Post on 19-Jul-2018
Bjorn EngquistEditor
Encyclopedia of Appliedand ComputationalMathematics
Volume 2
L–Z
With 361 Figures and 33 Tables
EditorBjorn EngquistUniversity of Texas at AustinAustin, TX, USA
ISBN 978-3-540-70528-4 ISBN 978-3-540-70529-1 (eBook)ISBN 978-3-540-70530-7 (print and electronic bundle)DOI 10.1007/ 978-3-540-70529-1
Library of Congress Control Number: 2015953230
Springer Heidelberg New York Dordrecht London© Springer-Verlag Berlin Heidelberg 2015This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors or theeditors give a warranty, express or implied, with respect to the material contained herein or for any errorsor omissions that may have been made.
Printed on acid-free paper
Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media (www.springer.com)
Preface
The scientific field of applied and computational mathematics has evolved andexpanded at a very rapid rate during the last few decades. Many subfields havematured, and it is therefore natural to consider publishing an encyclopedia for thefield. Traditional encyclopedias are, however, becoming part of history. For fast,simple, and up-to-date facts they cannot compete with web search engines andweb-based versions of the Wikipedia type. For much more extensive and completetreatments of topics, traditional monographs and review articles in specializedjournals are common. There is an advantage with web-based articles as in Wikipedia,which constantly evolves and adapts to changes. There is also an advantage witharticles that do not change, have known authors, and can be referred to in otherpublications. With the Encyclopedia for Applied and Computational Mathematics(EACM), we are aiming at achieving the best of these two models.
The goal with EACM is a publication with broad coverage by many articles, whichare quality controlled through a traditional peer review process. The articles can beformally cited, and the authors can take credit for their contributions. This publicationwill be frozen in its current form, obviously on paper as well as on the web. Inparallel there will be an electronic version, where the authors can make changesto their articles and where new articles will be added. After a couple of years, thisdynamic version will result in a publication of a new edition of EACM, which can bereferenced while the dynamic electronic version continues to evolve. The length ofthe articles is also chosen to fill the gap between the common shorter web versionsand more specialized longer publications. They are here typically between 5 and 10pages. A few are introductory overviews and a bit longer than the average article.
An encyclopedia will never be complete, and the decision to define the first editionat this time is based on a compromise between the desire of covering the field welland a timely published version. This first edition has 312 articles with the overallnumber of 1,575 pages in 2 volumes. There are few contributions with animations,which will appear in the electronic version. This will be expanded in the future.
The above discussion was about the “E” in EACM. Now we turn to rest of theacronym, “ACM.” Modern applied and computational mathematics is more applied,more computational, and more mathematical than ever. The exponential growth ofcomputational power has allowed for much more complex mathematical models,and these new models are typically more realistic for applications. It is naturalto include both applied and computational mathematics in the encyclopedia eventhough these two fields are philosophically different. In computational mathematics,algorithms are developed and analyzed and may in principle be independent fromapplications. The two fields are, however, now very tightly coupled in practice.
v
vi Preface
Almost all applied mathematics has some computational components, and mostcomputational mathematics is directly developed for applications.
Computation is now mentioned as the third pillar of science together with theclassical theory and experiments. Scientific progress of today is often based oncritical computational components, which can be seen in the growing contributionfrom computations in recent Nobel Prizes. The importance of mathematical modelingand scientific computing in engineering is even more obvious. The expression“Computational Science and Engineering” has emerged to describe a scientific fieldwhere computations have merged with applications.
Classical fields of applied mathematics, for example, asymptotic analysis andhomogenization, are today not so often used for achieving quantitative results.They are, however, very important in mathematical modeling and in deriving andunderstanding a variety of numerical techniques for multiscale simulations. This isexplained in different settings throughout EACM.
We mentioned above that modern applied and computational mathematics aremore mathematical than ever. In the early days, this coupling was natural. Manyalgorithms that are used today and also discussed in this encyclopedia have the namesof Newton and Gauss. However, during the century before the modern computer,mathematics in its pure form evolved rapidly and became more disconnected fromapplications. The computational tools of pen and paper, the slide rule, and simplemechanical devices stayed roughly the same. We got a clear division between pureand applied mathematics. This has changed with the emergence of the moderncomputer. Models based on much more sophisticated mathematics are now basesfor the quantitative computations and thus practical applications. There are manyexamples of this tighter coupling between applied and computational mathematicson the one hand and what we regard as pure mathematics on the other.
Harmonic analysis is a typical example. It had its origin in applied and compu-tational mathematics with the work of Fourier on heat conduction. However, onlyvery special cases can be studied in a quantitative way by hand. In the years after thisbeginning, there was substantial progress in pure directions of harmonic analysis. Theemergence of powerful computers and the fast Fourier transform (FFT) algorithmdrastically changed the scene. This resulted, for example, in wavelets, the inverseRadon transform, a variety of spectral techniques for PDEs, computational informa-tion, and sampling theory and compressed sensing. The reader will see illustrativeexamples in EACM. Partial differential equations have also had a recent developmentwhere ideas have bounced back and forth between applications, computations, andfundamental theory.
The field of applied and computational mathematics is of course not well defined.We will use the term in a broad sense, but we have not included areas that have theirown identity and where applied and computational mathematics is not what you thinkof even if in a strict sense applied and computational mathematics would be correct.Statistics is the most prominent example. This was an editorial decision. Throughthe process of producing the EACM, there has been a form of self-selection. Whensection editors and authors were asked to cover an area or a topic closer to the core ofapplied and computational mathematics, the success rate was very high. Examples arenumerical analysis and inverse problems. Many applied areas are also well covered,ranging from general topics in fluid and solid mechanics to computational aspects ofchemistry and the mathematics of atmosphere and ocean science.
Preface vii
In fields further from the core, the response was less complete. Examples ofthe latter are the mathematical aspects of computer science and physics, where theresearchers generally do not think of themselves as doing applied and computationalmathematics even when they are. The current encyclopedia naturally does not coverall topics that should ideally have their own articles. We hope to fill these holes in theevolving web-based version and then in the future editions.
