{"id":{"repo_id":"strathclyde","oai_identifier":"oai:strathclyde:z029p4825"},"canonical_url":"https://search.dev.ndltd.org/etd/strathclyde/oai:strathclyde:z029p4825","repository":{"repo_id":"strathclyde","name":"University of Strathclyde","base_url":"https://stax.strath.ac.uk/catalog/oai"},"display":{"title":"Anisotropic piecewise linear approximation","abstract":"The subject of this thesis includes the design of new partitioning methods for the approximation of a function f on a domain C Rd, d 2, by piecewise linear functions, and the derivation of errors estimations in Lp-norm and W1 p - seminorm. In the two-dimensional setting, we develop a construction of a sequence of anisotropic triangulations, where the approximation provided by the piecewise linear interpolant for a given f C2() with a positive definite Hessian, is asymptotically optimal in Lp-norm and in the same time optimal in W1 p - seminorm with respect to the number of degrees of freedom. As a preparation for this result, we review various local error bounds for the interpolation by linear polynomials on a triangle, and derive a number of new estimates of this type. In addition, for functions of d 2 variables, we propose a new approximation method, where several overlaying partitions of are designed such that the sum of piecewise constant or piecewise linear polynomials over these partitions provides a better approximation order than the one obtainable by using a single partition.","abstract_html":"The subject of this thesis includes the design of new partitioning methods for the approximation of a function f on a domain C Rd, d 2, by piecewise linear functions, and the derivation of errors estimations in Lp-norm and W1 p - seminorm. In the two-dimensional setting, we develop a construction of a sequence of anisotropic triangulations, where the approximation provided by the piecewise linear interpolant for a given f C2() with a positive definite Hessian, is asymptotically optimal in Lp-norm and in the same time optimal in W1 p - seminorm with respect to the number of degrees of freedom. As a preparation for this result, we review various local error bounds for the interpolation by linear polynomials on a triangle, and derive a number of new estimates of this type. In addition, for functions of d 2 variables, we propose a new approximation method, where several overlaying partitions of are designed such that the sum of piecewise constant or piecewise linear polynomials over these partitions provides a better approximation order than the one obtainable by using a single partition.","abstract_has_math":false,"creators":["Rabarison, Andrianarivo Fabien"],"institution":"University of Strathclyde","degree_name":"phd","degree_level":"doctoral-pg","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012","date_published":"2012","updated_at":"2026-07-24T04:53:12Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.48730/tkrb-hq53"],"render_values":[{"text":"10.48730/tkrb-hq53","href":"https://doi.org/10.48730/tkrb-hq53","code":true}]},{"key":"dc:identifier","label":"Identifier","values":["T13399"],"render_values":[{"text":"T13399","href":null,"code":true}]}]},"links":{"outbound_url":"https://stax.strath.ac.uk/concern/theses/z029p4825","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Rabarison, Andrianarivo Fabien"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012"]},{"key":"dc:date.issued","label":"Date","values":["2012"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Department of Mathematics and Statistics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Strathclyde"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral-pg"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["phd"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["T13399"]},{"key":"dc:identifier.doi","label":"DOI","values":["10.48730/tkrb-hq53"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://stax.strath.ac.uk/concern/theses/z029p4825"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The subject of this thesis includes the design of new partitioning methods for the approximation of a function f on a domain C Rd, d 2, by piecewise linear functions, and the derivation of errors estimations in Lp-norm and W1 p - seminorm. In the two-dimensional setting, we develop a construction of a sequence of anisotropic triangulations, where the approximation provided by the piecewise linear interpolant for a given f C2() with a positive definite Hessian, is asymptotically optimal in Lp-norm and in the same time optimal in W1 p - seminorm with respect to the number of degrees of freedom. As a preparation for this result, we review various local error bounds for the interpolation by linear polynomials on a triangle, and derive a number of new estimates of this type. In addition, for functions of d 2 variables, we propose a new approximation method, where several overlaying partitions of are designed such that the sum of piecewise constant or piecewise linear polynomials over these partitions provides a better approximation order than the one obtainable by using a single partition."]},{"key":"dc:description.abstract","label":"Abstract","values":["The subject of this thesis includes the design of new partitioning methods for the approximation of a function f on a domain C Rd, d 2, by piecewise linear functions, and the derivation of errors estimations in Lp-norm and W1 p - seminorm. In the two-dimensional setting, we develop a construction of a sequence of anisotropic triangulations, where the approximation provided by the piecewise linear interpolant for a given f C2() with a positive definite Hessian, is asymptotically optimal in Lp-norm and in the same time optimal in W1 p - seminorm with respect to the number of degrees of freedom. As a preparation for this result, we review various local error bounds for the interpolation by linear polynomials on a triangle, and derive a number of new estimates of this type. In addition, for functions of d 2 variables, we propose a new approximation method, where several overlaying partitions of are designed such that the sum of piecewise constant or piecewise linear polynomials over these partitions provides a better approximation order than the one obtainable by using a single partition."]},{"key":"dc:title","label":"Title","values":["Anisotropic piecewise linear approximation"]}]}],"canonical_facts":{"dc:creator":["Rabarison, Andrianarivo Fabien"],"dc:date":["2012"],"dc:date.issued":["2012"],"dc:description":["The subject of this thesis includes the design of new partitioning methods for the approximation of a function f on a domain C Rd, d 2, by piecewise linear functions, and the derivation of errors estimations in Lp-norm and W1 p - seminorm. In the two-dimensional setting, we develop a construction of a sequence of anisotropic triangulations, where the approximation provided by the piecewise linear interpolant for a given f C2() with a positive definite Hessian, is asymptotically optimal in Lp-norm and in the same time optimal in W1 p - seminorm with respect to the number of degrees of freedom. As a preparation for this result, we review various local error bounds for the interpolation by linear polynomials on a triangle, and derive a number of new estimates of this type. In addition, for functions of d 2 variables, we propose a new approximation method, where several overlaying partitions of are designed such that the sum of piecewise constant or piecewise linear polynomials over these partitions provides a better approximation order than the one obtainable by using a single partition."],"dc:description.abstract":["The subject of this thesis includes the design of new partitioning methods for the approximation of a function f on a domain C Rd, d 2, by piecewise linear functions, and the derivation of errors estimations in Lp-norm and W1 p - seminorm. In the two-dimensional setting, we develop a construction of a sequence of anisotropic triangulations, where the approximation provided by the piecewise linear interpolant for a given f C2() with a positive definite Hessian, is asymptotically optimal in Lp-norm and in the same time optimal in W1 p - seminorm with respect to the number of degrees of freedom. As a preparation for this result, we review various local error bounds for the interpolation by linear polynomials on a triangle, and derive a number of new estimates of this type. In addition, for functions of d 2 variables, we propose a new approximation method, where several overlaying partitions of are designed such that the sum of piecewise constant or piecewise linear polynomials over these partitions provides a better approximation order than the one obtainable by using a single partition."],"dc:identifier":["T13399"],"dc:identifier.doi":["10.48730/tkrb-hq53"],"dc:identifier.uri":["https://stax.strath.ac.uk/concern/theses/z029p4825"],"dc:publisher.department":["Department of Mathematics and Statistics"],"dc:publisher.institution":["University of Strathclyde"],"dc:title":["Anisotropic piecewise linear approximation"],"dc:type.qualificationlevel":["doctoral-pg"],"dc:type.qualificationname":["phd"]},"updated_at":"2026-07-24T04:53:12Z"}