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Monte Carlo and Quasi-Monte Carlo Methods 2006

Alexander Keller • Stefan Heinrich Harald Niederreiter Editors Monte Carlo and Quasi-Monte Carlo Methods 2006 123

Stefan Heinrich Department of Computer Science University of Kaiserslautern 67653 Kaiserslautern Germany heinrich@informatik.uni-kl.de Alexander Keller Department of Computer Science Ulm University Albert-Einstein-Allee 11 89069 Ulm Germany alexander.keller@uni-ulm.de Harald Niederreiter Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Republic of Singapore nied@math.nus.edu.sg ISBN 978-3-540-74495-5 e-ISBN 978-3-540-74496-2 Library of Congress Control Number: 2007936240 Mathematics Subject Classiﬁcation (2000): Primary: 11K45, 65-06, 65C05, 65C10 Secondary: 11K38, 65D18, 65D30, 65D32, 65R20, 91B28 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover Design: WMXDesign GmbH, Heidelberg Printed on acid-free paper 9 8 7 6 5 4 3 2 1 spinger.com

Preface This volume represents the refereed proceedings of the Seventh International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientiﬁc Computing, which was held at Ulm University, Germany, from 14–18 Au- gust 2006. The program of this conference was arranged by an international committee consisting of Ivan Dimov (Bulgarian Academy of Sciences), Henri Faure (CNRS Marseille), Paul Glasserman (Columbia University), Stefan Heinrich (co-chair, University of Kaiserslautern), Fred Hickernell (Illinois In- stitute of Technology), Alexander Keller (co-chair, University of Ulm), Pierre L’Ecuyer (Universit´e de Montr´eal), Michael Mascagni (The Florida State University), Peter Math´e (Weierstrass Institute for Applied Analysis and Stochastics), Harald Niederreiter (co-chair, National University of Singapore), Erich Novak (Friedrich-Schiller-Universit¨at Jena), Art Owen (Stanford Uni- versity), Klaus Ritter (TU Darmstadt), Ian Sloan (University of New South Wales), Denis Talay (INRIA Sophia Antipolis), and Henryk Wo´zniakowski (Columbia University). The local arrangements were in the hands of the PhD students Sabrina Dammertz, Holger Dammertz, Matthias Raab, Carsten W¨achter, the students Bernhard Finkbeiner, Leonhard Gr¨unschloss, Daniela Hauser, Johannes Hanika, Christian Kempter, Manuel Kugelmann, Sehera Nawaz, Daniel Seibert, and our secretary Claudia Wainczyk, all at the computer graphics group of Ulm University. This conference continued the tradition of biennial MCQMC conferences which was begun at the University of Nevada in Las Vegas, Nevada, USA, in June 1994 and followed by conferences at the University of Salzburg, Austria, in July 1996, the Claremont Colleges in Claremont, California, USA, in June 1998, Hong Kong Baptist University in Hong Kong, China, in November 2000, the National University of Singapore, Republic of Singapore, in November 2002, and at the Palais des Congr`es in Juan-les-Pins, France, in June 2004. The proceedings of these previous conferences were all published by Springer-Verlag, under the titles Monte Carlo and Quasi-Monte Carlo Meth- ods in Scientiﬁc Computing (H. Niederreiter and P.J.-S. Shiue, eds.), Monte

VI Preface Carlo and Quasi-Monte Carlo Methods 1996 (H. Niederreiter, P. Hellekalek, G. Larcher and P. Zinterhof, eds.), Monte Carlo and Quasi-Monte Carlo Meth- ods 1998 (H. Niederreiter and J. Spanier, eds.), Monte Carlo and Quasi-Monte Carlo Methods 2000 (K.-T. Fang, F.J. Hickernell and H. Niederreiter, eds.), Monte Carlo and Quasi-Monte Carlo Methods 2002 (H. Niederreiter, ed.), and Monte Carlo and Quasi-Monte Carlo Methods 2004 (H. Niederreiter and D. Talay, eds.). The next MCQMC conference will be held in Montr´eal, Canada, in July 2008. The program of the conference was rich and varied with over 120 talks being presented. Highlights were the invited plenary talks given by Ronald Cools (Katholieke Universiteit Leuven), Sergei Mikhailovitch Ermakov (Saint- Petersburg State University), Alan Genz (Washington State University), Frances Kuo (The University of New South Wales), Thomas M¨uller-Gronbach (Otto-von-Guericke-Universit¨at Magdeburg), Harald Niederreiter (National University of Singapore), Gilles Pag`es (Universit´es Paris VI et VII - CNRS), Karl Sabelfeld (Weierstraßinstitut f¨ur angewandte Analysis und Stochastik), and Peter Shirley (University of Utah) as well as the special sessions that were organized by designated chairpersons. The papers in this volume were carefully screened and cover both the theory and the applications of Monte Carlo and quasi-Monte Carlo methods. We gratefully acknowledge generous ﬁnancial support of the conference by the German Science Foundation (DFG) and the ﬁnancial backup by the mental images GmbH, Berlin. We also thank the anonymous reviewers for their reports and many others who contributed enormously to the excellent quality of the conference pre- sentations and to the high standards for publication in these proceedings by careful review of the abstracts and manuscripts that were submitted. Leonhard Gr¨unschloß provided invaluable help with editing the proceedings. Finally, we want to express our gratitude to Springer-Verlag for publishing this volume. Ulm, August 2007 Alexander Keller Stefan Heinrich Harald Niederreiter