Finally, I would like to thank all section editors and authors for their outstandingcontributions and their patience. I also hope that you will continue to improve EACMin its dynamic form and in future editions. Joachim Heinze and Martin Peters atSpringer initiated the process when they came with the idea of an encyclopedia.Martin Peters’ highly professional supervision of the development and publicationprocess has absolutely been critical. I am also very grateful for the excellent supportfrom Ruth Allewelt and Tina Shelton at Springer.
Austin, USA Bjorn EngquistSeptember 2015
About the Editor
Bjorn Engquist received his Ph.D. in Numerical Analysisfrom Uppsala University in the year 1975. He has beenProfessor of Mathematics at UCLA, Uppsala University,and the Royal Institute of Technology, Stockholm. Hewas Michael Henry Strater University Professor of Math-ematics and Applied and Computational Mathematics atPrinceton University and now holds the Computationaland Applied Mathematics Chair I at the University ofTexas at Austin.
He was Director of the Research Institute for IndustrialApplications of Scientific Computing and of the Centrefor Parallel Computers at the Royal Institute of Technol-
ogy, Stockholm. At Princeton University, he was Director of the Program in Appliedand Computational Mathematics and the Princeton Institute for ComputationalScience, and he is now the Director of the ICES Center for Numerical Analysis inAustin.
Engquist is a member of the American Association for the Advancement ofScience, the Royal Swedish Academy of Sciences, the Royal Swedish Academy ofEngineering Sciences, and the Norwegian Academy of Science and Letters. He was aGuggenheim fellow and received the first SIAM Prize in Scientific Computing 1982,the Celsius Medal 1992, the Henrici Prize 2011, the George David Birkhoff Prizein Applied Mathematics 2012, and the ICIAM Pioneer Prize 2015. He was an ICMspeaker in the years 1982 and 1998.
His research field is development, analysis, and application of numerical methodsfor differential equations. A particular focus has been multiscale problems andapplications to fluid mechanics and wave propagation. He has had 40 Ph.D. students.
ix
Section Editors
Mark AlberDepartment of Applied andComputational Mathematics and StatisticsUniversity of Notre DameNotre Dame, IN, USA
Ernst HairerSection de MathematiquesUniversite de GeneveGeneve, Switzerland
Johan HastadRoyal Insitute of TechnologyStockholm, Sweden
xi
xii Section Editors
Arieh IserlesDepartment of Applied Mathematicsand Theoretical PhysicsCentre for Mathematical SciencesUniversity of CambridgeCambridge, UK
Hans Petter LangtangenSimula Research LaboratoryCenter for Biomedical ComputingFornebu, Norway
Department of InformaticsUniversity of Oslo, Oslo, Norway
Claude Le BrisEcole des Ponts – INRIAParis, France
Christian LubichMathematisches InstitutUniversity of TubingenTubingen, Germany
Section Editors xiii
Andrew J. MajdaDepartment of Mathematics and ClimateAtmosphere, Ocean Science (CAOS)Courant Institute of Mathematical SciencesNew York UniversityNew York, NY, USA
Joyce R. McLaughlinDepartment of Mathematical SciencesRensselaer Polytechnic InstituteTroy, NY, USA
Risto NieminenSchool of ScienceAalto UniversityEspoo, Finland
J. Tinsley OdenInstitute for Computational Engineering and ScienceThe University of Texas at AustinAustin, TX, USA
xiv Section Editors
Aslak TveitoSimula Research LaboratoryCenter for Biomedical ComputingFornebu, Norway
Department of InformaticsUniversity of Oslo, Oslo, Norway
Contributors
Assyr Abdulle Mathematics Section, Ecole Polytechnique Federale de Lausanne(EPFL), Lausanne, Switzerland
Andrew Adamatzky Unconventional Computing Centre, University of the West ofEngland, Bristol, UK
Todd Arbogast Institute for Computational Engineering and Sciences, Universityof Texas, Austin, TX, USA
Douglas N. Arnold School of Mathematics, University of Minnesota, Minneapolis,MN, USA
Simon R. Arridge Department of Computer Science, Center for Medical ImageComputing, University College London, London, UK
Uri Ascher Department of Computer Science, University of British Columbia,Vancouver, BC, Canada
Kendall E. Atkinson Department of Mathematics and Department of ComputerScience, University of Iowa, Iowa City, IA, USA
Paul J. Atzberger Department of Mathematics, University of California SantaBarbara (UCSB), Santa Barbara, CA, USA
Florian Augustin Technische Universitat Munchen, Fakultat Mathematik, Munich,Germany
Winfried Auzinger Institute for Analysis und Scientific Computing, TechnischeUniversitat Wien, Wien, Austria
Owe Axelsson Division of Scientific Computing, Department of InformationTechnology, Uppsala University, Uppsala, Sweden
Institute of Genomics, ASCR, Ostrava, Czech Republic
Ruth E. Baker Centre for Mathematical Biology, Mathematical Institute, Universityof Oxford, Oxford, UK
Guillaume Bal Department of Applied Physics and Applied Mathematics,Columbia University, New York, NY, USA
Vijay Balasubramanian Department of Physics and Astronomy, Department ofNeuroscience, University of Pennsylvania, Philadelphia, PA, USA
xv
xvi Contributors
Roberto Barrio Departamento de Matematica Aplicada and IUMA, University ofZaragoza, Zaragoza, Spain
Timothy Barth NASA Ames Research Center, Moffett Field, CA, USA
Catherine A.A. Beauchemin Department of Physics, Ryerson University, Toronto,ON, Canada
Margaret Beck Department of Mathematics, Heriot-Watt University, Edinburgh,UK
Mikhail I. Belishev PDMI, Saint-Petersburg, Russia
Alfredo Bellen Department of Mathematics and Geosciences, University of Trieste,Trieste, Italy
Ted Belytschko Department of Mechanical Engineering, Northwestern University,Evanston, IL, USA
