Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity
Abstract In piezoelectric applications, especially when the devices are used as actuators, the piezoelectric materials are driven under large signals which cause a nonlinear behavior. One way to model the nonlinearities is by functional dependencies of the material parameters on the electric field s...
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| Autor*in: |
Lahmer, T. [verfasserIn] |
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| Format: |
Artikel |
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| Erschienen: |
2009 |
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| Anmerkung: |
© de Gruyter 2009 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of inverse and ill-posed problems - Walter de Gruyter GmbH & Co. KG, 1993, 17(2009), 6 vom: Aug., Seite 585-593 |
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| Übergeordnetes Werk: |
volume:17 ; year:2009 ; number:6 ; month:08 ; pages:585-593 |
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| DOI / URN: |
10.1515/JIIP.2009.036 |
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OLC2137266148 |
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| 520 | |a Abstract In piezoelectric applications, especially when the devices are used as actuators, the piezoelectric materials are driven under large signals which cause a nonlinear behavior. One way to model the nonlinearities is by functional dependencies of the material parameters on the electric field strength or the mechanical strain, respectively. The focus lies in the inverse problem, namely the identification of the parameter curves by appropriate measurements of charge signals over time. The problem is assumed to be ill-posed, since in general measured data are contaminated with noise. The solution process requires regularizing methods where modified Landweber iterations are in the focus. Implementations of modified Landweber iterations, namely the steepest descent and minimal error method can be shown to perform much faster than classical Landweber iterations due to the flexible handling of the relaxation parameter. In our application, parameter curve identification in nonlinear piezoelectricity, the sought-for quantities require to be discretized. Therefore, an iterative multilevel algorithm as proposed by Scherzer [Numer. Math.: 579–600, 1998] is investigated where the iterations begin with coarse discretizations of the parameter curves profitting from the inherent regularization property of coarse discretization. At an advanced state of the iterations the algorithm switches according to an inner discrepancy principle to finer levels of discretization. By this, a sufficient smooth resolution of the sought-for quantities can be achieved. Convergence results and the regularizing property of such an iterative multilevel algorithm are proven. Numerical identification results are presented at the end of this article. | ||
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10.1515/JIIP.2009.036 doi (DE-627)OLC2137266148 (DE-B1597)JIIP.2009.036-p DE-627 ger DE-627 rakwb 510 VZ 510 VZ 11 ssgn Lahmer, T. verfasserin aut Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity 2009 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © de Gruyter 2009 Abstract In piezoelectric applications, especially when the devices are used as actuators, the piezoelectric materials are driven under large signals which cause a nonlinear behavior. One way to model the nonlinearities is by functional dependencies of the material parameters on the electric field strength or the mechanical strain, respectively. The focus lies in the inverse problem, namely the identification of the parameter curves by appropriate measurements of charge signals over time. The problem is assumed to be ill-posed, since in general measured data are contaminated with noise. The solution process requires regularizing methods where modified Landweber iterations are in the focus. Implementations of modified Landweber iterations, namely the steepest descent and minimal error method can be shown to perform much faster than classical Landweber iterations due to the flexible handling of the relaxation parameter. In our application, parameter curve identification in nonlinear piezoelectricity, the sought-for quantities require to be discretized. Therefore, an iterative multilevel algorithm as proposed by Scherzer [Numer. Math.: 579–600, 1998] is investigated where the iterations begin with coarse discretizations of the parameter curves profitting from the inherent regularization property of coarse discretization. At an advanced state of the iterations the algorithm switches according to an inner discrepancy principle to finer levels of discretization. By this, a sufficient smooth resolution of the sought-for quantities can be achieved. Convergence results and the regularizing property of such an iterative multilevel algorithm are proven. Numerical identification results are presented at the end of this article. Enthalten in Journal of inverse and ill-posed problems Walter de Gruyter GmbH & Co. KG, 1993 17(2009), 6 vom: Aug., Seite 585-593 (DE-627)165676728 (DE-600)1160989-8 (DE-576)04851134X 0928-0219 nnns volume:17 year:2009 number:6 month:08 pages:585-593 https://doi.org/10.1515/JIIP.2009.036 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_60 GBV_ILN_70 GBV_ILN_134 GBV_ILN_267 GBV_ILN_2020 GBV_ILN_4277 AR 17 2009 6 08 585-593 |
