Iterative methods for ill-posed problems :: an introduction /
Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the assoc...
Gespeichert in:
| 1. Verfasser: | |
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| Weitere Verfasser: | , |
| Format: | Elektronisch E-Book |
| Sprache: | English Russian |
| Veröffentlicht: |
Berlin ; New York :
De Gruyter,
©2011.
|
| Schriftenreihe: | Inverse and ill-posed problems series ;
v. 54. |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions. Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces. |
| Beschreibung: | 1 online resource (xi, 136 pages) |
| Bibliographie: | Includes bibliographical references and index. |
| ISBN: | 9783110250657 3110250659 3110250640 9783110250640 |
| ISSN: | 1381-4524 ; |
Internformat
MARC
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| 100 | 1 | |a Bakushinskiĭ, A. B. |q (Anatoliĭ Borisovich) |1 https://id.oclc.org/worldcat/entity/E39PCjCVQ78YftvGfqxgkGxDpX |0 http://id.loc.gov/authorities/names/nr93006047 | |
| 240 | 1 | 0 | |a Iterativnye metody reshenii︠a︡ nekorrektnykh zadach. |l English |
| 245 | 1 | 0 | |a Iterative methods for ill-posed problems : |b an introduction / |c Anatoly B. Bakushinsky, Mikhail Yu. Kokurin, Alexandra Smirnova. |
| 260 | |a Berlin ; |a New York : |b De Gruyter, |c ©2011. | ||
| 300 | |a 1 online resource (xi, 136 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 490 | 1 | |a Inverse and ill-posed problems series, |x 1381-4524 ; |v 54 | |
| 504 | |a Includes bibliographical references and index. | ||
| 588 | 0 | |a Print version record. | |
| 505 | 0 | 0 | |g Machine generated contents note: |g 1. |t The regularity condition. Newton's method -- |g 1.1. |t Preliminary results -- |g 1.2. |t Linearization procedure -- |g 1.3. |t Error analysis -- |t Problems -- |g 2. |t The Gauss -- Newton method -- |g 2.1. |t Motivation -- |g 2.2. |t Convergence rates -- |t Problems -- |g 3. |t The gradient method -- |g 3.1. |t The gradient method for regular problems -- |g 3.2. |t Ill-posed case -- |t Problems -- |g 4. |t Tikhonov's scheme -- |g 4.1. |t The Tikhonov functional -- |g 4.2. |t Properties of a minimizing sequence -- |g 4.3. |t Other types of convergence -- |g 4.4. |t Equations with noisy data -- |t Problems -- |g 5. |t Tikhonov's scheme for linear equations -- |g 5.1. |t The main convergence result -- |g 5.2. |t Elements of spectral theory -- |g 5.3. |t Minimizing sequences for linear equations. |
| 505 | 0 | 0 | |g 5.4. |t A priori agreement between the regularization parameter and the error for equations with perturbed right-hand sides -- |g 5.5. |t The discrepancy principle -- |g 5.6. |t Approximation of a quasi-solution -- |t Problems -- |g 6. |t The gradient scheme for linear equations -- |g 6.1. |t The technique of spectral analysis -- |g 6.2. |t A priori stopping rule -- |g 6.3. |t A posteriori stopping rule -- |t Problems -- |g 7. |t Convergence rates for the approximation methods in the case of linear irregular equations -- |g 7.1. |t The source-type condition (STC) -- |g 7.2. |t STC for the gradient method -- |g 7.3. |t The saturation phenomena -- |g 7.4. |t Approximations in case of a perturbed STC -- |g 7.5. |t Accuracy of the estimates -- |t Problems -- |g 8. |t Equations with a convex discrepancy functional by Tikhonov's method -- |g 8.1. |t Some difficulties associated with Tikhonov's method in case of a convex discrepancy functional. |
| 505 | 0 | 0 | |g 8.2. |t An illustrative example -- |t Problems -- |g 9. |t Iterative regularization principle -- |g 9.1. |t The idea of iterative regularization -- |g 9.2. |t The iteratively regularized gradient method -- |t Problems -- |g 10. |t The iteratively regularized Gauss -- Newton method -- |g 10.1. |t Convergence analysis -- |g 10.2. |t Further properties of IRGN iterations -- |g 10.3. |t A unified approach to the construction of iterative methods for irregular equations -- |g 10.4. |t The reverse connection control -- |t Problems -- |g 11. |t The stable gradient method for irregular nonlinear equations -- |g 11.1. |t Solving an auxiliary finite dimensional problem by the gradient descent method -- |g 11.2. |t Investigation of a difference inequality -- |g 11.3. |t The case of noisy data -- |t Problems -- |g 12. |t Relative computational efficiency of iteratively regularized methods -- |g 12.1. |t Generalized Gauss -- Newton methods -- |g 12.2. |t A more restrictive source condition. |
