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  • 301.
    Bergsten, Christer
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Tvillingcirklar2009In: Normat - Nordisk matematisk tidskrift, ISSN 0801-3500, Vol. 57, no 1, p. 22-31Article in journal (Refereed)
    Abstract [en]

    This ties with previous articles on the arbelos of Archimedes, but instead of starting with two tangent circles, enclosed in a common tangent circle, no conditions are imposed on the two. We then get two cases, whether the circles intersect or not, and a detailed study is made on the nesuing circles. Much of the motivation for this paper is how the problems can be used to illustrate the power of various dynamic geometry software available on the market since the 80's.

  • 302.
    Bergsten, Christer
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Vad är en parabel?2006In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, Vol. 33, no 1, p. 45-48Article in journal (Other (popular science, discussion, etc.))
    Abstract [sv]

    Med parabeln som exempel visas hur den historiskt-kulturella utvecklingen av idéer, uttrycksformer, metoder och verktyg kan berika och utveckla ett matematiskt begrepp och hur detta kan uppfattas. Vilka didaktiska problem och möjligheter ger detta i dagens skola?

  • 303.
    Bergsten, Christer
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Why do students go to lectures?2011In: Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education / [ed] Marta Pytlak, Tim Rowland, Ewa Svoboda, Rzeszów, Poland: University of Rzeszów, Poland , 2011, p. 1960-1970Conference paper (Refereed)
    Abstract [en]

    This paper investigates, from a student perspective, the fact that, despite arguments put forward against the educational value of large group lectures, this teaching format prevails and attracts students in undergraduate mathematics education.

  • 304.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Educational Sciences.
    Botten, Geir
    Fuglestad, Anne-Berit
    Grevholm, Barbro
    Holden, Ingvill
    Lingefjärd, Thomas
    Background document: The education and competence development of mathematics teachers in Norway and Sweden2004In: Educating for the future: Proceedings of an International Symposium on Mathematics Teacher Education / [ed] Strässer, R., Brandell, G., Grevholm, B, Helenius, O., Göteborg: The Royal Swedish Academy of Sciences , 2004, p. 9-38Conference paper (Refereed)
    Abstract [en]

    In this document we present the study material, which was the starting point for the symposium. It consists of experienced teacher educators' and researchers' collected knowledge from classroom and teacher education experience as well as research from the authors and colleagues. The text describes the educational system and the teacher education in Sweden and Norway and recent reforms. It concludes with the identification of some critical issues, which have been, still are or have recently appeared within the educational and competency development of mathematics teachers in Norway and Sweden.

  • 305.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Educational Sciences.
    Dahland, GöteGöteborg University, Sweden.Grevholm, BarbroUniversity of Agder, Norway.
    Research and action in the mathematics classroom: proceedings of MADIF 2, the 2nd Swedish Mathematics Education Research Seminar, Göteborg, January 26-27, 20002002Collection (editor) (Refereed)
  • 306.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Educational Sciences.
    Engelbrecht, Johann
    University of Pretoria, South Africa.
    Kågesten, Owe
    Linköping University, Department of Science and Technology. Linköping University, Faculty of Science & Engineering.
    Conceptual and procedural approaches to mathematics in the engineering curriculum: Comparing views of junior and senior engineering students in two countries2017In: Eurasia Journal of Mathematics, Science and Technology Education, ISSN 1305-8223, ISSN 1305-8223, Vol. 33, no 3, p. 533-553Article in journal (Refereed)
    Abstract [en]

    One challenge for an optimal design of the mathematical components in engineering education curricula is to understand how the procedural and conceptual dimensions of mathematical work can be matched with different demands and contexts from the education and practice of engineers. The focus in this paper is on how engineering students respond to the conceptual-procedural distinction, comparing performance and confidence between second and fourth year groups of students in their answers to a questionnaire comprising conceptually and procedurally focused mathematics problems. We also compare these students’ conceptions on the role of conceptual and procedural mathematics problems within and outside their mathematics studies. Our data suggest that when mathematical knowledge is being recontextualised to engineering subjects or engineering design, a conceptual approach to mathematics is more essential than a procedural approach; working within the mathematical domain, however, the procedural aspects of mathematics are as essential as the conceptual aspects.

  • 307.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Educational Sciences.
    Engelbrecht, Johann
    University of Pretoria, South Africa.
    Kågesten, Owe
    Linköping University, Department of Science and Technology, Communications and Transport Systems. Linköping University, Faculty of Science & Engineering.
    Conceptual or procedural mathematics for engineering students – views of two qualified engineers from two countries2015In: International journal of mathematical education in science and technology, ISSN 0020-739X, E-ISSN 1464-5211, Vol. 46, no 7, p. 979-990Article in journal (Refereed)
    Abstract [en]

    This study forms part of a collaboration project between universities in South Africa and Sweden in which we investigate whether the emphasis in undergraduate mathematics courses for engineering students would benefit from being more conceptually oriented than a traditional more procedurally oriented way of teaching. In this paper, we report in some detail from two interviews with professional engineers, selected to represent two different ‘poles’ of engineering work. The aim was to explore different kinds of arguments regarding the role of mathematics in engineering work, as well as some common across contexts. Both interviewees feel that conceptual mathematics is more important for engineering work, although the role of the procedural aspect was seen by one of the interviewees also to be important, but in a very intricate way.

  • 308.
    Bergsten, Christer
    et al.
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Applied Mathematics.
    Fransson, Torbjörn
    Växjö universitet.
    Students interacting with an artefact designed to visualise three-dimensional analytic geometry2006In: CERME - Conference of the European Society for Research in Mathematics Education,2005, Barcelona: Universitat Ramon Llull , 2006, p. 71-Conference paper (Refereed)
    Abstract [en]

     To investigate students- ways of working with concrete materials in mathematics, a three-dimensional static artefact was constructed and made available to upper secondary students, with pre-knowledge only in two-dimensional coordinate geo¬metry, for solving problems about planes and straight lines in space. Artefact interactivity was generally high, even students also disregarded the model to work only numerically with the coordinates, building on know¬ledge about lines in two dimensions. The model was used when trying to convince other students in the group. 