Rafael D. Benguria Departamento de Fısica, Pontificia Universidad Catolica deChile, Santiago de Chile, Chile
Fredrik Bengzon Department of Mathematics and Mathematical Statistics, UmeaUniversity, Umea, Sweden
Jean–Paul Berrut Departement de Mathematiques, Universite de Fribourg,Fribourg/Perolles, Switzerland
Ake Bjorck Department of Mathematics, Linkoping University, Linkoping, Sweden
Petter E. Bjørstad Department of Informatics, University of Bergen, Bergen,Norway
Sergio Blanes Instituto de Matematica Multidisciplinar, Universitat Politecnica deValencia, Valencia, Spain
Pavel Bochev Computational Mathematics, Sandia National Laboratories,Albuquerque, NM, USA
Liliana Borcea Department of Mathematics, University of Michigan, Ann Arbor,MI, USA
Brett Borden Physics Department, Naval Postgraduate School, Monterey, CA, USA
John P. Boyd Department of Atmospheric, Oceanic and Space Science, Universityof Michigan, Ann Arbor, MI, USA
Michal Branicki School of Mathematics, The University of Edinburgh, Edinburgh,UK
Claude Brezinski Laboratoire Paul Painleve, UMR CNRS 8524, UFR deMathematiques Pures et Appliquees, Universite des Sciences et Technologies deLille, Villeneuve d’Ascq, France
Hermann Brunner Department of Mathematics, Hong Kong Baptist University,Kowloon Tong, Hong Kong SAR, China
Department of Mathematics and Statistics, Memorial University of Newfoundland,St. John’s, NL, Canada
Contributors xvii
Martin Buhmann Mathematisches Institut, Justus-Liebig-Universitat, Giessen,Germany
Martin Burger Institute for Computational and Applied Mathematics, WestfalischeWilhelms-Universitat (WWU) Munster, Munster, Germany
John C. Butcher Department of Mathematics, University of Auckland, Auckland,New Zealand
Michel Caffarel Laboratoire de Chimie et Physique Quantiques, IRSAMC,Universite de Toulouse, Toulouse, France
Russel Caflisch UCLA – Department of Mathematics, Institute for Pure and AppliedMathematics, Los Angeles, CA, USA
Xing Cai Simula Research Laboratory, Center for Biomedical Computing, Fornebu,Norway
University of Oslo, Oslo, Norway
Fioralba Cakoni Department of Mathematics, Rutgers University, New Brunswick,NJ, USA
Daniela Calvetti Department of Mathematics, Applied Mathematics, and Statistics,Case Western Reserve University, Cleveland, OH, USA
Mari Paz Calvo Departamento de Matematica Aplicada, Universidad de Valladolid,Valladolid, Spain
Eric Cances Ecole des Ponts ParisTech – INRIA, Universite Paris Est, CERMICS,Projet MICMAC, Marne-la-Vallee, Paris, France
Fernando Casas Departament de Matematiques and IMAC, Universitat Jaume I,Castellon, Spain
Jeff R. Cash Department of Mathematics, Imperial College, London, England
Carlos Castillo-Chavez Mathematical and Computational Modeling SciencesCenter, School of Human Evolution and Social Change, School of Sustainability,Arizona State University, Tempe, AZ, USA
Santa Fe Institute, Santa Fe, NM, USA
Isabelle Catto CEREMADE UMR 7534, CNRS and Universite Paris-Dauphine,Paris, France
Ondrej Certık Los Alamos National Laboratory, Los Alamos, NM, USA
Raymond Chan Department of Mathematics, The Chinese University of HongKong, Shatin, Hong Kong
Philippe Chartier INRIA-ENS Cachan, Rennes, France
Gui-Qiang G. Chen Mathematical Institute, University of Oxford, Oxford, UK
Jiun-Shyan Chen Department of Structural Engineering, University of California,San Diego, CA, USA
Margaret Cheney Department of Mathematics, Colorado State University, FortCollins, CO, USA
xviii Contributors
Christophe Chipot Laboratoire International Associe CNRS, UMR 7565,Universite de Lorraine, Vandœuvre-les-Nancy, France
Theoretical and Computational Biophysics Group, Beckman Institute for AdvancedScience and Technology, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, USA
Emiliano Cristiani Istituto per le Applicazioni del Calcolo “Mauro Picone”,Consiglio Nazionale delle Ricerche, Rome, RM, Italy
Daan Crommelin Scientific Computing Group, Centrum Wiskunde and Informatica(CWI), Amsterdam, The Netherlands
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam,The Netherlands
Felipe Cucker Department of Mathematics, City University of Hong Kong,Kowloon Tong, Hong Kong
Constantine M. Dafermos Division of Applied Mathematics, Brown University,Providence, RI, USA
Eric Darve Mechanical Engineering Department, Institute for Computational andMathematical Engineering, Stanford University, Stanford, CA, USA
Clint N. Dawson Institute for Computational Engineering and Sciences, Universityof Texas, Austin, TX, USA
Ben De Lacy Costello Unconventional Computing Centre, University of the Westof England, Bristol, UK
Jean-Pierre Dedieu Toulouse, France
Paul Dellar OCIAM, Mathematical Institute, Oxford, UK
Leszek F. Demkowicz Institute for Computational Engineering and Sciences(ICES), The University of Texas at Austin, Austin, TX, USA
Luca Dieci School of Mathematics, Georgia Institute of Technology, Atlanta,GA, USA
Bernard Ducomet Departement de Physique Theorique et Appliquee, CEA/DAMIle De France, Arpajon, France
Iain Duff Scientific Computing Department, STFC – Rutherford AppletonLaboratory, Oxfordshire, UK
CERFACS, Toulouse, France
Nira Dyn School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel
Bo Einarsson Linkoping University, Linkoping, Sweden
Heinz W. Engl Johann Radon Institute for Computational and Applied Mathematics(RICAM), Austrian Academy of Sciences, Linz, Austria
Charles L. Epstein Departments of Mathematics and Radiology, University ofPennsylvania, Philadelphia, PA, USA
Contributors xix
Maria J. Esteban CEREMADE, CNRS and Universite Paris-Dauphine, Paris,France
Adel Faridani Department of Mathematics, Oregon State University, Corvallis,OR, USA
Jean-Luc Fattebert Lawrence Livermore National Laboratory, Livermore,CA, USA