| 505 | 0 | 0 | |g 12.3. |t Comparison to iteratively regularized gradient scheme -- |t Problems -- |g 13. |t Numerical investigation of two-dimensional inverse gravimetry problem -- |g 13.1. |t Problem formulation -- |g 13.2. |t The algorithm -- |g 13.3. |t Simulations -- |t Problems -- |g 14. |t Iteratively regularized methods for inverse problem in optical tomography -- |g 14.1. |t Statement of the problem -- |g 14.2. |t Simple example -- |g 14.3. |t Forward simulation -- |g 14.4. |t The inverse problem -- |g 14.5. |t Numerical results -- |t Problems -- |g 15. |t Feigenbaum's universality equation -- |g 15.1. |t The universal constants -- |g 15.2. |t Ill-posedness -- |g 15.3. |t Numerical algorithm for 2 & le; z & le; 12 -- |g 15.4. |t Regularized method for z & ge; 13 -- |t Problems -- |g 16. |t Conclusion. |
| 520 | |a Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions. Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces. | ||
| 546 | |a English. | ||
| 650 | 0 | |a Differential equations, Partial |x Improperly posed problems. |0 http://id.loc.gov/authorities/subjects/sh85037914 | |
| 650 | 0 | |a Iterative methods (Mathematics) |0 http://id.loc.gov/authorities/subjects/sh85069058 | |
| 650 | 6 | |a Équations aux dérivées partielles |x Problèmes mal posés. | |
| 650 | 6 | |a Itération (Mathématiques) | |
| 650 | 7 | |a MATHEMATICS |x Differential Equations |x Partial. |2 bisacsh | |
| 650 | 7 | |a Differential equations, Partial |x Improperly posed problems |2 fast | |
| 650 | 7 | |a Iterative methods (Mathematics) |2 fast | |
| 650 | 7 | |a Hilbert-Raum |2 gnd |0 http://d-nb.info/gnd/4159850-7 | |
| 650 | 7 | |a Inkorrekt gestelltes Problem |2 gnd |0 http://d-nb.info/gnd/4186951-5 | |
| 650 | 7 | |a Iteration |2 gnd |0 http://d-nb.info/gnd/4123457-1 | |
| 650 | 7 | |a Operatorgleichung |2 gnd |0 http://d-nb.info/gnd/4043601-9 | |
| 700 | 1 | |a Kokurin, M. I︠U︡. |q (Mikhail I︠U︡rʹevich) |1 https://id.oclc.org/worldcat/entity/E39PCjvMhJRhdRd7HJYjPJprRq |0 http://id.loc.gov/authorities/names/nb2005003123 | |
| 700 | 1 | |a Smirnova, A. B. |q (Aleksandra Borisovna) |1 https://id.oclc.org/worldcat/entity/E39PCjJqVdbRCRbDwYKQF6vQC3 |0 http://id.loc.gov/authorities/names/n2010064084 | |
| 776 | 0 | 8 | |i Print version: |a Bakushinskiĭ, A.B. (Anatoliĭ Borisovich). |s Iterativnye metody reshenii︠a︡ nekorrektnykh zadach. English. |t Iterative methods for ill-posed problems. |d Berlin ; New York : De Gruyter, ©2011 |w (DLC) 2010038154 |
| 830 | 0 | |a Inverse and ill-posed problems series ; |v v. 54. |0 http://id.loc.gov/authorities/names/no95046818 | |
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Datensatz im Suchindex
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| author | Bakushinskiĭ, A. B. (Anatoliĭ Borisovich) |
| author2 | Kokurin, M. I︠U︡. (Mikhail I︠U︡rʹevich) Smirnova, A. B. (Aleksandra Borisovna) |
| author2_role | |
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| author_GND | http://id.loc.gov/authorities/names/nr93006047 http://id.loc.gov/authorities/names/nb2005003123 http://id.loc.gov/authorities/names/n2010064084 |
| author_facet | Bakushinskiĭ, A. B. (Anatoliĭ Borisovich) Kokurin, M. I︠U︡. (Mikhail I︠U︡rʹevich) Smirnova, A. B. (Aleksandra Borisovna) |
