Regularization theory for ill-posed problems :: selected topics /
Thismonograph is a valuable contribution to thehighly topical and extremly productive field ofregularisationmethods for inverse and ill-posed problems. The author is an internationally outstanding and acceptedmathematicianin this field. In his book he offers a well-balanced mixtureof basic and innov...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Berlin ; Boston :
Walter de Gruyter GmbH & Co. KG,
[2013]
|
| Series: | Inverse and ill-posed problems series.
|
| Subjects: | |
| Online Access: | DE-862 DE-863 |
| Summary: | Thismonograph is a valuable contribution to thehighly topical and extremly productive field ofregularisationmethods for inverse and ill-posed problems. The author is an internationally outstanding and acceptedmathematicianin this field. In his book he offers a well-balanced mixtureof basic and innovative aspects. He demonstrates new, differentiatedviewpoints, and important examples for applications. The bookdemontrates thecurrent developments inthe field of regularization theory, such as multiparameter regularization and regularization in learning theory. The book is written for graduate and PhDs. |
| Item Description: | 3.2.6 Generalization in the case of more than two regularization parameters. |
| Physical Description: | 1 online resource (xiv, 289 pages) : illustrations |
| Bibliography: | Includes bibliographical references and index. |
| ISBN: | 9783110286496 3110286491 |
Staff View
MARC
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| 100 | 1 | |a Lu, Shuai, |e author. | |
| 245 | 1 | 0 | |a Regularization theory for ill-posed problems : |b selected topics / |c by Shuai Lu, Sergei V. Pereverzev. |
| 264 | 1 | |a Berlin ; |a Boston : |b Walter de Gruyter GmbH & Co. KG, |c [2013] | |
| 264 | 4 | |c ©2013 | |
| 300 | |a 1 online resource (xiv, 289 pages) : |b illustrations | ||
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| 490 | 1 | |a Inverse and Ill-Posed Problems Series | |
| 504 | |a Includes bibliographical references and index. | ||
| 505 | 0 | |a Preface; 1 An introduction using classical examples; 1.1 Numerical differentiation. First look at the problem of regularization. The balancing principle; 1.1.1 Finite-difference formulae; 1.1.2 Finite-difference formulae for nonexact data. A priori choice of the stepsize; 1.1.3 A posteriori choice of the stepsize; 1.1.4 Numerical illustration; 1.1.5 The balancing principle in a general framework; 1.2 Stable summation of orthogonal series with noisy coefficients. Deterministic and stochastic noise models. Description of smoothness properties; 1.2.1 Summation methods. | |
| 505 | 8 | |a 1.2.2 Deterministic noise model1.2.3 Stochastic noise model; 1.2.4 Smoothness associated with a basis; 1.2.5 Approximation and stability properties of -methods; 1.2.6 Error bounds; 1.3 The elliptic Cauchy problem and regularization by discretization; 1.3.1 Natural linearization of the elliptic Cauchy problem; 1.3.2 Regularization by discretization; 1.3.3 Application in detecting corrosion; 2 Basics of single parameter regularization schemes; 2.1 Simple example for motivation; 2.2 Essentially ill-posed linear operator equations. Least-squares solution. General view on regularization. | |
| 505 | 8 | |a 2.3 Smoothness in the context of the problem. Benchmark accuracy levels for deterministic and stochastic data noise models2.3.1 The best possible accuracy for the deterministic noise model; 2.3.2 The best possible accuracy for the Gaussian white noise model; 2.4 Optimal order and the saturation of regularization methods in Hilbert spaces; 2.5 Changing the penalty term for variance reduction. Regularization in Hilbert scales; 2.6 Estimation of linear functionals from indirect noisy observations; 2.7 Regularization by finite-dimensional approximation. | |
| 505 | 8 | |a 2.8 Model selection based on indirect observation in Gaussian white noise2.8.1 Linear models given by least-squares methods; 2.8.2 Operator monotone functions; 2.8.3 The problem of model selection (continuation); 2.9 A warning example: an operator equation formulation is not always adequate (numerical differentiation revisited); 2.9.1 Numerical differentiation in variable Hilbert scales associated with designs; 2.9.2 Error bounds in L2; 2.9.3 Adaptation to the unknown bound of the approximation error; 2.9.4 Numerical differentiation in the space of continuous functions. | |