  • 309.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Frejd, Peter
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Educational Sciences.
    Preparing pre-service mathematics teachers for STEM education: an analysis of lesson proposals2019In: ZDM - the International Journal on Mathematics Education, ISSN 1863-9690, E-ISSN 1863-9704, Vol. 51, no 6, p. 941-953Article in journal (Refereed)
    Abstract [en]

    Based on an investigation of research literature on the role of mathematics within the context of STEM education and twenty-first century skills, this explorative study presents and analyses 19 pre-service teachers’ lesson proposals for innovative STEM activities in secondary mathematics classrooms. Drawing on a categorisation of twenty-first century skills and key concepts from Realistic Mathematics Education and Basil Bernstein’s writings, the analysis indicated a use of detailed instructions regarding what students should do but unspecified criteria for their expected knowledge productions. Rather than serving as a digital tool for problem solving, programming was used for the purpose of generalising students’ conceptual knowledge in mathematics. At the same time a focus on modelling and applications in the STEM activities tended to weaken the disciplinary character of mathematics. The learning of various twenty-first century skills was promoted to different extents. The study raises a discussion of affordances and constraints regarding students’ access to mathematical knowledge through different modes of integration of mathematics and other STEM subjects aiming to support the development of twenty-first century skills.

  • 310.
    Bergsten, Christer
    et al.
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Applied Mathematics.
    Grevholm, Barbro
    Challenges in mathematics education: Proceedings of MADIF 3 : the 3rd Swedish Mathematics Education Research Seminar, Norrköping, January 23-25, 20022003Conference proceedings (editor) (Refereed)
  • 311.
    Bergsten, Christer
    et al.
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Applied Mathematics.
    Grevholm, BarbroHøgskolen i Agder.
    Conceptions of mathematics: proceedings of norma 01 : the 3rd Nordic Conference on Mathematics Education, Kristianstad, June 8-12, 20012005Collection (editor) (Other academic)
    Abstract [en]

     Proceedings of Norma01, Third Nordic Conference on Mathematics Education, Kristianstad 2001

  • 312.
    Bergsten, Christer
    et al.
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Applied Mathematics.
    Grevholm, BarbroUniversitetet i Agder.
    Developing and researching quality in mathematics teaching and learning2007Collection (editor) (Refereed)
  • 313.
    Bergsten, Christer
    et al.
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Applied Mathematics.
    Grevholm, Barbro
    Knowledgeable teacher educators and linking practices2008In: The International Handbook of mathematics Teacher Education, Volume 4: The Mathematics Teacher Educator as a Developing Professional / [ed] Barbara Jaworski and Terry Woods, Rotterdam: Sense Publishers , 2008, 1, p. 221-246Chapter in book (Other academic)
    Abstract [en]

    The Handbook of Mathematics Teacher Education, the first of its kind, addresses the learning of mathematics teachers at all levels of schooling to teach mathematics, and the provision of activity and programmes in which this learning can take place. It consists of four volumes. Volume 4 of this handbook has the title The Mathematics Teacher Educator as a Developing Professional. The volume seeks to complement the other three volumes by focusing on knowledge and roles of teacher educators working with teachers in teacher education processes and practices. In this respect it is unique. Chapter authors represent a community of teacher educators world wide who can speak from practical, professional and theoretical viewpoints about what it means to promote teacher education practice. The volume is in 3 main sections. In the first we focus on Challenges to and Theory in Mathematics Teacher Education. Here authors write from perspectives of theory and/or challenge and relate this to examples and insights from their practice. The second section, Reflection On Developing as a Mathematics Teacher Educator has four autobiographical chapters in which authors delineate their experiences as teacher educators and relate these to theoretical and/or moral standpoints. In Section 3, Working With Prospective and Practising Teachers: What We Learn; What We Come to Know, authors write from perspectives on practice - in many cases, the practices in which they themselves have engaged - and relate this to theoretical perspectives and rationales for teacher education programmes. The volume also has an introductory chapter in which the purpose and content of the volume is set out, and a final chapter that syntheses themes and issues from the chapters as a whole, offering an overview of the field and suggesting future directions. Bibliographical Information for the complete set: VOLUME 1: Knowledge and Beliefs in Mathematics Teaching and Teaching Development Peter Sullivan, Monash University, Clayton, Australia and Terry Wood, Purdue University, West Lafayette, USA (eds.) paperback: 978-90-8790-541-5, hardback: 978-90-8790-542-2, ebook: 978-90-8790-543-9 VOLUME 2: Tools and Processes in Mathematics Teacher Education Dina Tirosh, Tel Aviv University, Israel and Terry Wood, Purdue University, West Lafayette, USA (eds.) paperback: 978-90-8790-544-6, hardback: 978-90-8790-545-3, ebook: 978-90-8790-546-0 VOLUME 3: Participants in Mathematics Teacher Education: Individuals, Teams, Communities and Networks Konrad Krainer, University of Klagenfurt, Austria and Terry Wood, Purdue University, West Lafayette, USA (eds.) paperback: 978-90-8790-547-7, hardback: 978-90-8790-548-4, ebook: 978-90-8790-549-1 VOLUME 4: The Mathematics Teacher Educator as a Developing Professional Barbara Jaworski, Loughborough University, UK and Terry Wood, Purdue University, West Lafayette, USA (eds.) paperback: 978-90-8790-550-7, hardback: 978-90-8790-551-4, ebook: 978-90-8790-552-1