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| contents | The regularity condition. Newton's method -- Preliminary results -- Linearization procedure -- Error analysis -- Problems -- The Gauss -- Newton method -- Motivation -- Convergence rates -- The gradient method -- The gradient method for regular problems -- Ill-posed case -- Tikhonov's scheme -- The Tikhonov functional -- Properties of a minimizing sequence -- Other types of convergence -- Equations with noisy data -- Tikhonov's scheme for linear equations -- The main convergence result -- Elements of spectral theory -- Minimizing sequences for linear equations. A priori agreement between the regularization parameter and the error for equations with perturbed right-hand sides -- The discrepancy principle -- Approximation of a quasi-solution -- The gradient scheme for linear equations -- The technique of spectral analysis -- A priori stopping rule -- A posteriori stopping rule -- Convergence rates for the approximation methods in the case of linear irregular equations -- The source-type condition (STC) -- STC for the gradient method -- The saturation phenomena -- Approximations in case of a perturbed STC -- Accuracy of the estimates -- Equations with a convex discrepancy functional by Tikhonov's method -- Some difficulties associated with Tikhonov's method in case of a convex discrepancy functional. An illustrative example -- Iterative regularization principle -- The idea of iterative regularization -- The iteratively regularized gradient method -- The iteratively regularized Gauss -- Newton method -- Convergence analysis -- Further properties of IRGN iterations -- A unified approach to the construction of iterative methods for irregular equations -- The reverse connection control -- The stable gradient method for irregular nonlinear equations -- Solving an auxiliary finite dimensional problem by the gradient descent method -- Investigation of a difference inequality -- The case of noisy data -- Relative computational efficiency of iteratively regularized methods -- Generalized Gauss -- Newton methods -- A more restrictive source condition. Comparison to iteratively regularized gradient scheme -- Numerical investigation of two-dimensional inverse gravimetry problem -- Problem formulation -- The algorithm -- Simulations -- Iteratively regularized methods for inverse problem in optical tomography -- Statement of the problem -- Simple example -- Forward simulation -- The inverse problem -- Numerical results -- Feigenbaum's universality equation -- The universal constants -- Ill-posedness -- Numerical algorithm for 2 & le; z & le; 12 -- Regularized method for z & ge; 13 -- Conclusion. |
| ctrlnum | (OCoLC)732957489 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
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| dewey-search | 515/.353 |
| dewey-sort | 3515 3353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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the error for equations with perturbed right-hand sides --</subfield><subfield code="g">5.5.</subfield><subfield code="t">The discrepancy principle --</subfield><subfield code="g">5.6.</subfield><subfield code="t">Approximation of a quasi-solution --</subfield><subfield code="t">Problems --</subfield><subfield code="g">6.</subfield><subfield code="t">The gradient scheme for linear equations --</subfield><subfield code="g">6.1.</subfield><subfield code="t">The technique of spectral analysis --</subfield><subfield code="g">6.2.</subfield><subfield code="t">A priori stopping rule --</subfield><subfield code="g">6.3.</subfield><subfield code="t">A posteriori stopping rule --</subfield><subfield code="t">Problems --</subfield><subfield code="g">7.</subfield><subfield code="t">Convergence rates for the approximation methods in the case of linear irregular equations --</subfield><subfield code="g">7.1.</subfield><subfield code="t">The source-type condition (STC) --</subfield><subfield code="g">7.2.</subfield><subfield code="t">STC for the gradient method --</subfield><subfield code="g">7.3.</subfield><subfield code="t">The saturation phenomena --</subfield><subfield code="g">7.4.</subfield><subfield code="t">Approximations in case of a perturbed STC --</subfield><subfield code="g">7.5.</subfield><subfield code="t">Accuracy of the estimates --</subfield><subfield code="t">Problems --</subfield><subfield code="g">8.</subfield><subfield code="t">Equations with a convex discrepancy functional by Tikhonov's method --</subfield><subfield code="g">8.1.</subfield><subfield code="t">Some difficulties associated with Tikhonov's method in case of a convex discrepancy functional.</subfield></datafield><datafield tag="505" ind1="0" ind2="0"><subfield code="g">8.2.