Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess
Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when th...
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| Autor*in: |
Klibanov, Michael V. [verfasserIn] Bakushinsky, Anatoly B. [verfasserIn] Beilina, Larisa [verfasserIn] |
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| Format: |
E-Artikel |
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| Erschienen: |
Walter de Gruyter GmbH & Co. KG ; 2011 |
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© de Gruyter 2011 |
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| Umfang: |
23 |
| Reproduktion: |
Walter de Gruyter Online Zeitschriften |
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| Übergeordnetes Werk: |
Enthalten in: Journal of inverse and ill-posed problems - Berlin : de Gruyter, 1993, 19(2011), 1 vom: 02. Mai, Seite 83-105 |
| Übergeordnetes Werk: |
volume:19 ; year:2011 ; number:1 ; day:02 ; month:05 ; pages:83-105 ; extent:23 |
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| DOI / URN: |
10.1515/jiip.2011.024 |
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| 520 | |a Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented. | ||
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10.1515/jiip.2011.024 doi artikel_Grundlieferung.pp (DE-627)NLEJ247091057 DE-627 ger DE-627 rakwb Klibanov, Michael V. verfasserin aut Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess Walter de Gruyter GmbH & Co. KG 2011 23 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © de Gruyter 2011 Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented. Walter de Gruyter Online Zeitschriften Uniqueness theorem Tikhonov functional a single value of the level of error Bakushinsky, Anatoly B. verfasserin aut Beilina, Larisa verfasserin aut Enthalten in Journal of inverse and ill-posed problems Berlin : de Gruyter, 1993 19(2011), 1 vom: 02. Mai, Seite 83-105 (DE-627)NLEJ248236091 (DE-600)2041913-2 1569-3945 nnns volume:19 year:2011 number:1 day:02 month:05 pages:83-105 extent:23 https://doi.org/10.1515/jiip.2011.024 Deutschlandweit zugänglich GBV_USEFLAG_U ZDB-1-DGR GBV_NL_ARTICLE AR 19 2011 1 02 05 83-105 23 |
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10.1515/jiip.2011.024 doi artikel_Grundlieferung.pp (DE-627)NLEJ247091057 DE-627 ger DE-627 rakwb Klibanov, Michael V. verfasserin aut Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess Walter de Gruyter GmbH & Co. KG 2011 23 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © de Gruyter 2011 Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented. Walter de Gruyter Online Zeitschriften Uniqueness theorem Tikhonov functional a single value of the level of error Bakushinsky, Anatoly B. verfasserin aut Beilina, Larisa verfasserin aut Enthalten in Journal of inverse and ill-posed problems Berlin : de Gruyter, 1993 19(2011), 1 vom: 02. Mai, Seite 83-105 (DE-627)NLEJ248236091 (DE-600)2041913-2 1569-3945 nnns volume:19 year:2011 number:1 day:02 month:05 pages:83-105 extent:23 https://doi.org/10.1515/jiip.2011.024 Deutschlandweit zugänglich GBV_USEFLAG_U ZDB-1-DGR GBV_NL_ARTICLE AR 19 2011 1 02 05 83-105 23 |
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10.1515/jiip.2011.024 doi artikel_Grundlieferung.pp (DE-627)NLEJ247091057 DE-627 ger DE-627 rakwb Klibanov, Michael V. verfasserin aut Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess Walter de Gruyter GmbH & Co. KG 2011 23 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © de Gruyter 2011 Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented. Walter de Gruyter Online Zeitschriften Uniqueness theorem Tikhonov functional a single value of the level of error Bakushinsky, Anatoly B. verfasserin aut Beilina, Larisa verfasserin aut Enthalten in Journal of inverse and ill-posed problems Berlin : de Gruyter, 1993 19(2011), 1 vom: 02. Mai, Seite 83-105 (DE-627)NLEJ248236091 (DE-600)2041913-2 1569-3945 nnns volume:19 year:2011 number:1 day:02 month:05 pages:83-105 extent:23 https://doi.org/10.1515/jiip.2011.024 Deutschlandweit zugänglich GBV_USEFLAG_U ZDB-1-DGR GBV_NL_ARTICLE AR 19 2011 1 02 05 83-105 23 |
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10.1515/jiip.2011.024 doi artikel_Grundlieferung.pp (DE-627)NLEJ247091057 DE-627 ger DE-627 rakwb Klibanov, Michael V. verfasserin aut Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess Walter de Gruyter GmbH & Co. KG 2011 23 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © de Gruyter 2011 Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented. Walter de Gruyter Online Zeitschriften Uniqueness theorem Tikhonov functional a single value of the level of error Bakushinsky, Anatoly B. verfasserin aut Beilina, Larisa verfasserin aut Enthalten in Journal of inverse and ill-posed problems Berlin : de Gruyter, 1993 19(2011), 1 vom: 02. Mai, Seite 83-105 (DE-627)NLEJ248236091 (DE-600)2041913-2 1569-3945 nnns volume:19 year:2011 number:1 day:02 month:05 pages:83-105 extent:23 https://doi.org/10.1515/jiip.2011.024 Deutschlandweit zugänglich GBV_USEFLAG_U ZDB-1-DGR GBV_NL_ARTICLE AR 19 2011 1 02 05 83-105 23 |
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Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented. © de Gruyter 2011 |
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Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented. © de Gruyter 2011 |
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Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented. © de Gruyter 2011 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">NLEJ247091057</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20220820030145.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">220814s2011 xx |||||o 00| ||und c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1515/jiip.2011.024</subfield><subfield code="2">doi</subfield></datafield><datafield tag="028" ind1="5" ind2="2"><subfield code="a">artikel_Grundlieferung.pp</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)NLEJ247091057</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Klibanov, Michael V.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="b">Walter de Gruyter GmbH & Co. KG</subfield><subfield code="c">2011</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">23</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© de Gruyter 2011</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented.</subfield></datafield><datafield tag="533" ind1=" " ind2=" "><subfield code="f">Walter de Gruyter Online Zeitschriften</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Uniqueness theorem</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Tikhonov functional</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">a single value of the level of error</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Bakushinsky, Anatoly B.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Beilina, Larisa</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Journal of inverse and ill-posed problems</subfield><subfield code="d">Berlin : de Gruyter, 1993</subfield><subfield code="g">19(2011), 1 vom: 02. Mai, Seite 83-105</subfield><subfield code="w">(DE-627)NLEJ248236091</subfield><subfield code="w">(DE-600)2041913-2</subfield><subfield code="x">1569-3945</subfield><subfield code="7">nnns</subfield></datafield><datafield tag="773" ind1="1" ind2="8"><subfield code="g">volume:19</subfield><subfield code="g">year:2011</subfield><subfield code="g">number:1</subfield><subfield code="g">day:02</subfield><subfield code="g">month:05</subfield><subfield code="g">pages:83-105</subfield><subfield code="g">extent:23</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">https://doi.org/10.1515/jiip.2011.024</subfield><subfield code="z">Deutschlandweit zugänglich</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_USEFLAG_U</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">ZDB-1-DGR</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_NL_ARTICLE</subfield></datafield><datafield tag="951" ind1=" " ind2=" "><subfield code="a">AR</subfield></datafield><datafield tag="952" ind1=" " ind2=" "><subfield code="d">19</subfield><subfield code="j">2011</subfield><subfield code="e">1</subfield><subfield code="b">02</subfield><subfield code="c">05</subfield><subfield code="h">83-105</subfield><subfield code="g">23</subfield></datafield></record></collection>
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