  • 314.
    Bergsten, Christer
    et al.
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Applied Mathematics.
    Grevholm, BarbroHøgskolen i Agder.
    Mathematics and language: proceedings of MADIF 4 : the 4th Swedish Mathematics Education Research Seminar, Malmö, January 21-22, 20042004Collection (editor) (Other academic)
    Abstract [en]

     Proceedings of the 4th Swedish Mathematics Education Research Seminar, Malmö 2004 

  • 315.
    Bergsten, Christer
    et al.
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Applied Mathematics.
    Grevholm, Barbro
    Realfag Høgskolen i Agder.
    The didactic divide and the education of teachers of mathematics in Sweden2004In: Nordisk matematikkdidaktikk, ISSN 1104-2176, Vol. 9, no 2, p. 123-144Article in journal (Refereed)
    Abstract [en]

    On a background of tradition versus renewal, a discussion on recent teacher education reforms in Sweden is given. A balance of different aspects of content and a need for research are seen as critical for the formation of mathematics teacher education. A conception of a -didactic divide- between disciplinary and pedagogical knowledge is used as an analytic tool to describe the rationale behind the design of the reforms. Empirical data from student teachers enrolled in the new teacher education programme highlight how the structure and content of the programme contribute to overcoming the divide.

  • 316.
    Bergsten, Christer
    et al.
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Applied Mathematics.
    Grevholm, Barbro
    Favilli, Franco
    Learning to teach mathematics: expanding the role of practicum as an integrated part of a teacher education programme2009In: The professional education and development of teachers of mathematics. The 15th ICMI Study, New York: Springer , 2009, 1, p. 57-70Chapter in book (Other academic)
    Abstract [en]

    The premise of the 15th ICMI Study is that teachers are key to students' opportunities to learn mathematics. What teachers of mathematics know, care about, and do is a product of their experiences and socialization, together with the impact of their professional education. The Professional Education and Development of Teachers of Mathematics assembles important new international work- development, research, theory and practice - concerning the professional education of teachers of mathematics. As it examines critical areas to reveal what is known and what significant questions and problems warrant collective attention, the volume also contributes to the strengthening of the international community of mathematics educators. The Professional Education and Development of Teachers of Mathematics is of interest to the mathematics education community as well as to other researchers, practitioners and policy makers concerned with the professional education of teachers.

  • 317.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Grevholm, BarbroUniversitetet i Agder.Lingefjärd, ThomasGöteborgs universitet.
    Perspectives on mathematical knowledge2009Collection (editor) (Other academic)
  • 318.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Grevholm, BarbroUniversitetet i Agder.Strömskag Måsöval, HeidiSör-Tröndelag University College.Rönning, FrodeSör-Tröndelag University College.
    Relating practice and reserch in mathematics education: Proceedings of the Fourth Nordic Conference on Mathematics Education2007Collection (editor) (Other academic)
  • 319.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Häggström, Johan
    Lindberg, Lisbeth
    Algebra för alla1997 (ed. 1)Book (Other academic)
    Abstract [sv]

    Bakom titeln Algebra för alla ligger tanken att alla lärare som arbetar med matematik i grundskola och gymnasieskola ska ha glädje av innehållet och målet att alla elever ska möta matematikens generaliserande kraft. Syftet med materialet är att avdramatisera arbetet med skolalgebra samt att uppmärksamma och stimulera möjligheterna att arbeta med olika uttrycksformer. I boken ingår ett stort antal elevaktiviteter och studieuppgifter för lärare samt litteraturförslag.

  • 320.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology. Linköping University, Faculty of Educational Sciences.
    Jablonka, Eva
    Luleå tekniska universitet.
    Commentary on Theories of mathematics education: Is plurality a problem?2010In: Theories of Mathematics Education: Seeking New Frontiers / [ed] B. Sriraman, B. & L. English, New York: Springer , 2010, p. 111-120Chapter in book (Other academic)
    Abstract [en]

    The chapter by Steve Lerman provides a powerful analysis that helps to make sense of the changing nature of the discourse in mathematics education as a research domain, of its knowledge structure and of the positions of researchers in relation to intellectual traditions ‘outside’ the field as well as in relation to changing pedagogic modes. The reading invited us to expand on the conceptualisation of knowledge structures and discourses in mathematics education as a research domain.

  • 321.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Jablonka, Eva
    Engineerings students' recognition of university mathematics and their conceptuaiization of the role of mathematics for their future career.2014In: Mathematics in undergraduate study programs: Challenges for research and for the dialogue between mathematics and the didactics of mathematics / [ed] R. Biehler, R. Hochmuth, C. Hoyles, P. W. Thompson, Oberwolfach, 2014, Vol. 56, p. 52-53Conference paper (Other academic)
  • 322.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Jablonka, Eva
    Luleå tekniska universitet.
    Interpreting students' reasoning through the lens of two different languages of description: integration or juxtaposition?2010In: Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education. January 28th - February 1st 2009, Lyon (France) / [ed] V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello, Lyon: INSTITUT NATIONAL DE RECHERCHE PÉDAGOGIQUE , 2010, p. 1555-1564Conference paper (Refereed)
    Abstract [en]

    This contribution exemplifies the interpretation of a common set of data by using two languages of description originating from different theoretical perspectives. One ac- count uses categories from a psychological and the other from a sociological per- spective. The interpretations result in different explanations for the students’ strug- gles with sense making. However, the results cannot be integrated into a combined insight, but only be juxtaposed.

  • 323.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Jablonka, Eva
    King's College London, UK.
    Mathematics as “Meta-Technology” and “Mind-Power”: Views of engineering students2013In: Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education, Ankara, Turkey: Middle East Technical University , 2013Conference paper (Refereed)
    Abstract [en]

    The paper reports an exploration of first year undergraduate engineering students’ rationales for choosing engineering programmes at master’s level and the potential gains for their future professions they see from studying mathematics. As a means for organising the data, we used some notions of Bourdieu’s theory of the economy of forms of social practice. The study aims to contribute to understanding differences in the students’ experiences of and interest in university mathematics.