</subfield><subfield code="t">An illustrative example --</subfield><subfield code="t">Problems --</subfield><subfield code="g">9.</subfield><subfield code="t">Iterative regularization principle --</subfield><subfield code="g">9.1.</subfield><subfield code="t">The idea of iterative regularization --</subfield><subfield code="g">9.2.</subfield><subfield code="t">The iteratively regularized gradient method --</subfield><subfield code="t">Problems --</subfield><subfield code="g">10.</subfield><subfield code="t">The iteratively regularized Gauss -- Newton method --</subfield><subfield code="g">10.1.</subfield><subfield code="t">Convergence analysis --</subfield><subfield code="g">10.2.</subfield><subfield code="t">Further properties of IRGN iterations --</subfield><subfield code="g">10.3.</subfield><subfield code="t">A unified approach to the construction of iterative methods for irregular equations --</subfield><subfield code="g">10.4.</subfield><subfield code="t">The reverse connection control --</subfield><subfield code="t">Problems --</subfield><subfield code="g">11.</subfield><subfield code="t">The stable gradient method for irregular nonlinear equations --</subfield><subfield code="g">11.1.</subfield><subfield code="t">Solving an auxiliary finite dimensional problem by the gradient descent method --</subfield><subfield code="g">11.2.</subfield><subfield code="t">Investigation of a difference inequality --</subfield><subfield code="g">11.3.</subfield><subfield code="t">The case of noisy data --</subfield><subfield code="t">Problems --</subfield><subfield code="g">12.</subfield><subfield code="t">Relative computational efficiency of iteratively regularized methods --</subfield><subfield code="g">12.1.</subfield><subfield code="t">Generalized Gauss -- Newton methods --</subfield><subfield code="g">12.2.</subfield><subfield code="t">A more restrictive source condition.</subfield></datafield><datafield tag="505" ind1="0" ind2="0"><subfield code="g">12.3.</subfield><subfield code="t">Comparison to iteratively regularized gradient scheme --</subfield><subfield code="t">Problems --</subfield><subfield code="g">13.</subfield><subfield code="t">Numerical investigation of two-dimensional inverse gravimetry problem --</subfield><subfield code="g">13.1.</subfield><subfield code="t">Problem formulation --</subfield><subfield code="g">13.2.</subfield><subfield code="t">The algorithm --</subfield><subfield code="g">13.3.</subfield><subfield code="t">Simulations --</subfield><subfield code="t">Problems --</subfield><subfield code="g">14.</subfield><subfield code="t">Iteratively regularized methods for inverse problem in optical tomography --</subfield><subfield code="g">14.1.</subfield><subfield code="t">Statement of the problem --</subfield><subfield code="g">14.2.</subfield><subfield code="t">Simple example --</subfield><subfield code="g">14.3.</subfield><subfield code="t">Forward simulation --</subfield><subfield code="g">14.4.</subfield><subfield code="t">The inverse problem --</subfield><subfield code="g">14.5.</subfield><subfield code="t">Numerical results --</subfield><subfield code="t">Problems --</subfield><subfield code="g">15.</subfield><subfield code="t">Feigenbaum's universality equation --</subfield><subfield code="g">15.1.</subfield><subfield code="t">The universal constants --</subfield><subfield code="g">15.2.</subfield><subfield code="t">Ill-posedness --</subfield><subfield code="g">15.3.</subfield><subfield code="t">Numerical algorithm for 2 & le; z & le; 12 --</subfield><subfield code="g">15.4.</subfield><subfield code="t">Regularized method for z & ge; 13 --</subfield><subfield code="t">Problems --</subfield><subfield code="g">16.</subfield><subfield code="t">Conclusion.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions. Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces.</subfield></datafield><datafield tag="546" ind1=" " ind2=" "><subfield code="a">English.</subfield></datafield><datafield tag="650" ind1=" " ind2="0"><subfield code="a">Differential equations, Partial</subfield><subfield code="x">Improperly posed problems.