Hans Georg Feichtinger Institute of Mathematics, University of Vienna, Vienna,Austria
Yusheng Feng NSF/CREST Center for Simulation, Visulization and Real-TimePrediction, The University of Texas at San Antonio, San Antonio, TX, USA
David V. Finch Department of Mathematics, Oregon State University, Corvallis,OR, USA
Mathias Fink Institut Langevin, ESPCI ParisTech, Paris, France
Michael S. Floater Department of Mathematics, University of Oslo, Oslo, Norway
Aaron L. Fogelson Departments of Mathematics and Bioengineering, University ofUtah, Salt Lake City, UT, USA
A.S. Fokas DAMTP Centre for Mathematical Sciences, University of Cambridge,Cambridge, UK
Massimo Fornasier Department of Mathematics, Technische Universitat Munchen,Garching bei Munchen, Germany
Bengt Fornberg Department of Applied Mathematics, University of Colorado,Boulder, CO, USA
Piero Colli Franzone Dipartimento di Matematica “F. Casorati”, Universita degliStudi di Pavia, Pavia, Italy
Avner Friedman Department of Mathematics, Ohio State University, Columbus,OH, USA
Dargan M.W. Frierson Department of Atmospheric Sciences, University ofWashington, Seattle, WA, USA
Gero Friesecke TU Munchen, Zentrum Mathematik, Garching, Munich, Germany
Martin J. Gander Section de Mathematiques, Universite de Geneve, Geneva,Switzerland
Carlos J. Garcıa-Cervera Mathematics Department, University of California,Santa Barbara, CA, USA
Edwin P. Gerber Center for Atmosphere Ocean Science, Courant Institute ofMathematical Sciences, New York University, New York, NY, USA
Dimitrios Giannakis Center for Atmosphere Ocean Science (CAOS), CourantInstitute of Mathematical Sciences, New York University, New York, NY, USA
Amparo Gil Departamento de Matematica Aplicada y Ciencias de la Computacion,Universidad de Cantabria, E.T.S. Caminos, Canales y Puertos, Santander, Spain
xx Contributors
Vivette Girault Laboratoire Jacques-Louis Lions, UPMC University of Paris 06 andCNRS, Paris, France
Ingrid Kristine Glad Department of Mathematics, University of Oslo, Oslo,Norway
Dominik Goddeke Applied Mathematics, TU Dortmund, Dortmund, Germany
Serdar Goktepe Department of Civil Engineering, Middle East TechnicalUniversity, Ankara, Turkey
Jerzy Gorecki Institute of Physical Chemistry and Warsaw University, Warsaw,Poland
Nicholas Ian Mark Gould Scientific Computing Department, Rutherford AppletonLaboratory, Oxfordshire, UK
Brian Granger Department of Physics, California Polytechnic State University, SanLuis Obispo, CA, USA
Frank R. Graziani Lawrence Livermore National Laboratory, Livermore, CA, USA
Andrey Gritsun Institute of Numerical Mathematics, Moscow, Russia
Martin Grohe Department of Computer Science, RWTH Aachen University,Aachen, Germany
Nicola Guglielmi Dipartimento di Matematica Pura e Applicata, Universitadell’Aquila, L’Aquila, Italy
Osman Guler Department of Mathematics and Statistics, University of MarylandBaltimore County, Baltimore, MD, USA
Jeremy Gunawardena Department of Systems Biology, Harvard Medical School,Boston, MA, USA
Michael Gunther Fachbereich Mathematik und Naturwissenschaften, BergischeUniversitat Wuppertal, Wuppertal, Germany
Max Gunzburger Department of Scientific Computing, Florida State University,Tallahassee, FL, USA
Bertil Gustafsson Department of Information Technology, Uppsala University,Uppsala, Sweden
Wolfgang Hackbusch Max-Planck-Institut fur Mathematik in den Naturwis-senschaften, Leipzig, Germany
George A. Hagedorn Department of Mathematics, Center for Statistical Mechanics,Mathematical Physics, and Theoretical Chemistry, Virginia Polytechnic Institute andState University, Blacksburg, VA, USA
Ernst Hairer Section de Mathematiques, Universite de Geneve, Geneve,Switzerland
Nicholas Hale Oxford Centre for Collaborative Applied Mathematics (OCCAM),Mathematical Institute, University of Oxford, Oxford, UK
Contributors xxi
Laurence Halpern Laboratoire Analyse, Geometrie and Applications, UMR 7539CNRS, Universite Paris, Villetaneuse, France
John Harlim Department of Mathematics and Department of Meteorology,Pennsylvania State University, State College, PA, USA
Frederic Hecht Laboratoire Jacques-Louis Lions, UPMC University of Paris 06 andCNRS, Paris, France
Dieter W. Heermann Institute for Theoretical Physics, Heidelberg University,Heidelberg, Germany
Gabor T. Herman Department of Computer Science, The Graduate Center of theCity University of New York, New York, NY, USA
Jan S. Hesthaven Division of Applied Mathematics, Brown University, Providence,RI, USA
Nicholas J. Higham School of Mathematics, The University of Manchester,Manchester, UK
Helge Holden Department of Mathematical Sciences, Norwegian University ofScience and Technology, Trondheim, Norway
Jan Homann Department of Physics and Astronomy, University of Pennsylvania,Philadelphia, PA, USA
Kai Hormann Universita della Svizzera italiana, Lugano, Switzerland
Bei Hu Department of Applied and Computational Mathematics and Statistics,University of Notre Dame, Notre Dame, IN, USA
Arne Bang Huseby Department of Mathematics, University of Oslo, Oslo, Norway
Daan Huybrechs Department of Computer Science, K.U. Leuven, Leuven, Belgium
Victor Isakov Department of Mathematics and Statistics, Wichita State University,Wichita, KS, USA
Arieh Iserles Department of Applied Mathematics and Theoretical Physics, Centrefor Mathematical Sciences, University of Cambridge, Cambridge, UK
Kazufumi Ito Center for Research in Scientific Computation and Department ofMathematics, North Carolina State University, Raleigh, NC, USA
Yasushi Ito Aviation Program Group, Japan Aerospace Exploration Agency,Mitaka, Tokyo, Japan
Zdzisław Jackiewicz Department of Mathematics and Statistics, Arizona StateUniversity, Tempe, AZ, USA
