The numerical solution of first kind integral equations
In a recent paper, Babolian and Delves (hereafter BD) described a Chebyshev series method for the solution of first kind Fredholm integral equations. The method imposes regularisation constraints directly on the Chebyshev polynomial expansion of the solution, and involves two regularisation paramete...
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| Veröffentlicht in: | Journal of computational and applied mathematics 1989-11, Vol.27 (3), p.363-387 |
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| container_title | Journal of computational and applied mathematics |
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| creator | Essah, W.A. Delves, L.M. |
| description | In a recent paper, Babolian and Delves (hereafter BD) described a Chebyshev series method for the solution of first kind Fredholm integral equations. The method imposes regularisation constraints directly on the Chebyshev polynomial expansion of the solution, and involves two regularisation parameters, Cf and r.
In this paper we develop a cross-validation algorithm capable of setting these parameters automatically. We show that the cross-validation scheme, coupled with the algorithm of BD leads to a stable regularised problem; and that the method can be implemented relatively inexpensively. Finally, we give a number of numerical examples showing that the method works well in practice. |
| doi_str_mv | 10.1016/0377-0427(89)90023-X |
| format | Article |
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In this paper we develop a cross-validation algorithm capable of setting these parameters automatically. We show that the cross-validation scheme, coupled with the algorithm of BD leads to a stable regularised problem; and that the method can be implemented relatively inexpensively. Finally, we give a number of numerical examples showing that the method works well in practice.</description><identifier>ISSN: 0377-0427</identifier><identifier>EISSN: 1879-1778</identifier><identifier>DOI: 10.1016/0377-0427(89)90023-X</identifier><identifier>CODEN: JCAMDI</identifier><language>eng</language><publisher>Amsterdam: Elsevier B.V</publisher><subject>augmented Galerkin method ; Chebyshev series ; constrained (or regularized) solution ; cross-validation technique ; Exact sciences and technology ; expansion method ; Fredholm integral equations of the first kind ; Integral equations, integral transforms ; Mathematics ; Numerical analysis ; Numerical analysis. Scientific computation ; numerical stability ; regularization estimators ; Sciences and techniques of general use</subject><ispartof>Journal of computational and applied mathematics, 1989-11, Vol.27 (3), p.363-387</ispartof><rights>1989</rights><rights>1990 INIST-CNRS</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c412t-8c0c388b982dd17fcc5599ebc1af2296545b8d268bc64ce2843aac41bb3268633</citedby><cites>FETCH-LOGICAL-c412t-8c0c388b982dd17fcc5599ebc1af2296545b8d268bc64ce2843aac41bb3268633</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/037704278990023X$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,3537,27901,27902,65534</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=6627020$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Essah, W.A.</creatorcontrib><creatorcontrib>Delves, L.M.</creatorcontrib><title>The numerical solution of first kind integral equations</title><title>Journal of computational and applied mathematics</title><description>In a recent paper, Babolian and Delves (hereafter BD) described a Chebyshev series method for the solution of first kind Fredholm integral equations. The method imposes regularisation constraints directly on the Chebyshev polynomial expansion of the solution, and involves two regularisation parameters, Cf and r.