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10.1515/JIIP.2009.036 doi (DE-627)OLC2137266148 (DE-B1597)JIIP.2009.036-p DE-627 ger DE-627 rakwb 510 VZ 510 VZ 11 ssgn Lahmer, T. verfasserin aut Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity 2009 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © de Gruyter 2009 Abstract In piezoelectric applications, especially when the devices are used as actuators, the piezoelectric materials are driven under large signals which cause a nonlinear behavior. One way to model the nonlinearities is by functional dependencies of the material parameters on the electric field strength or the mechanical strain, respectively. The focus lies in the inverse problem, namely the identification of the parameter curves by appropriate measurements of charge signals over time. The problem is assumed to be ill-posed, since in general measured data are contaminated with noise. The solution process requires regularizing methods where modified Landweber iterations are in the focus. Implementations of modified Landweber iterations, namely the steepest descent and minimal error method can be shown to perform much faster than classical Landweber iterations due to the flexible handling of the relaxation parameter. In our application, parameter curve identification in nonlinear piezoelectricity, the sought-for quantities require to be discretized. Therefore, an iterative multilevel algorithm as proposed by Scherzer [Numer. Math.: 579–600, 1998] is investigated where the iterations begin with coarse discretizations of the parameter curves profitting from the inherent regularization property of coarse discretization. At an advanced state of the iterations the algorithm switches according to an inner discrepancy principle to finer levels of discretization. By this, a sufficient smooth resolution of the sought-for quantities can be achieved. Convergence results and the regularizing property of such an iterative multilevel algorithm are proven. Numerical identification results are presented at the end of this article. Enthalten in Journal of inverse and ill-posed problems Walter de Gruyter GmbH & Co. KG, 1993 17(2009), 6 vom: Aug., Seite 585-593 (DE-627)165676728 (DE-600)1160989-8 (DE-576)04851134X 0928-0219 nnns volume:17 year:2009 number:6 month:08 pages:585-593 https://doi.org/10.1515/JIIP.2009.036 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_60 GBV_ILN_70 GBV_ILN_134 GBV_ILN_267 GBV_ILN_2020 GBV_ILN_4277 AR 17 2009 6 08 585-593 |
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10.1515/JIIP.2009.036 doi (DE-627)OLC2137266148 (DE-B1597)JIIP.2009.036-p DE-627 ger DE-627 rakwb 510 VZ 510 VZ 11 ssgn Lahmer, T. verfasserin aut Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity 2009 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © de Gruyter 2009 Abstract In piezoelectric applications, especially when the devices are used as actuators, the piezoelectric materials are driven under large signals which cause a nonlinear behavior. One way to model the nonlinearities is by functional dependencies of the material parameters on the electric field strength or the mechanical strain, respectively. The focus lies in the inverse problem, namely the identification of the parameter curves by appropriate measurements of charge signals over time. The problem is assumed to be ill-posed, since in general measured data are contaminated with noise. The solution process requires regularizing methods where modified Landweber iterations are in the focus. Implementations of modified Landweber iterations, namely the steepest descent and minimal error method can be shown to perform much faster than classical Landweber iterations due to the flexible handling of the relaxation parameter. In our application, parameter curve identification in nonlinear piezoelectricity, the sought-for quantities require to be discretized. Therefore, an iterative multilevel algorithm as proposed by Scherzer [Numer. Math.: 579–600, 1998] is investigated where the iterations begin with coarse discretizations of the parameter curves profitting from the inherent regularization property of coarse discretization. At an advanced state of the iterations the algorithm switches according to an inner discrepancy principle to finer levels of discretization. By this, a sufficient smooth resolution of the sought-for quantities can be achieved. Convergence results and the regularizing property of such an iterative multilevel algorithm are proven. Numerical identification results are presented at the end of this article. Enthalten in Journal of inverse and ill-posed problems Walter de Gruyter GmbH & Co. KG, 1993 17(2009), 6 vom: Aug., Seite 585-593 (DE-627)165676728 (DE-600)1160989-8 (DE-576)04851134X 0928-0219 nnns volume:17 year:2009 number:6 month:08 pages:585-593 https://doi.org/10.1515/JIIP.2009.036 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_60 GBV_ILN_70 GBV_ILN_134 GBV_ILN_267 GBV_ILN_2020 GBV_ILN_4277 AR 17 2009 6 08 585-593 |
