have been granted a photo. copy license by the CCC a separate system of payment has been arranged. Trademark Notice Product or corporate names may be trademarks or registered trademarks and are. used only for identification and explanation without intent to infringe. Visit the Taylor Francis Web site at,http www taylorandfrancis com. and the CRC Press Web site at,http www crcpress com. INTRODUCTION BRIEF OVERVIEW OF MONTE,CARLO METHODS 1. A LITTLE HISTORY FROM THE BUFFON NEEDLE TO,NEUTRON TRANSPORT 3. THREE TYPICAL PROBLEMS IN RANDOM,SIMULATION 10,B PROBLEM 1 NUMERICAL INTEGRATION.
QUADRATURE MONTE CARLO AND QUASI,MONTE CARLO METHODS 10. B PROBLEM 2 SIMULATION OF COMPLEX,DISTRIBUTIONS METROPOLIS HASTINGS. ALGORITHM GIBBS SAMPLER 18,B PROBLEM 3 STOCHASTIC OPTIMIZATION. SIMULATED ANNEALING AND THE ROBBINS,MONRO ALGORITHM 23. PART A TOOLBOX FOR STOCHASTIC,SIMULATION 29,Chapter 1 Generating random variables 31.
1 1 PSEUDORANDOM NUMBER GENERATOR 31,1 2 GENERATION OF ONE DIMENSIONAL RANDOM. VARIABLES 32,viii Contents,1 2 1 Inversion method 32. 1 2 2 Gaussian variables 36,1 3 ACCEPTANCE REJECTION METHODS 37. 1 3 1 Generation of conditional distribution 37,1 3 2 Generation of non conditional distributions. by the acceptance rejection method 38,1 3 3 Ratio of uniforms method 40.
1 4 OTHER TECHNIQUES FOR GENERATING,A RANDOM VECTO 42. 1 4 1 The Gaussian vector 43,1 4 2 Modeling of dependence using copulas 44. 1 5 EXERCISES 47,Chapter 2 Convergences and error estimates 49. 2 1 LAW OF LARGE NUMBERS 49,2 2 CENTRAL LIMIT THEOREM AND. CONSEQUENCES 52, 2 2 1 Central limit theorem in dimension 1 and beyond 52.
2 2 2 Asymptotic confidence regions and intervals 54. 2 2 3 Application to the evaluation of a function of. 2 2 4 Applications in the evaluation of sensitivity of. expectations 61,2 3 OTHER ASYMPTOTIC CONTROLS 65, 2 3 1 Berry Essen bounds and Edgeworth expansions 65. 2 3 2 Law of iterated logarithm 66,2 3 3 Almost sure central limit theorem 66. 2 4 NON ASYMPTOTIC ESTIMATES 67,2 4 1 About exponential inequalities 67. 2 4 2 Concentration inequalities in the case of,bounded random variables 69. 2 4 3 Uniform concentration inequalities 70,Contents ix.
2 4 4 Concentration inequalities in the case of,Gaussian noise 77. 2 5 EXERCISES 85,Chapter 3 Variance reduction 89,3 1 ANTITHETIC SAMPLING 89. 3 2 CONDITIONING AND STRATIFICATION 92,3 2 1 Conditioning technique 92. 3 2 2 Stratification technique 92,3 3 CONTROL VARIATES 94. 3 3 1 Concept 94,3 3 2 Optimal choice 95,3 4 IMPORTANCE SAMPLING 97.
3 4 1 Changes of probability measure basic notions. and applications to Monte Carlo methods 97,3 4 2 Changes of probability measure by a ne. transformations 103,3 4 3 Change of probability measure by Esscher. transform 107,3 4 4 Adaptive methods 110,3 5 EXERCISES 112. PART B SIMULATION OF LINEAR PROCESSES 115, Chapter 4 Stochastic di erential equations and Feynman. Kac formulas 117,4 1 BROWNIAN MOTION 119,4 1 1 A brief history 119.
4 1 2 Definition 119,4 1 3 Simulation 124,4 1 4 Heat equation 128. x Contents,4 1 5 Quadratic variation 131,4 2 STOCHASTIC INTEGRAL AND IT FORMULA 132. 4 2 1 Filtration and stopping times 133,4 2 2 Stochastic integral and its properties 134. 4 2 3 It process and It formula 137,4 3 STOCHASTIC DIFFERENTIAL EQUATIONS 138. 4 3 1 Definition existence uniqueness 138,4 3 2 Flow property and Markov property 139.
4 3 3 Examples 139,4 4 PROBABILISTIC REPRESENTATIONS OF. PARTIAL DIFFERENTIAL EQUATIONS,FEYNMAN KAC FORMULAS 142. 4 4 1 Infinitesimal generator 142, 4 4 2 Linear parabolic partial di erential equation. with Cauchy condition 144, 4 4 3 Linear elliptic partial di erential equation 148. 4 4 4 Linear parabolic partial di erential equation. with Cauchy Dirichlet condition 149, 4 4 5 Linear elliptic partial di erential equation with.