In this paper we develop a cross-validation algorithm capable of setting these parameters automatically. We show that the cross-validation scheme, coupled with the algorithm of BD leads to a stable regularised problem; and that the method can be implemented relatively inexpensively. Finally, we give a number of numerical examples showing that the method works well in practice.</description><subject>augmented Galerkin method</subject><subject>Chebyshev series</subject><subject>constrained (or regularized) solution</subject><subject>cross-validation technique</subject><subject>Exact sciences and technology</subject><subject>expansion method</subject><subject>Fredholm integral equations of the first kind</subject><subject>Integral equations, integral transforms</subject><subject>Mathematics</subject><subject>Numerical analysis</subject><subject>Numerical analysis. Scientific computation</subject><subject>numerical stability</subject><subject>regularization estimators</subject><subject>Sciences and techniques of general use</subject><issn>0377-0427</issn><issn>1879-1778</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1989</creationdate><recordtype>article</recordtype><recordid>eNp9kE1Lw0AQhhdRsFb_gYccRPQQ3Y_s10WQ4hcUvFTobdlsJrqaJu1uIvjv3djSo6eBmeedYR6Ezgm-IZiIW8ykzHFB5ZXS1xpjyvLlAZoQJXVOpFSHaLJHjtFJjJ8YY6FJMUFy8QFZO6wgeGebLHbN0Puuzbo6q32Iffbl2yrzbQ_vIc1hM9hxHk_RUW2bCGe7OkVvjw-L2XM-f316md3Pc1cQ2ufKYceUKrWiVUVk7RznWkPpiK0p1YIXvFQVFap0onBAVcGsTdGyZKkpGJuiy-3edeg2A8TerHx00DS2hW6IhnKOJeE8gcUWdKGLMUBt1sGvbPgxBJvRkhkVmFGBUdr8WTLLFLvY7bcxCaiDbZ2P-6wQVGKKE3a3xSD9-u0hmOg8tA4qH8D1pur8_3d-AURteuw</recordid><startdate>19891101</startdate><enddate>19891101</enddate><creator>Essah, W.A.</creator><creator>Delves, L.M.</creator><general>Elsevier B.V</general><general>Elsevier</general><scope>6I.</scope><scope>AAFTH</scope><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>19891101</creationdate><title>The numerical solution of first kind integral equations</title><author>Essah, W.A. ; Delves, L.M.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c412t-8c0c388b982dd17fcc5599ebc1af2296545b8d268bc64ce2843aac41bb3268633</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1989</creationdate><topic>augmented Galerkin method</topic><topic>Chebyshev series</topic><topic>constrained (or regularized) solution</topic><topic>cross-validation technique</topic><topic>Exact sciences and technology</topic><topic>expansion method</topic><topic>Fredholm integral equations of the first kind</topic><topic>Integral equations, integral transforms</topic><topic>Mathematics</topic><topic>Numerical analysis</topic><topic>Numerical analysis. Scientific computation</topic><topic>numerical stability</topic><topic>regularization estimators</topic><topic>Sciences and techniques of general use</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Essah, W.A.</creatorcontrib><creatorcontrib>Delves, L.M.</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Journal of computational and applied mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Essah, W.A.</au><au>Delves, L.M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The numerical solution of first kind integral equations</atitle><jtitle>Journal of computational and applied mathematics</jtitle><date>1989-11-01</date><risdate>1989</risdate><volume>27</volume><issue>3</issue><spage>363</spage><epage>387</epage><pages>363-387</pages><issn>0377-0427</issn><eissn>1879-1778</eissn><coden>JCAMDI</coden><abstract>In a recent paper, Babolian and Delves (hereafter BD) described a Chebyshev series method for the solution of first kind Fredholm integral equations. The method imposes regularisation constraints directly on the Chebyshev polynomial expansion of the solution, and involves two regularisation parameters, Cf and r.
In this paper we develop a cross-validation algorithm capable of setting these parameters automatically. We show that the cross-validation scheme, coupled with the algorithm of BD leads to a stable regularised problem; and that the method can be implemented relatively inexpensively. Finally, we give a number of numerical examples showing that the method works well in practice.</abstract><cop>Amsterdam</cop><pub>Elsevier B.V</pub><doi>10.1016/0377-0427(89)90023-X</doi><tpages>25</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | Journal of computational and applied mathematics, 1989-11, Vol.27 (3), p.363-387 |
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| language | eng |
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| source | Elsevier ScienceDirect Journals Complete; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | augmented Galerkin method Chebyshev series constrained (or regularized) solution cross-validation technique Exact sciences and technology expansion method Fredholm integral equations of the first kind Integral equations, integral transforms Mathematics Numerical analysis Numerical analysis. Scientific computation numerical stability regularization estimators Sciences and techniques of general use |
| title | The numerical solution of first kind integral equations |
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