  • 324.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Educational Sciences.
    Jablonka, Eva
    King's College London.
    The construction of the ‘transition problem’ by a group of mathematics lecturers2015In: Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education / [ed] Konrad Krainer, Nad'a Vondrová, Prague, Czech Republic: Charles University in Prague, Faculty of Education and ERME , 2015, p. 2053-2059Conference paper (Refereed)
    Abstract [en]

    This paper presents views of university staff about what has become called the ‘transition problem’ when students start studying mathematics at university. The data are from a focus group interview with eight experienced university lecturers at a Swedish university department that offers mathematics courses for engineering students. We use the portrayal of the problem in the literature as an axis for the discussion.

  • 325.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Jablonka, Eva
    Luleå tekniska universitet, Luelå.
    Ashjari, Hoda
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Recognising knowledge criteria in undergraduate mathematics education2012In: Evaluation and comparison of mathematical achievement: Dimensions and perspectives: Proceedings of MADIF 8, Umeå, January 24-­‐25, 2012 / [ed] Christer Bergsten, Eva Jablonka, Manya Raman, Linköping: Svensk förening för MatematikDidaktisk Forskning - SMDF, 2012, p. 101-110Chapter in book (Refereed)
    Abstract [en]

    As part of a larger study on the transition between upper secondary and tertiarymathematics education, this paper reports findings from an investigation of therelation between the students’ understanding of the criteria for legitimate mathematicalknowledge and their achievement during the first year in their undergraduatemathematics studies. As a methodology, we used interviews in whichengineering students were given excerpts from different, more or less formal,mathematics textbooks and asked whether they could rank these texts as being“more or less mathematical” and explain why. The results of our case studyindicate differences in the students’ views that are related to their achievement.

  • 326.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Educational Sciences.
    Jablonka, Eva
    King's College London.
    Ashjari, Hoda
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Övergången från skolans till högskolans matematik: en integrerad studie av en kulturkrock2015In: Resultatdialog 2015 / [ed] Petter Aasen, Eva Björck, Stockholm: Vetenskapsrådet , 2015, p. 29-41Chapter in book (Other academic)
    Abstract [sv]

    Projektets syfte var att utveckla en integrerad bild av matematiska, didaktiska och sociala dimensioner av matematikutbildning i samband med övergången gymnasium-högskola. Civilingenjörsstudenter intervjuades om sina matematikstudier i jämförelse med gymnasiets. Resultaten visar att de ser skillnader i matematiken på universitetet när det gäller kunskapskriterier, studietakt, undervisning och egna studievanor. Studenter medvetna om dessa skillnader lyckas bäst resultatmässigt. Matematikstudier upplevs annorlunda jämfört med andra ämnen vad gäller den tid och kraft som måste investeras. Intervjuade universitetslärare i matematik menar att övergångsproblemen främst rör studentens eget arbete och naturliga utveckling – dagens studenter måste "kämpa mer på vägen". Bland studenter som nämner eget studieansvar tycks ett fåtal ha haft "övergångsproblem": universitetsstudier beskrivs som en värdefull erfarenhet. De flesta anpassar sitt sätt att studera till de nya kraven men det behövs en ökad tydlighet gentemot de studerande vad dessa exakt innebär.

  • 327.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Jablonka, Eva
    Luleå tekniska universitet.
    Klisinska, Anna
    Luleå tekniska universitet.
    A remark on didactic transposition theory2010In: Mathematics and mathematics education: Cultural and social dimensions: Proceedings of MADIF7, The Seventh Mathematics Education Research Seminar, Stockholm, January 26-27. 2010 / [ed] C. Bergsten, E. Jablonka, T. Wedege, Linköping: Svensk förening för matematikdidaktisk forskning (SMDF) , 2010, p. 58-68Conference paper (Refereed)
    Abstract [en]

    With reference to a historical study on the relation between the production and distribution of mathematical knowledge, using calculus as an example, some assumptions in didactic transposition theory, as introduced by Yves Chevallard, are discussed. Given the prominent status of this theory, the paper intends to initiate a debate that could help lifting it out from its relative isolation within mathematics education as a research domain.

  • 328.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Jablonka, Eva
    Luleå tekniska universitet.
    Klisinska, Anna
    Luleå tekniska universitet.
    Reproduction and distribution of mathematical knowledge in higher education: constructing insiders and outsiders2010In: Mathematics Education and Society: Proceedings of the Sixth International Mathematics Education and Society Conference / [ed] U. Gellert, E. Jablonka, & C. Morgan, Berlin: Freie Universotät Berlin , 2010, p. 130-140Conference paper (Refereed)
    Abstract [en]

    The design and conduct of calculus courses has been and is an object of curriculardebates and reforms. By the reconstruction of the establishment of the mathematicalsub-area now called ‘calculus’ and its fundamental theorem as a piece ofinstitutionalized mathematical knowledge for the purpose of its reproduction, wereconsider the notion of knowledge recontextualization within the field of knowledgeproduction by showing that standardization of knowledge evolves in a dynamicrelationship with its production. Interviews with mathematicians suggest that they, asteachers, create different recontextualization principles. In undergraduate teachingcalculus they suggest to include the criteria of the field of knowledge production (e.g.proof) for the future ‘insiders’, while for those who will not pursue a career withinthis field, the ‘outsiders’, the criteria change towards computational efficiency.