Vincent Jacquemet Centre de Recherche, Hopital du Sacre-Coeur de Montreal,Montreal, QC, Canada
Department of Physiology, Universite de Montreal, Institut de Genie Biomedicaland Groupe de Recherche en Sciences et Technologies Biomedicales, Montreal, QC,Canada
xxii Contributors
Laurent O. Jay Department of Mathematics, The University of Iowa, Iowa City, IA,USA
Shi Jin Department of Mathematics and Institute of Natural Science, Shanghai JiaoTong University, Shanghai, China
Department of Mathematics, University of Wisconsin, Madison, WI, USA
Christopher R. Johnson Scientific Computing and Imaging Institute, University ofUtah, Warnock Engineering Building, Salt Lake City, UT, USA
Ansgar Jungel Institut fur Analysis und Scientific Computing, TechnischeUniversitat Wien, Wien, Austria
Rajiv K. Kalia Department of Computer Science, Department of Physics andAstronomy, and Department of Chemical Engineering and Materials Science,University of Southern California, Los Angeles, CA, USA
Erich L. Kaltofen Department of Mathematics, North Carolina State University,Raleigh, NC, USA
George Em Karniadakis Division of Applied Mathematics, Brown University,Providence, RI, USA
Boualem Khouider Department of Mathematics and Statistics, University ofVictoria, Victoria, BC, Canada
Isaac Klapper Department of Mathematical Sciences and Center for BiofilmEngineering, Montana State University, Bozeman, MT, USA
Rupert Klein FB Mathematik and Informatik, Freie Universitat Berlin, Berlin,Germany
Peter Kloeden FB Mathematik, J.W. Goethe-Universitat, Frankfurt am Main,Germany
Matthew G. Knepley Searle Chemistry Laboratory, Computation Institute,University of Chicago, Chicago, IL, USA
Huseyin Kocak Department of Computer Science, University of Miami, CoralGables, FL, USA
Jussi T. Koivumaki The Center for Biomedical Computing, Simula ResearchLaboratory, Lysaker, Norway
The Center for Cardiological Innovation, Oslo University Hospital, Oslo, Norway
Anatoly B. Kolomeisky Department of Chemistry-MS60, Rice University, Houston,TX, USA
Natalia L. Komarova Department of Mathematics, University of California Irvine,Irvine, CA, USA
Nikos Komodakis Ecole des Ponts ParisTech, Universite Paris-Est, Champs-sur-Marne, France
UMR Laboratoire d’informatique Gaspard-Monge, CNRS, Champs-sur-Marne,France
Contributors xxiii
Alper Korkmaz Department of Mathematics, Cankiri Karatekin University,Cankiri, Turkey
Gunilla Kreiss Division of Scientific Computing, Department of InformationTechnology, Uppsala University, Uppsala, Sweden
Peter Kuchment Mathematics Department, Texas A&M University, CollegeStation, TX, USA
M. Pawan Kumar Ecole Centrale Paris, Chatenay-Malabry, France
Equipe GALEN, INRIA Saclay, Ile-de-France, France
Angela Kunoth Institut fur Mathematik, Universitat Paderborn, Paderborn,Germany
Thomas Kurtz University of Wisconsin, Madison, WI, USA
Gitta Kutyniok Institut fur Mathematik, Technische Universitat Berlin, Berlin,Germany
Michael Kwok-Po Ng Department of Mathematics, Hong Kong Baptist University,Kowloon, Hong Kong
Hans Petter Langtangen Simula Research Laboratory, Center for BiomedicalComputing, Fornebu, Norway
Department of Informatics, University of Oslo, Oslo, Norway
Mats G. Larson Department of Mathematics and Mathematical Statistics, UmeaUniversity, Umea, Sweden
Matti Lassas Department of Mathematics and Statistics, University of Helsinki,Helsinki, Finland
Chun-Kong Law Department of Applied Mathematics, National Sun Yat-senUniversity, Kaohsiung, Taiwan
Armin Lechleiter Zentrum fur Technomathematik, University of Bremen, Bremen,Germany
Sunmi Lee School of Human Evolution and Social Change, Arizona StateUniversity, Tempe, AZ, USA
Department of Applied Mathematics, Kyung Hee University, Giheung-gu, Yongin-si,Gyeonggi-do, Korea
Ors Legeza Theoretical Solid State Physics, Hungarian Academy of Sciences,Budapest, Hungary
Benedict Leimkuhler Edinburgh University School of Mathematics, Edinburgh,Scotland, UK
Melvin Leok Department of Mathematics, University of California, San Diego,CA, USA
Randall J. LeVeque Department of Applied Mathematics, University ofWashington, Seattle, WA, USA
Adrian J. Lew Mechanical Engineering, Stanford University, Stanford, CA, USA
xxiv Contributors
Mathieu Lewin CNRS and Departement de Mathematiques, Universite de Cergy-Pontoise/Saint-Martin, Cergy-Pontoise, France
Tien-Yien Li Department of Mathematics, Michigan State University, East Lansing,MI, USA
Zhilin Li Center for Research in Scientific Computation and Department of Mathe-matics, North Carolina State University, Raleigh, NC, USA
Knut-Andreas Lie Department of Applied Mathematics, SINTEF ICT, Oslo,Norway
Guang Lin Fundamental and Computational Sciences Directorate, Pacific North-west National Laboratory, Richland, WA, USA
School of Mechanical Engineering, Purdue University, West Lafayette, IN, USA
Department of Mathematics, Purdue University, West Lafayette, IN, USA
Per Lotstedt Department of Information Technology, Uppsala University, Uppsala,Sweden
John S. Lowengrub Department of Mathematics, University of California, Irvine,CA, USA
Benzhuo Lu Institute of Computational Mathematics and Scientific/EngineeringComputing, Chinese Academy of Sciences, Beijing, China
Christian Lubich Mathematisches Institut, Universitat Tubingen, Tubingen,Germany
Franz Luef Department of Mathematics, University of California, Berkeley, CA,USA
Li-Shi Luo Department of Mathematics and Statistics, Old Dominion University,Norfolk, VA, USA
Beijing Computational Science Research Center, Beijing, China
Mitchell Luskin School of Mathematics, University of Minnesota, Minneapolis,MN, USA
Jianwei Ma Department of Mathematics, Harbin Institute of Technology, Harbin,China