</subfield><subfield code="0">http://id.loc.gov/authorities/subjects/sh85037914</subfield></datafield><datafield tag="650" ind1=" " ind2="0"><subfield code="a">Iterative methods (Mathematics)</subfield><subfield code="0">http://id.loc.gov/authorities/subjects/sh85069058</subfield></datafield><datafield tag="650" ind1=" " ind2="6"><subfield code="a">Équations aux dérivées partielles</subfield><subfield code="x">Problèmes mal posés.</subfield></datafield><datafield tag="650" ind1=" " ind2="6"><subfield code="a">Itération (Mathématiques)</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">MATHEMATICS</subfield><subfield code="x">Differential Equations</subfield><subfield code="x">Partial.</subfield><subfield code="2">bisacsh</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Differential equations, Partial</subfield><subfield code="x">Improperly posed problems</subfield><subfield code="2">fast</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Iterative methods (Mathematics)</subfield><subfield code="2">fast</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Hilbert-Raum</subfield><subfield code="2">gnd</subfield><subfield code="0">http://d-nb.info/gnd/4159850-7</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Inkorrekt gestelltes Problem</subfield><subfield code="2">gnd</subfield><subfield code="0">http://d-nb.info/gnd/4186951-5</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Iteration</subfield><subfield code="2">gnd</subfield><subfield code="0">http://d-nb.info/gnd/4123457-1</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Operatorgleichung</subfield><subfield code="2">gnd</subfield><subfield code="0">http://d-nb.info/gnd/4043601-9</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Kokurin, M. I︠U︡.</subfield><subfield code="q">(Mikhail I︠U︡rʹevich)</subfield><subfield code="1">https://id.oclc.org/worldcat/entity/E39PCjvMhJRhdRd7HJYjPJprRq</subfield><subfield code="0">http://id.loc.gov/authorities/names/nb2005003123</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Smirnova, A. B.</subfield><subfield code="q">(Aleksandra Borisovna)</subfield><subfield code="1">https://id.oclc.org/worldcat/entity/E39PCjJqVdbRCRbDwYKQF6vQC3</subfield><subfield code="0">http://id.loc.gov/authorities/names/n2010064084</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Print version:</subfield><subfield code="a">Bakushinskiĭ, A.B. (Anatoliĭ Borisovich).</subfield><subfield code="s">Iterativnye metody reshenii︠a︡ nekorrektnykh zadach. 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| id | ZDB-4-EBA-ocn732957489 |
| illustrated | Not Illustrated |
| indexdate | 2024-11-27T13:17:52Z |
| institution | BVB |
| isbn | 9783110250657 3110250659 3110250640 9783110250640 |
| issn | 1381-4524 ; |
| language | English Russian |
| lccn | 2010038154 |
| oclc_num | 732957489 |
| open_access_boolean | |
| owner | MAIN DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (xi, 136 pages) |
| psigel | ZDB-4-EBA |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | De Gruyter, |
| record_format | marc |
| series | Inverse and ill-posed problems series ; |
| series2 | Inverse and ill-posed problems series, |
| spelling | Bakushinskiĭ, A. B. (Anatoliĭ Borisovich) https://id.oclc.org/worldcat/entity/E39PCjCVQ78YftvGfqxgkGxDpX http://id.loc.gov/authorities/names/nr93006047 Iterativnye metody reshenii︠a︡ nekorrektnykh zadach. English Iterative methods for ill-posed problems : an introduction / Anatoly B. Bakushinsky, Mikhail Yu. Kokurin, Alexandra Smirnova. Berlin ; New York : De Gruyter, ©2011. 1 online resource (xi, 136 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier Inverse and ill-posed problems series, 1381-4524 ; 54 Includes bibliographical references and index. Print version record. Machine generated contents note: 1. The regularity condition. Newton's method -- 1.1. Preliminary results -- 1.2. Linearization procedure -- 1.3. Error analysis -- Problems -- 2. The Gauss -- Newton method -- 2.1. Motivation -- 2.2. Convergence rates -- Problems -- 3. The gradient method -- 3.1. The gradient method for regular problems -- 3.2. Ill-posed case -- Problems -- 4. Tikhonov's scheme -- 4.1. The Tikhonov functional -- 4.2. Properties of a minimizing sequence -- 4.3. Other types of convergence -- 4.4. Equations with noisy data -- Problems -- 5. Tikhonov's scheme for linear equations -- 5.1. The main convergence result -- 5.2. Elements of spectral theory -- 5.3. Minimizing sequences for linear equations. 