Contents Part I Invited Articles A Belgian View on Lattice Rules Ronald Cools and Dirk Nuyens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 MCQMC Algorithms for Solving some Classes of Equations Sergej Ermakov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 MCQMC Methods for Multivariate Statistical Distributions Alan Genz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Minimal Errors for Strong and Weak Approximation of Stochastic Diﬀerential Equations Thomas M¨uller-Gronbach and Klaus Ritter . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Nets, (t, s)-Sequences, and Codes Harald Niederreiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Quadratic Optimal Functional Quantization of Stochastic Processes and Numerical Applications Gilles Pag`es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Random Field Simulation and Applications Karl Sabelfeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Monte Carlo and Quasi-Monte Carlo Methods for Computer Graphics Peter Shirley, Dave Edwards, and Solomon Boulos . . . . . . . . . . . . . . . . . . . 167

VIII Contents Part II Contributed Articles Random Walk Algorithm for Estimating the Derivatives of Solution to the Elliptic BVP Alexander Burmistrov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Free-Knot Spline Approximation of Fractional Brownian Motion Jakob Creutzig and Mikhail Lifshits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Simulation on Rank-1 Lattices Holger Dammertz, Alexander Keller, and Sabrina Dammertz . . . . . . . . . . 205 Image Synthesis by Rank-1 Lattices Sabrina Dammertz and Alexander Keller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Continuous Runge-Kutta Methods for Stratonovich Stochastic Diﬀerential Equations Kristian Debrabant and Andreas R¨oßler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Issues on Computer Search for Large Order Multiple Recursive Generators Lih-Yuan Deng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Design and Implementation of Eﬃcient and Portable Multiple Recursive Generators with Few Zero Coeﬃcients Lih-Yuan Deng, Huajiang Li, Jyh-Jen Horng Shiau, and Gwei-Hung Tsai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Approximation of Functions Using Digital Nets Josef Dick, Peter Kritzer, and Frances Y. Kuo . . . . . . . . . . . . . . . . . . . . . . 275 Construction of Low-Discrepancy Point Sets of Small Size by Bracketing Covers and Dependent Randomized Rounding Benjamin Doerr and Michael Gnewuch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 A Coding Theoretic Approach to Building Nets with Well-Equidistributed Projections Yves Edel and Pierre L’Ecuyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Improvements on Low Discrepancy One-Dimensional Sequences and Two-Dimensional Point Sets Henri Faure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Improved Multilevel Monte Carlo Convergence using the Milstein Scheme Mike Giles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

Contents IX Generalized Tractability for Linear Functionals Michael Gnewuch and Henryk Wo´zniakowski . . . . . . . . . . . . . . . . . . . . . . . . 359 An Improved Implementation of Stochastic Particle Methods and Applications to Coagulation Equations Flavius Guia¸s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 (t, m, s)-Nets and Maximized Minimum Distance Leonhard Gr¨unschloß, Johannes Hanika, Ronnie Schwede, and Alexander Keller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Quasi-Monte Carlo Simulation of Discrete-Time Markov Chains on Multidimensional State Spaces Rami El Haddad, Christian L´ecot, and Pierre L’Ecuyer . . . . . . . . . . . . . . . 413 Computational Engine for a Virtual Tissue Simulator Carole Hayakawa, Jerome Spanier, and Vasan Venugopalan . . . . . . . . . . . 431 Randomized Approximation of Sobolev Embeddings Stefan Heinrich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Tractability of Linear Multivariate Problems in the Average Case Setting Fred Hickernell, Greg Wasilkowski, and Henryk Wo´zniakowski . . . . . . . . . 461 Zinterhof Sequences in GRID-Based Numerical Integration Heinz Hofbauer, Andreas Uhl, and Peter Zinterhof . . . . . . . . . . . . . . . . . . . 495 A Pragmatic View on Numerical Integration of Unbounded Functions Heinz Hofbauer, Andreas Uhl, and Peter Zinterhof . . . . . . . . . . . . . . . . . . . 511 Assessment of Genetic Association using Haplotypes Inferred with Uncertainty via Markov Chain Monte Carlo Raquel Iniesta and Victor Moreno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 The Generalized Gibbs Sampler and the Neighborhood Sampler Jonathan Keith, George Sofronov, and Dirk Kroese . . . . . . . . . . . . . . . . . . . 537 The Weighted Dyadic Diaphony of Digital Sequences Peter Kritzer and Friedrich Pillichshammer . . . . . . . . . . . . . . . . . . . . . . . . . 549 A New Criterion for Finiteness of Weight Estimator Variance in Statistical Simulation Ilya Medvedev and Gennadii Mikhailov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

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