Regularization Theory for Ill-posed Problems: Selected Topics
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Berlin
De Gruyter
2013
|
| Series: | Inverse and ill-posed problems series
|
| Subjects: | |
| Online Access: | FAW01 FAW02 Volltext |
| Item Description: | Print version record. - 3.2.6 Generalization in the case of more than two regularization parameters |
| Physical Description: | 1 online resource (304 pages) |
| ISBN: | 3110286467 3110286491 9783110286465 9783110286496 |
Staff View
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| 500 | |a Print version record. - 3.2.6 Generalization in the case of more than two regularization parameters | ||
| 505 | 8 | |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 | |
| 505 | 8 | |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 | |
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Record in the Search Index
| _version_ | 1804175398067503104 |
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| any_adam_object | |
| author | Lu, Shuai |
| author_facet | Lu, Shuai |
| author_role | aut |
| author_sort | Lu, Shuai |
| author_variant | s l sl |
| building | Verbundindex |
| bvnumber | BV043037066 |
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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 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 |
| ctrlnum | (OCoLC)858762149 (DE-599)BVBBV043037066 |
| 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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| id | DE-604.BV043037066 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:15:39Z |
| institution | BVB |
| isbn | 3110286467 3110286491 9783110286465 9783110286496 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028461714 |
| oclc_num | 858762149 |
| open_access_boolean | |
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| owner_facet | DE-1046 DE-1047 |
| physical | 1 online resource (304 pages) |
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| publishDate | 2013 |
| publishDateSearch | 2013 |
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| publisher | De Gruyter |
| record_format | marc |
| series2 | Inverse and ill-posed problems series |
| spelling | Lu, Shuai Verfasser aut Regularization Theory for Ill-posed Problems Selected Topics Berlin De Gruyter 2013 1 online resource (304 pages) txt rdacontent c rdamedia cr rdacarrier Inverse and ill-posed problems series Print version record. - 3.2.6 Generalization in the case of more than two regularization parameters 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 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 Numerical analysis / Improperly posed problems MATHEMATICS / Numerical Analysis bisacsh Numerical analysis / Improperly posed problems fast Numerical differentiation fast Numerical differentiation Numerical analysis Improperly posed problems Inkorrekt gestelltes Problem (DE-588)4186951-5 gnd rswk-swf Regularisierungsverfahren (DE-588)4846428-4 gnd rswk-swf Inverses Problem (DE-588)4125161-1 gnd rswk-swf Inverses Problem (DE-588)4125161-1 s Inkorrekt gestelltes Problem (DE-588)4186951-5 s Regularisierungsverfahren (DE-588)4846428-4 s 1\p DE-604 Pereverzev, Sergei V. Sonstige oth Erscheint auch als Druck-Ausgabe Lu, Shuai Regularization Theory for Ill-posed Problems : Selected Topics http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=641765 Aggregator Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lu, Shuai Regularization Theory for Ill-posed Problems Selected Topics 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 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 Numerical analysis / Improperly posed problems MATHEMATICS / Numerical Analysis bisacsh Numerical analysis / Improperly posed problems fast Numerical differentiation fast Numerical differentiation Numerical analysis Improperly posed problems Inkorrekt gestelltes Problem (DE-588)4186951-5 gnd Regularisierungsverfahren (DE-588)4846428-4 gnd Inverses Problem (DE-588)4125161-1 gnd |
| subject_GND | (DE-588)4186951-5 (DE-588)4846428-4 (DE-588)4125161-1 |
| 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 |
| title_fullStr | Regularization Theory for Ill-posed Problems Selected Topics |
| title_full_unstemmed | Regularization Theory for Ill-posed Problems Selected Topics |
| title_short | Regularization Theory for Ill-posed Problems |
| title_sort | regularization theory for ill posed problems selected topics |
| title_sub | Selected Topics |
| topic | Numerical analysis / Improperly posed problems MATHEMATICS / Numerical Analysis bisacsh Numerical analysis / Improperly posed problems fast Numerical differentiation fast Numerical differentiation Numerical analysis Improperly posed problems Inkorrekt gestelltes Problem (DE-588)4186951-5 gnd Regularisierungsverfahren (DE-588)4846428-4 gnd Inverses Problem (DE-588)4125161-1 gnd |
| topic_facet | Numerical analysis / Improperly posed problems MATHEMATICS / Numerical Analysis Numerical differentiation Numerical analysis Improperly posed problems Inkorrekt gestelltes Problem Regularisierungsverfahren Inverses Problem |
| url | http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=641765 |
| work_keys_str_mv | AT lushuai regularizationtheoryforillposedproblemsselectedtopics AT pereverzevsergeiv regularizationtheoryforillposedproblemsselectedtopics |