  • 329.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Jablonka, EvaLuleå tekniska universitet.Raman, ManyaUmeå universitet.
    Evaluation and comparison of mathematical achievement2012Collection (editor) (Other academic)
  • 330.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics.
    Jablonka, EvaLuleå tekniska universitet.Wedege, TineMalmö högskola.
    Mathematics and mathematics education: Cultural and social dimensions: Proceedings of MADIF7, The Seventh Mathematics Education Research Seminar, Stockholm, January 26-27, 20102010Collection (editor) (Refereed)
  • 331.
    Bergsten, Christer
    et al.
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Applied Mathematics.
    Lennerstad, Håkan
    Matematiska språk. Sju essäer om symbolspråkets roll i matematiken2008Collection (editor) (Other academic)
    Abstract [sv]

    Språk är ytterst centralt för oss människor i nästan allt vi gör. Hur är förhåller sig matematik till språk? Man kan med fog hävda att matematik är ett ämne med ett stort antal språkliga sidor. I denna bok tar sju personer från skilda områden upp matematikens språkligheter ur sina respektive perspektiv. Resultatet är ett brett och fascinerande landskap, där matematik som rent räknande är en smal och porlande rännil.Författarna som beskriver sin syn på matematikens språkliga kvaliteter är: Christer Kiselman, professor i matematik, Uppsala universitet; Håkan Lennerstad, docent i tillämpad matematik, Blekinge Tekniska högskola; Lars Mouwitz, matematiklärare och doktor i teknologi och yrkeskunnande, Nationellt Centrum för Matematikutbildning, Göteborg; Madeleine Löwing, doktor i matematikdidaktik, Göteborgs universitet; Bo Göranzon, professor i yrkeskunnande och teknologi, Kungliga Tekniska Högskolan, KTH i Stockholm; Östen Dahl, professor i lingvistik, Stockholms universitet; Christer Bergsten, docent i matematikdidaktik, Linköpings universitet

  • 332.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Presmeg, Norma
    Preference for visual methods: An international study1995In: Proceedings of the 19th PME Conference, Recife, Brazil, July 22-27, 1995 / [ed] Meira, L., Carraher, D., Recife, Brazil: Universidade Federal de Pernambuco , 1995, Vol. 3, p. 58-65Conference paper (Refereed)
    Abstract [en]

    There have been assertions in recent literature on visualization that students are reluctant to visualize when doing mathematics, particularly at the high school and college levels. While there is research evidence that in the classes of 'nonvisual' teachers even 'visual' students will suppress their preferred visual cognitive modes in favor of nonvisual methods used by their teachers, our data show that it is simplistic to claim that students are reluctant to visualize. In the present international study, the same instrument for measuring preference for visual methods in solving nonroutine mathematical problems was administered to students in three countries, South Africa, Sweden and the United States. Some results are analyzed here.

  • 333.
    Bergsten, Christer
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Educational Sciences.
    Sriraman, BharathUniversity of Montana.
    Refractions of mathematics education: Festschrift for Eva Jablonka2015Collection (editor) (Other academic)
    Abstract [en]

    The diversity of research in mathematics education has been addressed as both, a problem and a strength. When manifested through adherence to different intellectual roots and theoretical orientations, diversions constitute ‘refractions’ of mathematics education. The collection and analysis of empirical data in a study are by necessity refracted through the specific analytical lens employed, as well as the aim of the study itself. Refractions can also refer to looking at old phenomena through new lenses.The chapters in this book are refracted through philosophical, political, mathematical and personal lenses by distinguished authors in the field, addressing issues about the elusive experience of doing mathematics, purification of texts, refractions, mathematics and ethnomathematics, political messages in textbook tasks, mathematics education policy debate, the political in mathematics education research, philosophy and mathematics, meanings and representations, identity of mathematical modeling, and dilemmas in the teaching of calculus.An ancient Sanskrit adage states that Knowledge is something that grows when shared, but shrinks when hoarded. Academics engaged in the generation of new Knowledge are blessed with both the time and the freedom to engage in pursuits that allow for intellectual pleasure. As a phenomenon of the Zeitgeist many have succumbed to the increased corporatization of academic work, engaging in activities for monetary and self advancement purposes. Are there any real intellectuals left in academia, a là Adorno, Bourdieu, Chomsky, Foucault, among others? This Festschrift is dedicated to academics that don't bother with self promotion or aggrandizement of themselves or their ideas in simplistic terms.

  • 334.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    A spectral method for solving the sideways heat equation1999In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 15, no 4, p. 891-906Article in journal (Refereed)
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is the model of a problem where one wants to determine the temperature on the surface of a body, using interior measurements. Mathematically it can be formulated as a Cauchy problem for the heat equation, where the data are given along the line x = 1, and a solution is sought in the interval 0 ≤ x < 1.

    The problem is ill-posed, in the sense that the solution does not depend continuously on the data. Continuous dependence of the data is restored by replacing the time derivative in the heat equation with a bounded spectral-based approximation. The cut-off level in the spectral approximation acts as a regularization parameter. Error estimates for the regularized solution are derived and a procedure for selecting an appropriate regularization parameter is given. The discretized problem is an initial value problem for an ordinary differential equation in the space variable, which can be solved using standard numerical methods, for example a Runge-Kutta method. As test problems we take equations with constant and variable coefficients.

  • 335.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    A survey of methods for determinig surface temperatures using interior measurements2001In: Trends in Heat, Mass & Momentum Transfer, ISSN 0973-2446, Vol. 7, no pp, p. 105-128Article in journal (Refereed)
  • 336.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics, Computationel Mathematics. Linköping University, The Institute of Technology.
    An Inverse Heat Conduction Problem and Improving Shielded Thermocouple Accuracy2012In: Numerical Heat Transfer, Part A Applications, ISSN 1040-7782, E-ISSN 1521-0634, Vol. 61, no 10, p. 754-763Article in journal (Refereed)
    Abstract [en]

    A shielded thermocouple is a measurement device used for monitoring the temperature in chemically, or mechanically, hostile environments. The sensitive parts of the thermocouple are protected by a shielding layer. In order to improve the accuracy of the measurement device, we study an inverse heat conduction problem where the temperature on the surface of the shielding layer is sought, given measured temperatures in the interior of the thermocouple. The procedure is well suited for real-time applications where newly collected data is continuously used to compute current estimates of the surface temperature. Mathematically we can formulate the problem as a Cauchy problem for the heat equation, in cylindrical coordinates, where data is given along the line r = r 1 and the solution is sought at r 1 < r ≤ r 2. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. Thus, regularization techniques are needed. The ill–posedness of the problem is analyzed and a numerical method is proposed. Numerical experiments demonstrate that the proposed method works well.