Dirichlet condition 153,4 5 PROBABILISTIC FORMULAS FOR THE. GRADIENTS 153,4 5 1 Pathwise di erentiation method 154. 4 5 2 Likelihood method 155,4 6 EXERCISES 156,Chapter 5 Euler scheme for stochastic di erential. equations 163,5 1 DEFINITION AND SIMULATION 164, 5 1 1 Definition as an It process quadratic moments 164. 5 1 2 Simulation 166,Contents xi,5 1 3 Application to computation of di usion.
expectation discretization error and statistical,5 2 STRONG CONVERGENCE 170. 5 3 WEAK CONVERGENCE 173,5 3 1 Convergence at order 1 173. 5 3 2 Extensions 176,5 4 SIMULATION OF STOPPED PROCESSES 178. 5 4 1 Discrete approximation of exit time 179,5 4 2 Brownian bridge method 181. 5 4 3 Boundary shifting method 184,5 5 EXERCISES 186.
Chapter 6 Statistical error in the simulation of stochastic. di erential equations 191,6 1 ASYMPTOTIC ANALYSIS NUMBER OF. SIMULATIONS AND TIME STEP 191,6 2 NON ASYMPTOTIC ANALYSIS OF THE. STATISTICAL ERROR IN THE EULER SCHEME 194,6 3 MULTI LEVEL METHOD 197. 6 4 UNBIASED SIMULATION USING A RANDOMIZED,MULTI LEVEL METHOD 202. 6 5 VARIANCE REDUCTION METHODS 206,6 5 1 Control variates 206.
6 5 2 Importance sampling 207,6 6 EXERCISES 208,PART C SIMULATION OF NON LINEAR. PROCESSES 211, Chapter 7 Backward stochastic di erential equations 213. xii Contents,7 1 EXAMPLES 214,7 1 1 Examples coming from reaction di usion. equations 214, 7 1 2 Examples coming from stochastic modeling 217. 7 2 FEYNMAN KAC FORMULAS 221,7 2 1 A general result 221.
7 2 2 Toy model 224,7 3 TIME DISCRETIZATION AND DYNAMIC. PROGRAMMING EQUATION 227,7 3 1 Discretization of the problem 227. 7 3 2 Error analysis 228,7 4 OTHER DYNAMIC PROGRAMMING EQUATIONS 231. 7 5 ANOTHER PROBABILISTIC REPRESENTATION,VIA BRANCHING PROCESSES 233. 7 6 EXERCISES 236,Chapter 8 Simulation by empirical regression 241.
8 1 THE DIFFICULTIES OF A NAIVE APPROACH 241,8 2 APPROXIMATION OF CONDITIONAL. EXPECTATIONS BY LEAST SQUARES METHODS 244,8 2 1 Empirical regression 245. 8 2 2 SVD method 246,8 2 3 Example of approximation space the local. polynomials 249, 8 2 4 Error estimations robust with respect to the. 8 2 5 Adjustment of the parameters in the case of lo. cal polynomials 252,8 2 6 Proof of the error estimations 254.
8 3 APPLICATION TO THE RESOLUTION OF THE,DYNAMIC PROGRAMMING EQUATION BY. EMPIRICAL REGRESSION 257,8 3 1 Learning sample and approximation space 257. Contents xiii, 8 3 2 Calculation of the empirical regression functions 258. 8 3 3 Equation of the error propagation 260,8 3 4 Optimal adjustment of the convergence. parameters in the case of local polynomials 266,8 4 EXERCISES 267.
Chapter 9 Interacting particles and non linear equations. in the McKean sense 273,9 1 HEURISTICS 273, 9 1 1 Macroscopic scale versus microscopic scale 273. 9 1 2 Examples and applications 275,9 2 EXISTENCE AND UNIQUENESS OF NON LINEAR. DIFFUSIONS 278,9 3 CONVERGENCE OF THE SYSTEM OF,INTERACTING DIFFUSIONS PROPAGATION OF. CHAOS AND SIMULATION 279, Appendix A Reminders and complementary results 285. A 1 ABOUT CONVERGENCES 285, A 1 1 Convergence a s in probability and in L1 285.
A 1 2 Convergence in distribution 286,A 2 SEVERAL USEFUL INEQUALITIES 287. A 2 1 Inequalities for moments 287, A 2 2 Inequalities in the deviation probabilities 290. Monte Carlo Methods and Stochastic Processes From Linear to Non Linear EMMANUEL GOBET ECOLE POLYTECHNIQUE UNIVERSITY PARIS SACLAY CMAP PALAISEAU CEDEX FRANCE

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