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10.1515/JIIP.2009.036 doi (DE-627)OLC2137266148 (DE-B1597)JIIP.2009.036-p DE-627 ger DE-627 rakwb 510 VZ 510 VZ 11 ssgn Lahmer, T. verfasserin aut Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity 2009 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © de Gruyter 2009 Abstract In piezoelectric applications, especially when the devices are used as actuators, the piezoelectric materials are driven under large signals which cause a nonlinear behavior. One way to model the nonlinearities is by functional dependencies of the material parameters on the electric field strength or the mechanical strain, respectively. The focus lies in the inverse problem, namely the identification of the parameter curves by appropriate measurements of charge signals over time. The problem is assumed to be ill-posed, since in general measured data are contaminated with noise. The solution process requires regularizing methods where modified Landweber iterations are in the focus. Implementations of modified Landweber iterations, namely the steepest descent and minimal error method can be shown to perform much faster than classical Landweber iterations due to the flexible handling of the relaxation parameter. In our application, parameter curve identification in nonlinear piezoelectricity, the sought-for quantities require to be discretized. Therefore, an iterative multilevel algorithm as proposed by Scherzer [Numer. Math.: 579–600, 1998] is investigated where the iterations begin with coarse discretizations of the parameter curves profitting from the inherent regularization property of coarse discretization. At an advanced state of the iterations the algorithm switches according to an inner discrepancy principle to finer levels of discretization. By this, a sufficient smooth resolution of the sought-for quantities can be achieved. Convergence results and the regularizing property of such an iterative multilevel algorithm are proven. Numerical identification results are presented at the end of this article. Enthalten in Journal of inverse and ill-posed problems Walter de Gruyter GmbH & Co. KG, 1993 17(2009), 6 vom: Aug., Seite 585-593 (DE-627)165676728 (DE-600)1160989-8 (DE-576)04851134X 0928-0219 nnns volume:17 year:2009 number:6 month:08 pages:585-593 https://doi.org/10.1515/JIIP.2009.036 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_60 GBV_ILN_70 GBV_ILN_134 GBV_ILN_267 GBV_ILN_2020 GBV_ILN_4277 AR 17 2009 6 08 585-593 |
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10.1515/JIIP.2009.036 doi (DE-627)OLC2137266148 (DE-B1597)JIIP.2009.036-p DE-627 ger DE-627 rakwb 510 VZ 510 VZ 11 ssgn Lahmer, T. verfasserin aut Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity 2009 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © de Gruyter 2009 Abstract In piezoelectric applications, especially when the devices are used as actuators, the piezoelectric materials are driven under large signals which cause a nonlinear behavior. One way to model the nonlinearities is by functional dependencies of the material parameters on the electric field strength or the mechanical strain, respectively. The focus lies in the inverse problem, namely the identification of the parameter curves by appropriate measurements of charge signals over time. The problem is assumed to be ill-posed, since in general measured data are contaminated with noise. The solution process requires regularizing methods where modified Landweber iterations are in the focus. Implementations of modified Landweber iterations, namely the steepest descent and minimal error method can be shown to perform much faster than classical Landweber iterations due to the flexible handling of the relaxation parameter. In our application, parameter curve identification in nonlinear piezoelectricity, the sought-for quantities require to be discretized. Therefore, an iterative multilevel algorithm as proposed by Scherzer [Numer. Math.: 579–600, 1998] is investigated where the iterations begin with coarse discretizations of the parameter curves profitting from the inherent regularization property of coarse discretization. At an advanced state of the iterations the algorithm switches according to an inner discrepancy principle to finer levels of discretization. By this, a sufficient smooth resolution of the sought-for quantities can be achieved. Convergence results and the regularizing property of such an iterative multilevel algorithm are proven. Numerical identification results are presented at the end of this article. Enthalten in Journal of inverse and ill-posed problems Walter de Gruyter GmbH & Co. KG, 1993 17(2009), 6 vom: Aug., Seite 585-593 (DE-627)165676728 (DE-600)1160989-8 (DE-576)04851134X 0928-0219 nnns volume:17 year:2009 number:6 month:08 pages:585-593 https://doi.org/10.1515/JIIP.2009.036 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_60 GBV_ILN_70 GBV_ILN_134 GBV_ILN_267 GBV_ILN_2020 GBV_ILN_4277 AR 17 2009 6 08 585-593 |