  • 337.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Boundary identification for an elliptic equation2001Report (Other academic)
    Abstract [en]

    We consider an inverse problem for the two dimensional steady state heat equation. More precisely, the heat equation is valid in a domain Ω, that is a subset of the unit square, temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and x = 1 are assumed to be insulated. From these we wish to determine the temperature in the domain Ω. Furthermore, a part of the boundry ∂Ω is considered to be unknown, and must also be determined.

    The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equation, which can be solved using standard numerical methods.

    The inverse problem, that we consider in this paper, arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an invers heat conduction problem.

  • 338.
    Berntsson, Fredrik
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Boundary identification for an elliptic equation2002In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 18, no 6, p. 1579-1592Article in journal (Refereed)
    Abstract [en]

    We consider an inverse problem for the two-dimensional steady-state heat equation. More precisely, the heat equation is valid in a domain O, that is a subset of the unit square. Temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and 1 are assumed to be insulated. From these we wish to determine the temperature in the domain O. Furthermore, a part of the boundary ?O is considered to be unknown, and must also be determined. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least-squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equations, which can be solved using standard numerical methods. The inverse problem that we consider in this paper arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an inverse heat conduction problem.

  • 339.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Numerical Methods for an Inverse Heat Conduction Problem1998Conference paper (Other academic)
  • 340.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Numerical methods for inverse heat conduction problems2001Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    In many industrial applications one wishes to determine the temperature history on the surface of a body, where the surface itself is inaccessible for measurements. The sideways heat equation is a model of this situation. In a one-dimensional setting this is formulated mathematically as a Cauchy problem for the heat equation, where temperature and heat--flux data are available along the line x=1, and a solution is sought for 0 ≤ x< 1. This problem is ill-posed in the sense that the solution does not depend continuously on the data. Stability can be restored by replacing the time derivative in the heat equation by a bounded approximation. We consider both spectral and wavelet approximations of the derivative. The resulting problem is a system of ordinary differential equations in the space variable, that can be solved using standard methods, e.g. a Runge-Kutta method. The methods are analyzed theoretically, and error estimates are derived, that can be used for selecting the appropriate level of regularization. The numerical implementation of the proposed methods is discussed. Numerical experiments demonstrate that the proposed methods work well, and can be implemented efficiently. Furthermore, the numerical methods can easily be adapted to solve problems with variable coefficients, and also non-linear equations. As test problems we take model equations, with constant and variable coefficients. Also, we solve problems from applications, with actual measured data.

    Inverse problems for the stationary heat equation are also discussed. Suppose that the Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are given on the outer boundary, and we wish to compute the steady state temperature on the inner boundary. A standard approach is to discretize the equation by finite differences, and use Tikhonov's method for stabilizing the discrete problem, which leads to a large sparse least squares problem. Alternatively, we propose to use a conformal mapping to transform the domain into an annulus, where the equivalent problem can be solved using separation of variables. The ill-posedness is dealt with by filtering away high frequencies from the solution. Numerical results using both methods are presented. A closely related problem is that of determining the stationary temperature inside a body, from temperature and heat-flux measurements on a part of the boundary. In practical applications it is sometimes the case that the domain, where the differential equation is valid, is partly unknown. In such cases we want to determine not only the temperature, but also the shape of the boundary of the domain. This problem arises, for instance, in iron production, where the walls of a melting furnace is subject to both physical and chemical wear. In order to avoid a situation where molten metal breaks out through the walls the thickness of the walls should be constantly monitored. This is done by solving an inverse problem for the stationary heat equation, where temperature and heat-flux data are available at certain locations inside the walls of the furnace. Numerical results are presented also for this problem.

    List of papers
    1. Wavelet and Fourier methods for solving the sideways heat equation
    Open this publication in new window or tab >>Wavelet and Fourier methods for solving the sideways heat equation
    2000 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 21, no 6, p. 2187-2205Article in journal (Refereed) Published
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted for 0 ≤ x < 1.

    The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g., a Runge–Kutta method.

    We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations with constant and variable coefficients. We also solve a problem from an industrial application with actual measured data.

    National Category
    Engineering and Technology
    Identifiers
    urn:nbn:se:liu:diva-47660 (URN)10.1137/S1064827597331394 (DOI)
    Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13
    2. A spectral method for solving the sideways heat equation
    Open this publication in new window or tab >>A spectral method for solving the sideways heat equation
    1999 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 15, no 4, p. 891-906Article in journal (Refereed) Published
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is the model of a problem where one wants to determine the temperature on the surface of a body, using interior measurements. Mathematically it can be formulated as a Cauchy problem for the heat equation, where the data are given along the line x = 1, and a solution is sought in the interval 0 ≤ x < 1.

    The problem is ill-posed, in the sense that the solution does not depend continuously on the data. Continuous dependence of the data is restored by replacing the time derivative in the heat equation with a bounded spectral-based approximation. The cut-off level in the spectral approximation acts as a regularization parameter. Error estimates for the regularized solution are derived and a procedure for selecting an appropriate regularization parameter is given. The discretized problem is an initial value problem for an ordinary differential equation in the space variable, which can be solved using standard numerical methods, for example a Runge-Kutta method. As test problems we take equations with constant and variable coefficients.