| 505 | 8 | |a 2.9.5 Relation to the Savitzky-Golay method. Numerical examples3 Multiparameter regularization; 3.1 When do we really need multiparameter regularization?; 3.2 Multiparameter discrepancy principle; 3.2.1 Model function based on the multiparameter discrepancy principle; 3.2.2 A use of the model function to approximate one set of parameters satisfying the discrepancy principle; 3.2.3 Properties of the model function approximation; 3.2.4 Discrepancy curve and the convergence analysis; 3.2.5 Heuristic algorithm for the model function approximation of the multiparameter discrepancy principle. | |
| 500 | |a 3.2.6 Generalization in the case of more than two regularization parameters. | ||
| 520 | |a Thismonograph is a valuable contribution to thehighly topical and extremly productive field ofregularisationmethods for inverse and ill-posed problems. The author is an internationally outstanding and acceptedmathematicianin this field. In his book he offers a well-balanced mixtureof basic and innovative aspects. He demonstrates new, differentiatedviewpoints, and important examples for applications. The bookdemontrates thecurrent developments inthe field of regularization theory, such as multiparameter regularization and regularization in learning theory. The book is written for graduate and PhDs. | ||
| 588 | 0 | |a Print version record. | |
| 546 | |a English. | ||
| 650 | 0 | |a Numerical differentiation. |0 http://id.loc.gov/authorities/subjects/sh85093243 | |
| 650 | 0 | |a Numerical analysis |x Improperly posed problems. |0 http://id.loc.gov/authorities/subjects/sh85093240 | |
| 650 | 6 | |a Dérivation numérique. | |
| 650 | 6 | |a Analyse numérique |x Problèmes mal posés. | |
| 650 | 7 | |a MATHEMATICS |x Numerical Analysis. |2 bisacsh | |
| 650 | 7 | |a Numerical analysis |x Improperly posed problems |2 fast | |
| 650 | 7 | |a Numerical differentiation |2 fast | |
| 655 | 0 | |a Electronic books. | |
| 700 | 1 | |a Pereverzev, Sergei V., |e author. |1 https://id.oclc.org/worldcat/entity/E39PCjxc6ywK48gkrGtxTrhd33 |0 http://id.loc.gov/authorities/names/n93055459 | |
| 776 | 0 | 8 | |i Print version: |a Lu, Shuai. |t Regularization Theory for Ill-posed Problems : Selected Topics. |d Berlin : De Gruyter, ©2013 |z 9783110286465 |
| 830 | 0 | |a Inverse and ill-posed problems series. |0 http://id.loc.gov/authorities/names/no95046818 | |
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Record in the Search Index
| DE-BY-FWS_katkey | ZDB-4-EBA-ocn858762149 |
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| author | Lu, Shuai Pereverzev, Sergei V. |
| author_GND | http://id.loc.gov/authorities/names/n93055459 |
| author_facet | Lu, Shuai Pereverzev, Sergei V. |
| author_role | aut aut |
| author_sort | Lu, Shuai |
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| contents | Preface; 1 An introduction using classical examples; 1.1 Numerical differentiation. First look at the problem of regularization. The balancing principle; 1.1.1 Finite-difference formulae; 1.1.2 Finite-difference formulae for nonexact data. A priori choice of the stepsize; 1.1.3 A posteriori choice of the stepsize; 1.1.4 Numerical illustration; 1.1.5 The balancing principle in a general framework; 1.2 Stable summation of orthogonal series with noisy coefficients. Deterministic and stochastic noise models. Description of smoothness properties; 1.2.1 Summation methods. 1.2.2 Deterministic noise model1.2.3 Stochastic noise model; 1.2.4 Smoothness associated with a basis; 1.2.5 Approximation and stability properties of -methods; 1.2.6 Error bounds; 1.3 The elliptic Cauchy problem and regularization by discretization; 1.3.1 Natural linearization of the elliptic Cauchy problem; 1.3.2 Regularization by discretization; 1.3.3 Application in detecting corrosion; 2 Basics of single parameter regularization schemes; 2.1 Simple example for motivation; 2.2 Essentially ill-posed linear operator equations. Least-squares solution. General view on regularization. 2.3 Smoothness in the context of the problem. Benchmark accuracy levels for deterministic and stochastic data noise models2.3.1 The best possible accuracy for the deterministic noise model; 2.3.2 The best possible accuracy for the Gaussian white noise model; 2.4 Optimal order and the saturation of regularization methods in Hilbert spaces; 2.5 Changing the penalty term for variance reduction. Regularization in Hilbert scales; 2.6 Estimation of linear functionals from indirect noisy observations; 2.7 Regularization by finite-dimensional approximation. 2.8 Model selection based on indirect observation in Gaussian white noise2.8.1 Linear models given by least-squares methods; 2.8.2 Operator monotone functions; 2.8.3 The problem of model selection (continuation); 2.9 A warning example: an operator equation formulation is not always adequate (numerical differentiation revisited); 2.9.1 Numerical differentiation in variable Hilbert scales associated with designs; 2.9.2 Error bounds in L2; 2.9.3 Adaptation to the unknown bound of the approximation error; 2.9.4 Numerical differentiation in the space of continuous functions. 