Encyclopedia of Applied and ComputationalMathematics

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