    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-68233 (URN)10.1088/0266-5611/15/4/305 (DOI)
    Available from: 2011-05-13 Created: 2011-05-13 Last updated: 2017-12-11
    3. An inverse heat conduction problem and an application to heat treatment of aluminium
    Open this publication in new window or tab >>An inverse heat conduction problem and an application to heat treatment of aluminium
    2000 (English)In: Inverse Problems in Engineering Mechanics II / [ed] Masataka Tanaka, G.S. Dulikravich, 2000, p. 99-106Conference paper, Published paper (Refereed)
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem where one wants to determine the temperature on the surface of a body using internal measurements. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We discuss the nature of the ill-posedness as well as methods for restoring stability with respect to measurement errors.

    Successful heat treatment requires good control of the temperature and cooling rates during the process. In an experiment a aluminium block, of the alloy AA7010, was cooled rapidly by spraying water on one surface. Thermocouples inside the block recorded the temperature, and we demonstrate that it is possible to find the temperature distribution in the region between the thermocouple and the surface, by solving numerically the sideways heat equation.

    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-68235 (URN)978-0-08-043693-7 (ISBN)008053516X (ISBN)
    Conference
    International Symposium on Inverse Problems in Engineering Mechanics, Nagano, Japan, March 2000
    Available from: 2011-05-13 Created: 2011-05-13 Last updated: 2014-12-15
    4. Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation
    Open this publication in new window or tab >>Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation
    2001 (English)Report (Other academic)
    Abstract [en]

    Numerical procedures for solving a non-Characteristic Cauchy problem for the heat equation are discussed. More precisely we consider a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted 0 ≤ x <1. The problem is often referred to as the sideways heat equation.

    The problem is analyzed, using both Fourier analysis and the singular value decomposition, and is found to be severely ill-posed. The literature is vast, and many authors have proposed numerical methods that regularize the IHCP. In this paper we attempt to give an overview that covers the most popular methods that have been considered.

    Numerical examples that illustrate the numerical algorithms are given.

    Publisher
    p. 33
    Series
    LiTH-MAT-R, ISSN 0348-2960 ; 17
    National Category
    Engineering and Technology
    Identifiers
    urn:nbn:se:liu:diva-88733 (URN)
    Available from: 2013-02-15 Created: 2013-02-15 Last updated: 2013-02-15
    5. Numerical solution of a Cauchy problem for the Laplace equation
    Open this publication in new window or tab >>Numerical solution of a Cauchy problem for the Laplace equation
    2001 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 17, no 4, p. 839-853Article in journal (Refereed) Published
    Abstract [en]

    We consider a two-dimensional steady state heat conduction problem. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and we wish to compute the temperature on the inner boundary. This Cauchy problem is ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We consider two numerical methods for solving this problem. A standard approach is to discretize the differential equation by finite differences, and use Tikhonov regularization on the discrete problem, which leads to a large sparse least squares problem. We propose to use a conformal mapping that maps the region onto an annulus, where the equivalent problem is solved using a technique based on the fast Fourier transform. The ill-posedness is dealt with by filtering away high frequencies in the solution. Numerical results using both methods are given.

    National Category
    Engineering and Technology
    Identifiers
    urn:nbn:se:liu:diva-47296 (URN)10.1088/0266-5611/17/4/316 (DOI)
    Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13
    6. Boundary identification for an elliptic equation
    Open this publication in new window or tab >>Boundary identification for an elliptic equation
    2001 (English)Report (Other academic)
    Abstract [en]

    We consider an inverse problem for the two dimensional steady state heat equation. More precisely, the heat equation is valid in a domain Ω, that is a subset of the unit square, temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and x = 1 are assumed to be insulated. From these we wish to determine the temperature in the domain Ω. Furthermore, a part of the boundry ∂Ω is considered to be unknown, and must also be determined.

    The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equation, which can be solved using standard numerical methods.

    The inverse problem, that we consider in this paper, arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an invers heat conduction problem.

    Publisher
    p. 16
    Series
    LiTH-MAT-R, ISSN 0348-2960 ; 23
    National Category
    Engineering and Technology
    Identifiers
    urn:nbn:se:liu:diva-88734 (URN)
    Available from: 2013-02-15 Created: 2013-02-15 Last updated: 2013-02-15
  • 341.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation2001Report (Other academic)
    Abstract [en]

    Numerical procedures for solving a non-Characteristic Cauchy problem for the heat equation are discussed. More precisely we consider a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted 0 ≤ x <1. The problem is often referred to as the sideways heat equation.

    The problem is analyzed, using both Fourier analysis and the singular value decomposition, and is found to be severely ill-posed. The literature is vast, and many authors have proposed numerical methods that regularize the IHCP. In this paper we attempt to give an overview that covers the most popular methods that have been considered.

    Numerical examples that illustrate the numerical algorithms are given.

  • 342.
    Berntsson, Fredrik
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Sequential solution of the sideways heat equation by windowing of the data2003In: Inverse Problems in Engineering, ISSN 1068-2767, E-ISSN 1029-0281, Vol. 11, no 2, p. 91-103Article in journal (Refereed)
    Abstract [en]

    The sideways heat equation is a one-dimensional model of a problem, where one wants to determine the temperature on the surface of a body using interior measurements. More precisely, we consider a heat conduction problem, where temperature and heat-flux data are available along the line x = 1 and the solution is sought in the interval 0 = x < 1. The problem is ill-posed in the sense that the solution does not depend continuously on the data. Stability can be restored by replacing the time derivative in the heat equation with a bounded spectral approximation. The cut off level in the spectral approximation acts as a regularization parameter, that controls the degree of smoothness in the solution. In certain applications one wants to solve the sideways heat equation in real time, i.e. to constantly update the solution as new measurements are recorded. For this case sequential solution methods are required.