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Abstract In piezoelectric applications, especially when the devices are used as actuators, the piezoelectric materials are driven under large signals which cause a nonlinear behavior. One way to model the nonlinearities is by functional dependencies of the material parameters on the electric field strength or the mechanical strain, respectively. The focus lies in the inverse problem, namely the identification of the parameter curves by appropriate measurements of charge signals over time. The problem is assumed to be ill-posed, since in general measured data are contaminated with noise. The solution process requires regularizing methods where modified Landweber iterations are in the focus. Implementations of modified Landweber iterations, namely the steepest descent and minimal error method can be shown to perform much faster than classical Landweber iterations due to the flexible handling of the relaxation parameter. In our application, parameter curve identification in nonlinear piezoelectricity, the sought-for quantities require to be discretized. Therefore, an iterative multilevel algorithm as proposed by Scherzer [Numer. Math.: 579–600, 1998] is investigated where the iterations begin with coarse discretizations of the parameter curves profitting from the inherent regularization property of coarse discretization. At an advanced state of the iterations the algorithm switches according to an inner discrepancy principle to finer levels of discretization. By this, a sufficient smooth resolution of the sought-for quantities can be achieved. Convergence results and the regularizing property of such an iterative multilevel algorithm are proven. Numerical identification results are presented at the end of this article. © de Gruyter 2009 |
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Abstract In piezoelectric applications, especially when the devices are used as actuators, the piezoelectric materials are driven under large signals which cause a nonlinear behavior. One way to model the nonlinearities is by functional dependencies of the material parameters on the electric field strength or the mechanical strain, respectively. The focus lies in the inverse problem, namely the identification of the parameter curves by appropriate measurements of charge signals over time. The problem is assumed to be ill-posed, since in general measured data are contaminated with noise. The solution process requires regularizing methods where modified Landweber iterations are in the focus. Implementations of modified Landweber iterations, namely the steepest descent and minimal error method can be shown to perform much faster than classical Landweber iterations due to the flexible handling of the relaxation parameter. In our application, parameter curve identification in nonlinear piezoelectricity, the sought-for quantities require to be discretized. Therefore, an iterative multilevel algorithm as proposed by Scherzer [Numer. Math.: 579–600, 1998] is investigated where the iterations begin with coarse discretizations of the parameter curves profitting from the inherent regularization property of coarse discretization. At an advanced state of the iterations the algorithm switches according to an inner discrepancy principle to finer levels of discretization. By this, a sufficient smooth resolution of the sought-for quantities can be achieved. Convergence results and the regularizing property of such an iterative multilevel algorithm are proven. Numerical identification results are presented at the end of this article. © de Gruyter 2009 |
| abstract_unstemmed |
Abstract In piezoelectric applications, especially when the devices are used as actuators, the piezoelectric materials are driven under large signals which cause a nonlinear behavior. One way to model the nonlinearities is by functional dependencies of the material parameters on the electric field strength or the mechanical strain, respectively. The focus lies in the inverse problem, namely the identification of the parameter curves by appropriate measurements of charge signals over time. The problem is assumed to be ill-posed, since in general measured data are contaminated with noise. The solution process requires regularizing methods where modified Landweber iterations are in the focus. Implementations of modified Landweber iterations, namely the steepest descent and minimal error method can be shown to perform much faster than classical Landweber iterations due to the flexible handling of the relaxation parameter. In our application, parameter curve identification in nonlinear piezoelectricity, the sought-for quantities require to be discretized. Therefore, an iterative multilevel algorithm as proposed by Scherzer [Numer. Math.: 579–600, 1998] is investigated where the iterations begin with coarse discretizations of the parameter curves profitting from the inherent regularization property of coarse discretization. At an advanced state of the iterations the algorithm switches according to an inner discrepancy principle to finer levels of discretization. By this, a sufficient smooth resolution of the sought-for quantities can be achieved. Convergence results and the regularizing property of such an iterative multilevel algorithm are proven. Numerical identification results are presented at the end of this article. © de Gruyter 2009 |
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