2.9.5 Relation to the Savitzky-Golay method. Numerical examples3 Multiparameter regularization; 3.1 When do we really need multiparameter regularization?; 3.2 Multiparameter discrepancy principle; 3.2.1 Model function based on the multiparameter discrepancy principle; 3.2.2 A use of the model function to approximate one set of parameters satisfying the discrepancy principle; 3.2.3 Properties of the model function approximation; 3.2.4 Discrepancy curve and the convergence analysis; 3.2.5 Heuristic algorithm for the model function approximation of the multiparameter discrepancy principle. |
| ctrlnum | (OCoLC)858762149 |
| dewey-full | 518.53 518/.53 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.53 518/.53 |
| dewey-search | 518.53 518/.53 |
| dewey-sort | 3518.53 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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KG,</subfield><subfield code="c">[2013]</subfield></datafield><datafield tag="264" ind1=" " ind2="4"><subfield code="c">©2013</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">1 online resource (xiv, 289 pages) :</subfield><subfield code="b">illustrations</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">computer</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">online resource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Inverse and Ill-Posed Problems Series</subfield></datafield><datafield tag="504" ind1=" " ind2=" "><subfield code="a">Includes bibliographical references and index.</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">Preface; 1 An introduction using classical examples; 1.1 Numerical differentiation. First look at the problem of regularization. The balancing principle; 1.1.1 Finite-difference formulae; 1.1.2 Finite-difference formulae for nonexact data. A priori choice of the stepsize; 1.1.3 A posteriori choice of the stepsize; 1.1.4 Numerical illustration; 1.1.5 The balancing principle in a general framework; 1.2 Stable summation of orthogonal series with noisy coefficients. Deterministic and stochastic noise models. 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Benchmark accuracy levels for deterministic and stochastic data noise models2.3.1 The best possible accuracy for the deterministic noise model; 2.3.2 The best possible accuracy for the Gaussian white noise model; 2.4 Optimal order and the saturation of regularization methods in Hilbert spaces; 2.5 Changing the penalty term for variance reduction. Regularization in Hilbert scales; 2.6 Estimation of linear functionals from indirect noisy observations; 2.7 Regularization by finite-dimensional approximation.</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">2.8 Model selection based on indirect observation in Gaussian white noise2.8.1 Linear models given by least-squares methods; 2.8.2 Operator monotone functions; 2.8.3 The problem of model selection (continuation); 2.9 A warning example: an operator equation formulation is not always adequate (numerical differentiation revisited); 2.9.1 Numerical differentiation in variable Hilbert scales associated with designs; 2.9.2 Error bounds in L2; 2.9.3 Adaptation to the unknown bound of the approximation error; 2.9.4 Numerical differentiation in the space of continuous functions.</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">2.9.5 Relation to the Savitzky-Golay method. 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| genre | Electronic books. |
| genre_facet | Electronic books. |
| id | ZDB-4-EBA-ocn858762149 |
| illustrated | Illustrated |
| indexdate | 2025-04-11T08:41:35Z |
| institution | BVB |
| isbn | 9783110286496 3110286491 |
| language | English |
| lccn | 2013014716 |
| oclc_num | 858762149 |
| open_access_boolean | |
| owner | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| physical | 1 online resource (xiv, 289 pages) : illustrations |
| psigel | ZDB-4-EBA FWS_PDA_EBA ZDB-4-EBA |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Walter de Gruyter GmbH & Co. KG, |
| record_format | marc |
| series | Inverse and ill-posed problems series. |
| series2 | Inverse and Ill-Posed Problems Series |