  • 343.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Simulation Tools for Injection Moulding1997Conference paper (Other academic)
  • 344.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Baravdish, George
    Linköping University, Department of Science and Technology, Communications and Transport Systems. Linköping University, The Institute of Technology.
    Coefficient identification in PDEs applied to image inpainting2014In: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 242, p. 227-235Article in journal (Refereed)
    Abstract [en]

    In this paper, we introduce the concept of parameter identification problems, which are inverse problems, as a methodology to inpainting. More specifically, as a first study in this new direction, we generalize the method of harmonic inpainting by studying an elliptic equation in divergence form where we assume that the diffusion coefficient is unknown. As a first step, this unknown coefficient is determined from the information obtained by the known data in the image. Next, we fill in the region with missing data by solving an elliptic equation in divergence form using this obtained diffusion coefficient. An error analysis shows that this approach is promising and our numerical experiments produces better results than the harmonic inpainting.

  • 345.
    Berntsson, Fredrik
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Elden, Lars
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Numerical Solution of Cauchy Problems for Elliptic Equations in "Rectangle-like" Geometries2005In: FEMLAB Conference,2005, Stockholm: Comsol AB , 2005Conference paper (Other academic)
    Abstract [en]

    We consider two dimensional inverse steady state heat conduction problems in complex geometries. The coefficients of the elliptic equation are assumed to be non-constant. Cauchy data are given on one part of the boundary and we want to find the solution in the whole domain. The problem is ill--posed in the sense that the solution does not depend continuously on the data. Using an orthogonal coordinate transformation the domain is mapped onto a rectangle. The Cauchy problem can then be solved by replacing one derivative by a bounded approximation. The resulting well--posed problem can then be solved by a method of lines. A bounded approximation of the derivative can be obtained by differentiating a cubic spline, that approximate the function in the least squares sense. This particular approximation of the derivative is computationally efficient and flexible in the sense that its easy to handle different kinds of boundary conditions. This inverse problem arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. Temperature and heat--flux data are collected by several thermocouples located inside the walls. The shape of the interface between the molten iron and the walls can then be determined by solving an inverse heat conduction problem. In our work we make extensive use of Femlab for creating test problems. By using Femlab we solve relatively complex model problems for the purpose of creating numerical test data used for validating our methods. For the types of problems we are intressted in numerical artefacts appear, near corners in the domain, in the gradients that Femlab calculates. We demonstrate why this happen and also how we deal with the problem.

  • 346.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Eldén, Lars
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    An inverse heat conduction problem and an application to heat treatment of aluminium2000In: Inverse Problems in Engineering Mechanics II / [ed] Masataka Tanaka, G.S. Dulikravich, 2000, p. 99-106Conference paper (Refereed)
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem where one wants to determine the temperature on the surface of a body using internal measurements. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We discuss the nature of the ill-posedness as well as methods for restoring stability with respect to measurement errors.

    Successful heat treatment requires good control of the temperature and cooling rates during the process. In an experiment a aluminium block, of the alloy AA7010, was cooled rapidly by spraying water on one surface. Thermocouples inside the block recorded the temperature, and we demonstrate that it is possible to find the temperature distribution in the region between the thermocouple and the surface, by solving numerically the sideways heat equation.

  • 347.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Eldén, Lars
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Numerical solution of a Cauchy problem for the Laplace equation2001In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 17, no 4, p. 839-853Article in journal (Refereed)
    Abstract [en]

    We consider a two-dimensional steady state heat conduction problem. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and we wish to compute the temperature on the inner boundary. This Cauchy problem is ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We consider two numerical methods for solving this problem. A standard approach is to discretize the differential equation by finite differences, and use Tikhonov regularization on the discrete problem, which leads to a large sparse least squares problem. We propose to use a conformal mapping that maps the region onto an annulus, where the equivalent problem is solved using a technique based on the fast Fourier transform. The ill-posedness is dealt with by filtering away high frequencies in the solution. Numerical results using both methods are given.

  • 348.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Eldén, Lars
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Numerical Solution of an Inverse Steady State Heat Conduction Problem2000Conference paper (Other academic)
  • 349.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Eldén, Lars
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Numerical Solution of Cauchy Problems for Elliptic Equations in ``Rectangle-like'' Geometries2005In: Proceedings for the FEMLAB Conference 2005, 2005Conference paper (Other academic)
    Abstract [en]

    We consider two dimensional inverse steady state heat conductionproblems in complex geometries. The coefficients of the elliptic equation are assumed to be non-constant. Cauchy data are given on onepart of the boundary and we want to find the solution in the wholedomain. The problem is ill--posed in the sense that the solution doesnot depend continuously on the data.

    Using an orthogonal coordinate transformation the domain is mappedonto a rectangle. The Cauchy problem can then be solved by replacing one derivative by a bounded approximation. The resulting well--posed problem can then be solved by a method of lines. A bounded approximation of the derivative can be obtained by differentiating a cubic spline, that approximate the function in theleast squares sense. This particular approximation of the derivativeis computationally efficient and flexible in the sense that its easy to handle different kinds of boundary conditions.This inverse problem arises in iron production, where the walls of amelting furnace are subject to physical and chemical wear. Temperature and heat--flux data are collected by several thermocouples locatedinside the walls. The shape of the interface between the molten ironand the walls can then be determined by solving an inverse heatconduction problem.  In our work we make extensive use of Femlab for creating testproblems. By using FEMLAB we solve relatively complex model problems for the purpose of creating numerical test data used for validating our methods. For the types of problems we are intressted in numerical artefacts appear, near corners in the domain, in the gradients that Femlab calculates. We demonstrate why this happen and also how we deal with the problem.

  • 350.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Eldén, Lars
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Spectral and Wavelet Methods for Solving an Inverse Heat Conduction Problem1998In: Inverse Problems in Engineering Mechanics: International Symposium on Inverse Problems in Engineering Mechanics, 1998 (ISIP 98) / [ed] M. Tanaka and G.S. Dulikravich, Oxford: Elsevier Science , 1998, p. 3-10Conference paper (Other academic)
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