5.4. A priori agreement between the regularization parameter and the error for equations with perturbed right-hand sides -- 5.5. The discrepancy principle -- 5.6. Approximation of a quasi-solution -- Problems -- 6. The gradient scheme for linear equations -- 6.1. The technique of spectral analysis -- 6.2. A priori stopping rule -- 6.3. A posteriori stopping rule -- Problems -- 7. Convergence rates for the approximation methods in the case of linear irregular equations -- 7.1. The source-type condition (STC) -- 7.2. STC for the gradient method -- 7.3. The saturation phenomena -- 7.4. Approximations in case of a perturbed STC -- 7.5. Accuracy of the estimates -- Problems -- 8. Equations with a convex discrepancy functional by Tikhonov's method -- 8.1. Some difficulties associated with Tikhonov's method in case of a convex discrepancy functional. 8.2. An illustrative example -- Problems -- 9. Iterative regularization principle -- 9.1. The idea of iterative regularization -- 9.2. The iteratively regularized gradient method -- Problems -- 10. The iteratively regularized Gauss -- Newton method -- 10.1. Convergence analysis -- 10.2. Further properties of IRGN iterations -- 10.3. A unified approach to the construction of iterative methods for irregular equations -- 10.4. The reverse connection control -- Problems -- 11. The stable gradient method for irregular nonlinear equations -- 11.1. Solving an auxiliary finite dimensional problem by the gradient descent method -- 11.2. Investigation of a difference inequality -- 11.3. The case of noisy data -- Problems -- 12. Relative computational efficiency of iteratively regularized methods -- 12.1. Generalized Gauss -- Newton methods -- 12.2. A more restrictive source condition. 12.3. Comparison to iteratively regularized gradient scheme -- Problems -- 13. Numerical investigation of two-dimensional inverse gravimetry problem -- 13.1. Problem formulation -- 13.2. The algorithm -- 13.3. Simulations -- Problems -- 14. Iteratively regularized methods for inverse problem in optical tomography -- 14.1. Statement of the problem -- 14.2. Simple example -- 14.3. Forward simulation -- 14.4. The inverse problem -- 14.5. Numerical results -- Problems -- 15. Feigenbaum's universality equation -- 15.1. The universal constants -- 15.2. Ill-posedness -- 15.3. Numerical algorithm for 2 & le; z & le; 12 -- 15.4. Regularized method for z & ge; 13 -- Problems -- 16. Conclusion. Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions. Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces. English. Differential equations, Partial Improperly posed problems. http://id.loc.gov/authorities/subjects/sh85037914 Iterative methods (Mathematics) http://id.loc.gov/authorities/subjects/sh85069058 Équations aux dérivées partielles Problèmes mal posés. Itération (Mathématiques) MATHEMATICS Differential Equations Partial. bisacsh Differential equations, Partial Improperly posed problems fast Iterative methods (Mathematics) fast Hilbert-Raum gnd http://d-nb.info/gnd/4159850-7 Inkorrekt gestelltes Problem gnd http://d-nb.info/gnd/4186951-5 Iteration gnd http://d-nb.info/gnd/4123457-1 Operatorgleichung gnd http://d-nb.info/gnd/4043601-9 Kokurin, M. I︠U︡. (Mikhail I︠U︡rʹevich) https://id.oclc.org/worldcat/entity/E39PCjvMhJRhdRd7HJYjPJprRq http://id.loc.gov/authorities/names/nb2005003123 Smirnova, A. B. (Aleksandra Borisovna) https://id.oclc.org/worldcat/entity/E39PCjJqVdbRCRbDwYKQF6vQC3 http://id.loc.gov/authorities/names/n2010064084 Print version: Bakushinskiĭ, A.B. (Anatoliĭ Borisovich). Iterativnye metody reshenii︠a︡ nekorrektnykh zadach. English. Iterative methods for ill-posed problems. Berlin ; New York : De Gruyter, ©2011 (DLC) 2010038154 Inverse and ill-posed problems series ; v. 54. http://id.loc.gov/authorities/names/no95046818 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=388248 Volltext |