Yvon Maday Sorbonne Universites, UPMC Univ Paris 06, UMR 7598, LaboratoireJacques-Louis Lions, Paris, France
Institut Universitaire de France and Division of Applied Maths, Brown University,Providence, RI, USA
Philip K. Maini Centre for Mathematical Biology, Mathematical Institute,University of Oxford, Oxford, UK
Bradley T. Mallison Chevron Energy Technology Company, San Ramon, CA, USA
Francisco Marcellan Departamento de Matematicas, Universidad Carlos III deMadrid, Leganes, Spain
Contributors xxv
Per-Gunnar Martinsson Department of Applied Mathematics, University ofColorado, Boulder, CO, USA
Francesca Mazzia Dipartimento di Matematica, Universita degli Studi di BariAldo Moro, Bari, Italy
Robert I. McLachlan Institute of Fundamental Sciences, Massey University,Palmerston North, New Zealand
Joyce R. McLaughlin Department of Mathematical Sciences, RensselaerPolytechnic Institute, Troy, NY, USA
Volker Mehrmann Institut fur Mathematik, MA 4-5 TU, Berlin, Germany
Jens Markus Melenk Institute for Analysis and Scientific Computing, ViennaUniversity of Technology, Wien, Austria
Benedetta Mennucci Department of Chemistry, University of Pisa, Pisa, Italy
Roeland Merks Life Sciences (MAC-4), Centrum Wiskunde and Informatica(CWI), Netherlands Consortium for Systems Biology/Netherlands Institute forSystems Biology (NCSB-NISB), Amsterdam, The Netherlands
Aaron Meurer Department of Mathematics, New Mexico State University, LasCruces, NM, USA
Juan C. Meza School of Natural Sciences, University of California, Merced, CA,USA
Owen D. Miller Department of Electrical Engineering and Computer Sciences,University of California, Berkeley, CA, USA
Graeme Milton Department of Mathematics, The University of Utah, Salt LakeCity, UT, USA
J.D. Mireles James Department of Mathematics, Rutgers, The State University ofNew Jersey, Piscataway, NJ, USA
Konstantin Mischaikow Department of Mathematics, Rutgers, The StateUniversity of New Jersey, Piscataway, NJ, USA
Nicolas Moes Ecole Centrale de Nantes, GeM Institute, UMR CNRS 6183, Nantes,France
Mohammad Motamed Division of Mathematics and Computational Sciences andEngineering (MCSE), King Abdullah University of Science and Technology, Thuwal,Saudi Arabia
Hans Z. Munthe-Kaas Department of Mathematics, University of Bergen, Bergen,Norway
Ander Murua Konputazio Zientziak eta A.A. Saila, Informatika Fakultatea,UPV/EHU, Donostia/San Sebastian, Spain
Aiichiro Nakano Department of Computer Science, Department of Physics andAstronomy, and Department of Chemical Engineering and Materials Science,University of Southern California, Los Angeles, CA, USA
xxvi Contributors
Frank Natterer Department of Mathematics and Computer Science, Institute ofComputational Mathematics and Instrumental, University of Munster, Munster,Germany
Philip C. Nelson Department of Physics and Astronomy, University of Pennsylvania,Philadelphia, PA, USA
Qing Nie Department of Mathematics, University of California, Irvine, CA, USA
Harald Niederreiter RICAM, Austrian Academy of Sciences, Linz, Austria
H. Frederik Nijhout Duke University, Durham, NC, USA
Fabio Nobile EPFL Lausanne, Lausanne, Switzerland
Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Milan, Italy
Sarah L. Noble School of Electrical and Computer Engineering, Purdue University,West Lafayette, IN, USA
Clifford J. Nolan Department of Mathematics and Statistics, University ofLimerick, Limerick, Ireland
Ken-ichi Nomura Department of Computer Science, Department of Physics andAstronomy, and Department of Chemical Engineering and Materials Science,University of Southern California, Los Angeles, CA, USA
Jan Martin Nordbotten Department of Mathematics, University of Bergen,Bergen, Norway
W.L. Oberkampf Georgetown, TX, USA
J. Tinsley Oden Institute for Computational Engineering and Sciences, The Univer-sity of Texas at Austin, Austin, TX, USA
Roger Ohayon Structural Mechanics and Coupled Systems Laboratory, LMSSC,Conservatoire National des Arts et Metiers (CNAM), Paris, France
Luke Olson Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, USA
Sheehan Olver School of Mathematics and Statistics, The University of Sydney,Sydney, NSW, Australia
Robert O’Malley Department of Applied Mathematics, University of Washington,Seattle, WA, USA
Ahmet Omurtag Bio-Signal Group Inc., Brooklyn, NY, USA
Department of Physiology and Pharmacology, State University of New York,Downstate Medical Center, Brooklyn, NY, USA
Christoph Ortner Mathematics Institute, University of Warwick, Coventry, UK
Alexander Ostermann Institut fur Mathematik, Universitat Innsbruck, Innsbruck,Austria
Jose-Angel Oteo Departament de Fısica Teorica, Universitat de Valencia, Valencia,Spain
Contributors xxvii
Hans G. Othmer Department of Mathematics, University of Minnesota,Minneapolis, MN, USA
Gianluca Panati Dipartimento di Matematica, Universit di Roma “La Sapienza”,Rome, Italy
Alexander Panfilov Department of Physics and Astronomy, Gent University, Gent,Belgium
Mateusz Paprocki refptr.pl, Wroclaw, Poland
Nikos Paragios Ecole des Ponts ParisTech, Universite Paris-Est, Champs-sur-Marne, France
Ecole Centrale Paris, Chatenay-Malabry, France
Equipe GALEN, INRIA Saclay, Ile-de-France, France
Lorenzo Pareschi Department of Mathematics, University of Ferrara, Ferrara, Italy
John E. Pask Lawrence Livermore National Laboratory, Livermore, CA, USA
Geir K. Pedersen Department of Mathematics, University of Oslo, Oslo, Norway
Michele Piana Dipartimento di Matematica, Universita di Genova, CNR – SPIN,Genova, Italy
Olivier Pinaud Department of Mathematics, Colorado State University, FortCollins, CO, USA
Gernot Plank Institute of Biophysics, Medical University of Graz, Graz, Austria
Oxford e-Research Centre, University of Oxford, Oxford, UK
