Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity
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 o...
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Lahmer, T. [verfasserIn] |
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Walter de Gruyter GmbH & Co. KG ; 2009 |
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© de Gruyter 2009 |
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Walter de Gruyter Online Zeitschriften |
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Enthalten in: Journal of inverse and ill-posed problems - Berlin : de Gruyter, 1993, 17(2009), 6 vom: 19. Aug., Seite 585-593 |
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volume:17 ; year:2009 ; number:6 ; day:19 ; month:08 ; pages:585-593 ; extent:9 |
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| DOI / URN: |
10.1515/JIIP.2009.036 |
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| 520 | |a 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 artikel_Grundlieferung.pp (DE-627)NLEJ247090298 DE-627 ger DE-627 rakwb Lahmer, T. verfasserin aut Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity Walter de Gruyter GmbH & Co. KG 2009 9 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © de Gruyter 2009 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. Walter de Gruyter Online Zeitschriften Inverse problem piezoelectricity multilevel modified Landweber Enthalten in Journal of inverse and ill-posed problems Berlin : de Gruyter, 1993 17(2009), 6 vom: 19. Aug., Seite 585-593 (DE-627)NLEJ248236091 (DE-600)2041913-2 1569-3945 nnns volume:17 year:2009 number:6 day:19 month:08 pages:585-593 extent:9 https://doi.org/10.1515/JIIP.2009.036 Deutschlandweit zugänglich GBV_USEFLAG_U ZDB-1-DGR GBV_NL_ARTICLE AR 17 2009 6 19 08 585-593 9 |
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10.1515/JIIP.2009.036 doi artikel_Grundlieferung.pp (DE-627)NLEJ247090298 DE-627 ger DE-627 rakwb Lahmer, T. verfasserin aut Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity Walter de Gruyter GmbH & Co. KG 2009 9 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © de Gruyter 2009 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. Walter de Gruyter Online Zeitschriften Inverse problem piezoelectricity multilevel modified Landweber Enthalten in Journal of inverse and ill-posed problems Berlin : de Gruyter, 1993 17(2009), 6 vom: 19. Aug., Seite 585-593 (DE-627)NLEJ248236091 (DE-600)2041913-2 1569-3945 nnns volume:17 year:2009 number:6 day:19 month:08 pages:585-593 extent:9 https://doi.org/10.1515/JIIP.2009.036 Deutschlandweit zugänglich GBV_USEFLAG_U ZDB-1-DGR GBV_NL_ARTICLE AR 17 2009 6 19 08 585-593 9 |
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10.1515/JIIP.2009.036 doi artikel_Grundlieferung.pp (DE-627)NLEJ247090298 DE-627 ger DE-627 rakwb Lahmer, T. verfasserin aut Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity Walter de Gruyter GmbH & Co. KG 2009 9 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © de Gruyter 2009 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. Walter de Gruyter Online Zeitschriften Inverse problem piezoelectricity multilevel modified Landweber Enthalten in Journal of inverse and ill-posed problems Berlin : de Gruyter, 1993 17(2009), 6 vom: 19. Aug., Seite 585-593 (DE-627)NLEJ248236091 (DE-600)2041913-2 1569-3945 nnns volume:17 year:2009 number:6 day:19 month:08 pages:585-593 extent:9 https://doi.org/10.1515/JIIP.2009.036 Deutschlandweit zugänglich GBV_USEFLAG_U ZDB-1-DGR GBV_NL_ARTICLE AR 17 2009 6 19 08 585-593 9 |
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10.1515/JIIP.2009.036 doi artikel_Grundlieferung.pp (DE-627)NLEJ247090298 DE-627 ger DE-627 rakwb Lahmer, T. verfasserin aut Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity Walter de Gruyter GmbH & Co. KG 2009 9 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © de Gruyter 2009 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. Walter de Gruyter Online Zeitschriften Inverse problem piezoelectricity multilevel modified Landweber Enthalten in Journal of inverse and ill-posed problems Berlin : de Gruyter, 1993 17(2009), 6 vom: 19. Aug., Seite 585-593 (DE-627)NLEJ248236091 (DE-600)2041913-2 1569-3945 nnns volume:17 year:2009 number:6 day:19 month:08 pages:585-593 extent:9 https://doi.org/10.1515/JIIP.2009.036 Deutschlandweit zugänglich GBV_USEFLAG_U ZDB-1-DGR GBV_NL_ARTICLE AR 17 2009 6 19 08 585-593 9 |
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10.1515/JIIP.2009.036 doi artikel_Grundlieferung.pp (DE-627)NLEJ247090298 DE-627 ger DE-627 rakwb Lahmer, T. verfasserin aut Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity Walter de Gruyter GmbH & Co. KG 2009 9 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © de Gruyter 2009 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. Walter de Gruyter Online Zeitschriften Inverse problem piezoelectricity multilevel modified Landweber Enthalten in Journal of inverse and ill-posed problems Berlin : de Gruyter, 1993 17(2009), 6 vom: 19. Aug., Seite 585-593 (DE-627)NLEJ248236091 (DE-600)2041913-2 1569-3945 nnns volume:17 year:2009 number:6 day:19 month:08 pages:585-593 extent:9 https://doi.org/10.1515/JIIP.2009.036 Deutschlandweit zugänglich GBV_USEFLAG_U ZDB-1-DGR GBV_NL_ARTICLE AR 17 2009 6 19 08 585-593 9 |
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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 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 |
| abstractGer |
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 |
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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Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity |
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