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[{"key": "dc.contributor.advisor", "value": "Salo, Mikko", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Nurminen, Janne", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2020-03-25T05:19:18Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2020-03-25T05:19:18Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2020", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/68306", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa esitell\u00e4\u00e4n s\u00e4hk\u00f6isen impedanssitomografian matemaattista mallia sek\u00e4 johtavuusyht\u00e4l\u00f6\u00f6n liittyv\u00e4\u00e4 inversio-ongelmaa. Tutkielman p\u00e4\u00e4tuloksena osoitetaan, ett\u00e4 kappaleen reunalla teht\u00e4vien mittauksien avulla voidaan m\u00e4\u00e4ritt\u00e4\u00e4 kappaleen sis\u00e4ll\u00e4 oleva johtavuus. S\u00e4hk\u00f6inen impedanssitomografia on siis kuvantamismenetelm\u00e4, jonka avulla jonkin kappaleen pinnalla teht\u00e4vist\u00e4 s\u00e4hk\u00f6isist\u00e4 mittauksista pyrit\u00e4\u00e4n selvitt\u00e4m\u00e4\u00e4n kappaleen sis\u00e4ist\u00e4 rakennetta.\n\nKyseisen inversio-ongelman muotoilemiseen tarvitaan esitiedoiksi teoriaa Sobolev-avaruuksista sek\u00e4 osittaisdifferentiaaliyht\u00e4l\u00f6iden heikoista ratkaisuista. Heikkojen ratkaisujen teoria pohjautuu nimenomaan Sobolev-avaruuksien teoriaan. Osittaisdifferentiaaliyht\u00e4l\u00f6iden heikkojen ratkaisujen teorian avulla voidaan antaa m\u00e4\u00e4ritelm\u00e4 niin sanotulle Dirichlet-to-Neumann -kuvaukselle, jonka voidaan ajatella sis\u00e4lt\u00e4v\u00e4n tiedot kappaleen reunalla teht\u00e4vist\u00e4 mittauksista.\n\nKun tiedet\u00e4\u00e4n, mit\u00e4 Dirichlet-to-Neumann -kuvaukset ovat, voidaan muotoilla s\u00e4hk\u00f6iseen kuvantamismenetelm\u00e4\u00e4n liittyv\u00e4 inversio-ongelma johtavuusyht\u00e4l\u00f6n tapauksessa: Olkoon \\gamma positiivinen oleellisesti rajoitettu funktio avaruuden \\R^n avoimessa ja rajoitetussa joukossa. M\u00e4\u00e4rit\u00e4 Dirichlet-to-Neumann -kuvauksesta johtavuus kyseisess\u00e4 joukossa.\nTutkielman p\u00e4\u00e4tulos liittyy t\u00e4h\u00e4n inversio-ongelmaan, jossa osoitetaan, ett\u00e4 reunalla teht\u00e4v\u00e4t s\u00e4hk\u00f6iset mittaukset, eli Dirichlet-to-Neumann -kuvaus, m\u00e4\u00e4r\u00e4\u00e4v\u00e4t tuntemattoman johtavuuden arvon reunalla. \n\nT\u00e4m\u00e4n lis\u00e4ksi tutkielmassa esitell\u00e4\u00e4n hieman Schr\u00f6dingerin yht\u00e4l\u00f6\u00e4. Schr\u00f6dingerin yht\u00e4l\u00f6lle muotoillaan siihen liittyv\u00e4 inversio-ongelma ja vastaava Dirichlet-to-Neumann -kuvaus. Lis\u00e4ksi todistetaan samankaltainen tulos kuin johtavuusyht\u00e4l\u00f6lle, eli reunalla teht\u00e4v\u00e4t mittaukset m\u00e4\u00e4r\u00e4\u00e4v\u00e4t tuntemattoman potentiaalin arvon reunalla.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2020-03-25T05:19:18Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2020-03-25T05:19:18Z (GMT). No. of bitstreams: 0\n Previous issue date: 2020", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "60", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "inversio-ongelmat", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Calder\u00f3n ongelma", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "osittaisdifferentiaaliyht\u00e4l\u00f6t", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "s\u00e4hk\u00f6inen impedanssitomografia", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "johtavuusyht\u00e4l\u00f6", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Schr\u00f6dingerin yht\u00e4l\u00f6", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "S\u00e4hk\u00f6isen impedanssitomografian inversio-ongelma", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202003252523", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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