| spellingShingle | Bakushinskiĭ, A. B. (Anatoliĭ Borisovich) Iterative methods for ill-posed problems : an introduction / Inverse and ill-posed problems series ; The regularity condition. Newton's method -- Preliminary results -- Linearization procedure -- Error analysis -- Problems -- The Gauss -- Newton method -- Motivation -- Convergence rates -- The gradient method -- The gradient method for regular problems -- Ill-posed case -- Tikhonov's scheme -- The Tikhonov functional -- Properties of a minimizing sequence -- Other types of convergence -- Equations with noisy data -- Tikhonov's scheme for linear equations -- The main convergence result -- Elements of spectral theory -- Minimizing sequences for linear equations. A priori agreement between the regularization parameter and the error for equations with perturbed right-hand sides -- The discrepancy principle -- Approximation of a quasi-solution -- The gradient scheme for linear equations -- The technique of spectral analysis -- A priori stopping rule -- A posteriori stopping rule -- Convergence rates for the approximation methods in the case of linear irregular equations -- The source-type condition (STC) -- STC for the gradient method -- The saturation phenomena -- Approximations in case of a perturbed STC -- Accuracy of the estimates -- Equations with a convex discrepancy functional by Tikhonov's method -- Some difficulties associated with Tikhonov's method in case of a convex discrepancy functional. An illustrative example -- Iterative regularization principle -- The idea of iterative regularization -- The iteratively regularized gradient method -- The iteratively regularized Gauss -- Newton method -- Convergence analysis -- Further properties of IRGN iterations -- A unified approach to the construction of iterative methods for irregular equations -- The reverse connection control -- The stable gradient method for irregular nonlinear equations -- Solving an auxiliary finite dimensional problem by the gradient descent method -- Investigation of a difference inequality -- The case of noisy data -- Relative computational efficiency of iteratively regularized methods -- Generalized Gauss -- Newton methods -- A more restrictive source condition. Comparison to iteratively regularized gradient scheme -- Numerical investigation of two-dimensional inverse gravimetry problem -- Problem formulation -- The algorithm -- Simulations -- Iteratively regularized methods for inverse problem in optical tomography -- Statement of the problem -- Simple example -- Forward simulation -- The inverse problem -- Numerical results -- Feigenbaum's universality equation -- The universal constants -- Ill-posedness -- Numerical algorithm for 2 & le; z & le; 12 -- Regularized method for z & ge; 13 -- Conclusion. Differential equations, Partial Improperly posed problems. http://id.loc.gov/authorities/subjects/sh85037914 Iterative methods (Mathematics) http://id.loc.gov/authorities/subjects/sh85069058 Équations aux dérivées partielles Problèmes mal posés. Itération (Mathématiques) MATHEMATICS Differential Equations Partial. bisacsh Differential equations, Partial Improperly posed problems fast Iterative methods (Mathematics) fast Hilbert-Raum gnd http://d-nb.info/gnd/4159850-7 Inkorrekt gestelltes Problem gnd http://d-nb.info/gnd/4186951-5 Iteration gnd http://d-nb.info/gnd/4123457-1 Operatorgleichung gnd http://d-nb.info/gnd/4043601-9 |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85037914 http://id.loc.gov/authorities/subjects/sh85069058 http://d-nb.info/gnd/4159850-7 http://d-nb.info/gnd/4186951-5 http://d-nb.info/gnd/4123457-1 http://d-nb.info/gnd/4043601-9 |
| title | Iterative methods for ill-posed problems : an introduction / |