| spelling | Lu, Shuai, author. Regularization theory for ill-posed problems : selected topics / by Shuai Lu, Sergei V. Pereverzev. Berlin ; Boston : Walter de Gruyter GmbH & Co. KG, [2013] ©2013 1 online resource (xiv, 289 pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier Inverse and Ill-Posed Problems Series Includes bibliographical references and index. Preface; 1 An introduction using classical examples; 1.1 Numerical differentiation. First look at the problem of regularization. The balancing principle; 1.1.1 Finite-difference formulae; 1.1.2 Finite-difference formulae for nonexact data. A priori choice of the stepsize; 1.1.3 A posteriori choice of the stepsize; 1.1.4 Numerical illustration; 1.1.5 The balancing principle in a general framework; 1.2 Stable summation of orthogonal series with noisy coefficients. Deterministic and stochastic noise models. Description of smoothness properties; 1.2.1 Summation methods. 1.2.2 Deterministic noise model1.2.3 Stochastic noise model; 1.2.4 Smoothness associated with a basis; 1.2.5 Approximation and stability properties of -methods; 1.2.6 Error bounds; 1.3 The elliptic Cauchy problem and regularization by discretization; 1.3.1 Natural linearization of the elliptic Cauchy problem; 1.3.2 Regularization by discretization; 1.3.3 Application in detecting corrosion; 2 Basics of single parameter regularization schemes; 2.1 Simple example for motivation; 2.2 Essentially ill-posed linear operator equations. Least-squares solution. General view on regularization. 2.3 Smoothness in the context of the problem. Benchmark accuracy levels for deterministic and stochastic data noise models2.3.1 The best possible accuracy for the deterministic noise model; 2.3.2 The best possible accuracy for the Gaussian white noise model; 2.4 Optimal order and the saturation of regularization methods in Hilbert spaces; 2.5 Changing the penalty term for variance reduction. Regularization in Hilbert scales; 2.6 Estimation of linear functionals from indirect noisy observations; 2.7 Regularization by finite-dimensional approximation. 2.8 Model selection based on indirect observation in Gaussian white noise2.8.1 Linear models given by least-squares methods; 2.8.2 Operator monotone functions; 2.8.3 The problem of model selection (continuation); 2.9 A warning example: an operator equation formulation is not always adequate (numerical differentiation revisited); 2.9.1 Numerical differentiation in variable Hilbert scales associated with designs; 2.9.2 Error bounds in L2; 2.9.3 Adaptation to the unknown bound of the approximation error; 2.9.4 Numerical differentiation in the space of continuous functions. 2.9.5 Relation to the Savitzky-Golay method. Numerical examples3 Multiparameter regularization; 3.1 When do we really need multiparameter regularization?; 3.2 Multiparameter discrepancy principle; 3.2.1 Model function based on the multiparameter discrepancy principle; 3.2.2 A use of the model function to approximate one set of parameters satisfying the discrepancy principle; 3.2.3 Properties of the model function approximation; 3.2.4 Discrepancy curve and the convergence analysis; 3.2.5 Heuristic algorithm for the model function approximation of the multiparameter discrepancy principle. 3.2.6 Generalization in the case of more than two regularization parameters. Thismonograph is a valuable contribution to thehighly topical and extremly productive field ofregularisationmethods for inverse and ill-posed problems. The author is an internationally outstanding and acceptedmathematicianin this field. In his book he offers a well-balanced mixtureof basic and innovative aspects. He demonstrates new, differentiatedviewpoints, and important examples for applications. The bookdemontrates thecurrent developments inthe field of regularization theory, such as multiparameter regularization and regularization in learning theory. The book is written for graduate and PhDs. Print version record. English. Numerical differentiation. http://id.loc.gov/authorities/subjects/sh85093243 Numerical analysis Improperly posed problems. http://id.loc.gov/authorities/subjects/sh85093240 Dérivation numérique. Analyse numérique Problèmes mal posés. MATHEMATICS Numerical Analysis. bisacsh Numerical analysis Improperly posed problems fast Numerical differentiation fast Electronic books. Pereverzev, Sergei V., author. https://id.oclc.org/worldcat/entity/E39PCjxc6ywK48gkrGtxTrhd33 http://id.loc.gov/authorities/names/n93055459 Print version: Lu, Shuai. Regularization Theory for Ill-posed Problems : Selected Topics. Berlin : De Gruyter, ©2013 9783110286465 Inverse and ill-posed problems series. http://id.loc.gov/authorities/names/no95046818 |