Gerlind Plonka Institute for Numerical and Applied Mathematics, University ofGottingen, Gottingen, Germany
Aleksander S. Popel Systems Biology Laboratory, Department of BiomedicalEngineering, School of Medicine, The Johns Hopkins University, Baltimore, MD,USA
Jason S. Prentice Department of Physics and Astronomy, University ofPennsylvania, Philadelphia, PA, USA
Luigi Preziosi Department of Mathematics, Politecnico di Torino, Torino, Italy
Andrea Prosperetti Department of Mechanical Engineering, Johns HopkinsUniversity, Baltimore, MD, USA
Department of Applied Sciences, University of Twente, Enschede, The Netherlands
Serge Prudhomme Department of Mathematics and Industrial Engineering, EcolePolytechnique de Montreal, Montreal, QC, Canada
Amina A. Qutub Department of Bioengineering, Rice University, Houston,TX, USA
Venkat Raman Aerospace Engineering, University of Michigan, Ann Arbor,MI, USA
xxviii Contributors
Ronny Ramlau Institute for Industrial Mathematics, Kepler University Linz,Linz, Austria
Rakesh Ranjan NSF/CREST Center for Simulation, Visulization and Real-TimePrediction, The University of Texas at San Antonio, San Antonio, TX, USA
Thilina Rathnayake Department of Computer Science, University of Moratuwa,Moratuwa, Sri Lanka
Holger Rauhut Lehrstuhl C fur Mathematik (Analysis), RWTH Aachen University,Aachen, Germany
Stephane Redon Laboratoire Jean Kuntzmann, NANO-D – INRIA Grenoble –Rhone-Alpes, Saint Ismier, France
Michael C. Reed Department of Mathematics, Duke University, Durham, NC, USA
Peter Rentrop Technische Universitat Munchen, Fakultat Mathematik, Munich,Germany
Nils Henrik Risebro Department of Mathematics, University of Oslo, Oslo, Norway
Philip L. Roe Department of Aerospace Engineering, University of Michigan, AnnArbor, MI, USA
Thorsten Rohwedder Institut fur Mathematik, Technische Universitat Berlin,Berlin, Germany
Dana Ron School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel
Einar M. Rønquist Department of Mathematical Sciences, Norwegian Universityof Science and Technology, Trondheim, Norway
Jose Ros Departament de Fısica Teorica and IFIC, Universitat de Valencia-CSIC,Valencia, Spain
Christopher J. Roy Aerospace and Ocean Engineering Department, Virginia Tech,Blacksburg, VA, USA
Ulrich Rude Department of Computer Science, University Erlangen-Nuremberg,Erlangen, Germany
Siegfried M. Rump Institute for Reliable Computing, Hamburg University ofTechnology, Hamburg, Germany
Faculty of Science and Engineering, Waseda University, Tokyo, Japan
Ann E. Rundell School of Electrical and Computer Engineering, Purdue University,West Lafayette, IN, USA
Weldon School of Biomedical Engineering, Purdue University, West Lafayette, IN,USA
Robert D. Russell Department of Mathematics, Simon Fraser University, Burnaby,BC, Canada
Lenya Ryzhik Department of Mathematics, Stanford University, Stanford, CA,USA
Contributors xxix
Yousef Saad Department of Computer Science and Engineering, University ofMinnesota, Minneapolis, MN, USA
Paul E. Sacks Department of Mathematics, Iowa State University, Ames, IA, USA
Mikko Salo Department of Mathematics and Statistics, University of Jyvaskyla,Jyvaskyla, Finland
Bjorn Sandstede Division of Applied Mathematics, Brown University, Providence,RI, USA
J.M. Sanz-Serna Departamento de Matematica Aplicada, Universidad deValladolid, Valladolid, Spain
Murat Sari Department of Mathematics, Pamukkale University, Denizli, Turkey
Trond Saue Laboratoire de Chimie et Physique Quantiques, CNRS/UniversiteToulouse III, Toulouse, France
Robert Schaback Institut fur Numerische und Angewandte Mathematik (NAM),Georg-August-Universitat Gottingen, Gottingen, Germany
Otmar Scherzer Computational Science Center, University of Vienna, Vienna,Austria
Johann Radon Institute for Computational and Applied Mathematics (RICAM),Austrian Academy of Sciences, Linz, Austria
Tamar Schlick Department of Chemistry, New York University, New York, NY,USA
Reinhold Schneider Institut fur Mathematik, Technische Universitat Berlin, Berlin,Germany
John C. Schotland Department of Mathematics and Department of Physics,University of Michigan, Ann Arbor, MI, USA
Christoph Schwab Seminar for Applied Mathematics (SAM), ETH Zurich, ETHZentrum, Zurich, Switzerland
Javier Segura Departamento de Matematicas, Estadıstica y Computacion,Universidad de Cantabria, Santander, Spain
Eric Sere CEREMADE, Universite Paris-Dauphine, Paris, France
James A. Sethian Department of Mathematics, University of California, Berkeley,CA, USA
Mathematics Department, Lawrence Berkeley National Laboratory, Berkeley, CA,USA
Rudiger Seydel Mathematisches Institut, Universitat zu Koln, Koln, Germany
Lawrence F. Shampine Department of Mathematics, Southern MethodistUniversity, Dallas, TX, USA
Qin Sheng Department of Mathematics, Baylor University, Waco, TX,USA
xxx Contributors
Chi-Wang Shu Division of Applied Mathematics, Brown University, Providence,RI, USA
Avram Sidi Computer Science Department, Technion – Israel Institute ofTechnology, Haifa, Israel
David J. Silvester School of Mathematics, University of Manchester, Manchester,UK
Bernd Simeon Department of Mathematics, Felix-Klein-Zentrum, TU Kaiserslautern,Kaiserslautern, Germany
Kristina D. Simmons Department of Neuroscience, University of Pennsylvania,Philadelphia, PA, USA
Mourad Sini Johann Radon Institute for Computational and Applied Mathematics(RICAM), Austrian Academy of Sciences, Linz, Austria
Ralph Sinkus CRB3, Centre de Recherches Biomedicales Bichat-Beaujon, HopitalBeaujon, Clichy, France
Ian H. Sloan School of Mathematics and Statistics, University of New South Wales,Sydney, NSW, Australia
Barry Smith Mathematics and Computer Science Division, Argonne NationalLaboratory, Argonne, IL, USA