| title_alt | Iterativnye metody reshenii︠a︡ nekorrektnykh zadach. The regularity condition. Newton's method -- Preliminary results -- Linearization procedure -- Error analysis -- Problems -- The Gauss -- Newton method -- Motivation -- Convergence rates -- The gradient method -- The gradient method for regular problems -- Ill-posed case -- Tikhonov's scheme -- The Tikhonov functional -- Properties of a minimizing sequence -- Other types of convergence -- Equations with noisy data -- Tikhonov's scheme for linear equations -- The main convergence result -- Elements of spectral theory -- Minimizing sequences for linear equations. A priori agreement between the regularization parameter and the error for equations with perturbed right-hand sides -- The discrepancy principle -- Approximation of a quasi-solution -- The gradient scheme for linear equations -- The technique of spectral analysis -- A priori stopping rule -- A posteriori stopping rule -- Convergence rates for the approximation methods in the case of linear irregular equations -- The source-type condition (STC) -- STC for the gradient method -- The saturation phenomena -- Approximations in case of a perturbed STC -- Accuracy of the estimates -- Equations with a convex discrepancy functional by Tikhonov's method -- Some difficulties associated with Tikhonov's method in case of a convex discrepancy functional. An illustrative example -- Iterative regularization principle -- The idea of iterative regularization -- The iteratively regularized gradient method -- The iteratively regularized Gauss -- Newton method -- Convergence analysis -- Further properties of IRGN iterations -- A unified approach to the construction of iterative methods for irregular equations -- The reverse connection control -- The stable gradient method for irregular nonlinear equations -- Solving an auxiliary finite dimensional problem by the gradient descent method -- Investigation of a difference inequality -- The case of noisy data -- Relative computational efficiency of iteratively regularized methods -- Generalized Gauss -- Newton methods -- A more restrictive source condition. Comparison to iteratively regularized gradient scheme -- Numerical investigation of two-dimensional inverse gravimetry problem -- Problem formulation -- The algorithm -- Simulations -- Iteratively regularized methods for inverse problem in optical tomography -- Statement of the problem -- Simple example -- Forward simulation -- The inverse problem -- Numerical results -- Feigenbaum's universality equation -- The universal constants -- Ill-posedness -- Numerical algorithm for 2 & le; z & le; 12 -- Regularized method for z & ge; 13 -- Conclusion. |
| title_auth | Iterative methods for ill-posed problems : an introduction / |
| title_exact_search | Iterative methods for ill-posed problems : an introduction / |
| title_full | Iterative methods for ill-posed problems : an introduction / Anatoly B. Bakushinsky, Mikhail Yu. Kokurin, Alexandra Smirnova. |
| title_fullStr | Iterative methods for ill-posed problems : an introduction / Anatoly B. Bakushinsky, Mikhail Yu. Kokurin, Alexandra Smirnova. |
| title_full_unstemmed | Iterative methods for ill-posed problems : an introduction / Anatoly B. Bakushinsky, Mikhail Yu. Kokurin, Alexandra Smirnova. |
| title_short | Iterative methods for ill-posed problems : |
| title_sort | iterative methods for ill posed problems an introduction |
| title_sub | an introduction / |
| topic | Differential equations, Partial Improperly posed problems. http://id.loc.gov/authorities/subjects/sh85037914 Iterative methods (Mathematics) http://id.loc.gov/authorities/subjects/sh85069058 Équations aux dérivées partielles Problèmes mal posés. Itération (Mathématiques) MATHEMATICS Differential Equations Partial. bisacsh Differential equations, Partial Improperly posed problems fast Iterative methods (Mathematics) fast Hilbert-Raum gnd http://d-nb.info/gnd/4159850-7 Inkorrekt gestelltes Problem gnd http://d-nb.info/gnd/4186951-5 Iteration gnd http://d-nb.info/gnd/4123457-1 Operatorgleichung gnd http://d-nb.info/gnd/4043601-9 |
| topic_facet | Differential equations, Partial Improperly posed problems. Iterative methods (Mathematics) Équations aux dérivées partielles Problèmes mal posés. Itération (Mathématiques) MATHEMATICS Differential Equations Partial. Differential equations, Partial Improperly posed problems Hilbert-Raum Inkorrekt gestelltes Problem Iteration Operatorgleichung |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=388248 |
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