| spellingShingle | Lu, Shuai Pereverzev, Sergei V. Regularization theory for ill-posed problems : selected topics / Inverse and ill-posed problems series. Preface; 1 An introduction using classical examples; 1.1 Numerical differentiation. First look at the problem of regularization. The balancing principle; 1.1.1 Finite-difference formulae; 1.1.2 Finite-difference formulae for nonexact data. A priori choice of the stepsize; 1.1.3 A posteriori choice of the stepsize; 1.1.4 Numerical illustration; 1.1.5 The balancing principle in a general framework; 1.2 Stable summation of orthogonal series with noisy coefficients. Deterministic and stochastic noise models. Description of smoothness properties; 1.2.1 Summation methods. 1.2.2 Deterministic noise model1.2.3 Stochastic noise model; 1.2.4 Smoothness associated with a basis; 1.2.5 Approximation and stability properties of -methods; 1.2.6 Error bounds; 1.3 The elliptic Cauchy problem and regularization by discretization; 1.3.1 Natural linearization of the elliptic Cauchy problem; 1.3.2 Regularization by discretization; 1.3.3 Application in detecting corrosion; 2 Basics of single parameter regularization schemes; 2.1 Simple example for motivation; 2.2 Essentially ill-posed linear operator equations. Least-squares solution. General view on regularization. 2.3 Smoothness in the context of the problem. Benchmark accuracy levels for deterministic and stochastic data noise models2.3.1 The best possible accuracy for the deterministic noise model; 2.3.2 The best possible accuracy for the Gaussian white noise model; 2.4 Optimal order and the saturation of regularization methods in Hilbert spaces; 2.5 Changing the penalty term for variance reduction. Regularization in Hilbert scales; 2.6 Estimation of linear functionals from indirect noisy observations; 2.7 Regularization by finite-dimensional approximation. 2.8 Model selection based on indirect observation in Gaussian white noise2.8.1 Linear models given by least-squares methods; 2.8.2 Operator monotone functions; 2.8.3 The problem of model selection (continuation); 2.9 A warning example: an operator equation formulation is not always adequate (numerical differentiation revisited); 2.9.1 Numerical differentiation in variable Hilbert scales associated with designs; 2.9.2 Error bounds in L2; 2.9.3 Adaptation to the unknown bound of the approximation error; 2.9.4 Numerical differentiation in the space of continuous functions. 2.9.5 Relation to the Savitzky-Golay method. Numerical examples3 Multiparameter regularization; 3.1 When do we really need multiparameter regularization?; 3.2 Multiparameter discrepancy principle; 3.2.1 Model function based on the multiparameter discrepancy principle; 3.2.2 A use of the model function to approximate one set of parameters satisfying the discrepancy principle; 3.2.3 Properties of the model function approximation; 3.2.4 Discrepancy curve and the convergence analysis; 3.2.5 Heuristic algorithm for the model function approximation of the multiparameter discrepancy principle. Numerical differentiation. http://id.loc.gov/authorities/subjects/sh85093243 Numerical analysis Improperly posed problems. http://id.loc.gov/authorities/subjects/sh85093240 Dérivation numérique. Analyse numérique Problèmes mal posés. MATHEMATICS Numerical Analysis. bisacsh Numerical analysis Improperly posed problems fast Numerical differentiation fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85093243 http://id.loc.gov/authorities/subjects/sh85093240 |
| title | Regularization theory for ill-posed problems : selected topics / |
| title_auth | Regularization theory for ill-posed problems : selected topics / |
| title_exact_search | Regularization theory for ill-posed problems : selected topics / |
| title_full | Regularization theory for ill-posed problems : selected topics / by Shuai Lu, Sergei V. Pereverzev. |
| title_fullStr | Regularization theory for ill-posed problems : selected topics / by Shuai Lu, Sergei V. Pereverzev. |
| title_full_unstemmed | Regularization theory for ill-posed problems : selected topics / by Shuai Lu, Sergei V. Pereverzev. |
| title_short | Regularization theory for ill-posed problems : |
| title_sort | regularization theory for ill posed problems selected topics |
| title_sub | selected topics / |
| topic | Numerical differentiation. http://id.loc.gov/authorities/subjects/sh85093243 Numerical analysis Improperly posed problems. http://id.loc.gov/authorities/subjects/sh85093240 Dérivation numérique. Analyse numérique Problèmes mal posés. MATHEMATICS Numerical Analysis. bisacsh Numerical analysis Improperly posed problems fast Numerical differentiation fast |
| topic_facet | Numerical differentiation. Numerical analysis Improperly posed problems. Dérivation numérique. Analyse numérique Problèmes mal posés. MATHEMATICS Numerical Analysis. Numerical analysis Improperly posed problems Numerical differentiation Electronic books. |
| work_keys_str_mv | AT lushuai regularizationtheoryforillposedproblemsselectedtopics AT pereverzevsergeiv regularizationtheoryforillposedproblemsselectedtopics |