Moshe Sniedovich Department of Mathematics and Statistics, The University ofMelbourne, Melbourne, VIC, Australia
Gustaf Soderlind Centre for Mathematical Sciences, Numerical Analysis, LundUniversity, Lund, Sweden
Christian Soize Laboratoire Modelisation et Simulation Multi-Echelle, MSMEUMR 8208 CNRS, Universite Paris-Est, Marne-la-Vallee, France
Jan Philip Solovej Department of Mathematics, University of Copenhagen,Copenhagen, Denmark
Erkki Somersalo Department of Mathematics, Applied Mathematics and Statistics,Case Western Reserve University, Cleveland, OH, USA
Thomas Sonar Computational Mathematics, TU Braunschweig, Braunschweig,Germany
Euan A. Spence Department of Mathematical Sciences, University of Bath, Bath,UK
Samuel N. Stechmann Department of Mathematics, University of Wisconsin–Madison, Madison, WI, USA
Plamen Stefanov Department of Mathematics, Purdue University, West Lafayette,IN, USA
Gabriel Stoltz Universite Paris Est, CERMICS, Projet MICMAC Ecole des Ponts,ParisTech – INRIA, Marne-la-Vallee, France
Contributors xxxi
Arne Storjohann David R. Cheriton School of Computer Science, University ofWaterloo, Waterloo, ON, Canada
Endre Suli Mathematical Institute, University of Oxford, Oxford, UK
Joakim Sundnes Simula Research Laboratory, Lysaker, Norway
Denis Talay INRIA Sophia Antipolis, Valbonne, France
Martin A. Tanner Department of Statistics, Northwestern University, Evanston, IL,USA
Vladimir Temlyakov Department of Mathematics, University of South Carolina,Columbia, SC, USA
Steklov Institute of Mathematics, Moscow, Russia
Nico M. Temme Centrum voor Wiskunde and Informatica (CWI), Amsterdam, TheNetherlands
Raul Tempone Division of Mathematics and Computational Sciences and Engineer-ing (MCSE), King Abdullah University of Science and Technology, Thuwal, SaudiArabia
Luis Tenorio Mathematical and Computer Sciences, Colorado School of Mines,Golden, CO, USA
Kukatharmini Tharmaratnam Department of Mathematics, University of Oslo,Oslo, Norway
Florian Theil Mathematics Institute, University of Warwick, Coventry, UK
Francoise Tisseur School of Mathematics, The University of Manchester,Manchester, UK
Gasper Tkacik Department of Physics and Astronomy, University of Pennsylvania,Philadelphia, PA, USA
Øystein Trasdahl Department of Mathematical Sciences, Norwegian University ofScience and Technology, Trondheim, Norway
M.V. Tretyakov School of Mathematical Sciences, University of Nottingham,Nottingham, UK
Yen-Hsi Tsai Department of Mathematics, Center for Numerical Analysis, Institutefor Computational Engineering and Science, University of Texas, Austin, TX, USA
Xuemin Tu Department of Mathematics, University of Kansas, Lawrence, KS, USA
Stefan Turek Applied Mathematics, TU Dortmund, Dortmund, Germany
Gabriel Turinici Departement MIDO, CEREMADE, Universite Paris-Dauphine,Paris, France
Aslak Tveito Simula Research Laboratory, Center for Biomedical Computing,Fornebu, Norway
Department of Informatics, University of Oslo, Oslo, Norway
xxxii Contributors
Gunther Uhlmann Department of Mathematics, University of Washington, Seattle,WA, USA
Erik S. Van Vleck Department of Mathematics, University of Kansas, Lawrence,KS, USA
Robert J. Vanderbei Department of Operations Research and FinancialEngineering, Princeton University, Princeton, NJ, USA
Priya Vashishta Department of Computer Science, Department of Physics andAstronomy, and Department of Chemical Engineering and Materials Science, Uni-versity of Southern California, Los Angeles, CA, USA
Gilles Vilmart Departement de Mathematiques, Ecole Normale Superieure deCachan, antenne de Bretagne, INRIA Rennes, IRMAR, CNRS, UEB, Bruz, France
Gerhard Wanner Section de Mathematiques, Universite de Geneve, Geneve,Switzerland
Andy Wathen Mathematical Institute, Oxford University, Oxford, UK
Christian Wieners Karlsruhe Institute of Technology, Institute for Applied andNumerical Mathematics, Karlsruhe, Germany
Ragnar Winther Center of Mathematics for Applications, University of Oslo, Oslo,Norway
Henryk Wozniakowski Department of Computer Science, Columbia University,New York, NY, USA
Institute of Applied Mathematics, University of Warsaw, Warsaw, Poland
Luiz Carlos Wrobel School of Engineering and Design, Brunel University London,Uxbridge, Middlesex, UK
M. Wu Department of Mathematics, University of California, Irvine, CA, USA
Christos Xenophontos Department of Mathematics and Statistics, University ofCyprus, Nicosia, Cyprus
Dongbin Xiu Department of Mathematics and Scientific Computing and ImagingInstitute, University of Utah, Salt Lake City, UT, USA
Eli Yablonovitch Department of Electrical Engineering and Computer Sciences,University of California, Berkeley, CA, USA
Chao Yang Computational Research Division, MS-50F, Lawrence BerkeleyNational Laboratory, Berkeley, CA, USA
Robert Young Department of Mathematics, University of Toronto, Toronto, ON,Canada
Harry Yserentant Institut fur Mathematik, Technische Universitat Berlin, Berlin,Germany
Contributors xxxiii
Ya-xiang Yuan State Key Laboratory of Scientific/Engineering Computing, Instituteof Computational Mathematics and Scientific/Engineering Computing, Academy ofMathematics and Systems Science, Chinese Academy of Sciences, Beijing, P.R.China
Yong-Tao Zhang Department of Applied and Computational Mathematics andStatistics, University of Notre Dame, Notre Dame, IN, USA
Ding-Xuan Zhou Department of Mathematics, City University of Hong Kong,Hong Kong, China
Ting Zhou Department of Mathematics, Northeastern University, Boston, MA, USA
Tarek I. Zohdi Department of Mechanical Engineering, University of California,Berkeley, CA, USA
Enrique Zuazua BCAM – Basque Center for Applied Mathematics, Bilbao, BasqueCountry, Spain
Ikerbasque – Basque Foundation for Science, Bilbao, Basque Country, Spain