On the computation of Proper Generalized Decomposition modes of parametric elliptic problems

Abstract In a recent paper (Azaïez et al. in SIAM J Math Anal 50(5):5426–5445, 10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence...
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Autor*in:	Azaïez, M. [verfasserIn] Chacón Rebollo, T. Gómez Mármol, M.

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2019

Schlagwörter:	Proper Generalized Decomposition Computation of PGD modes Convergence rate analysis

Anmerkung:	© Sociedad Española de Matemática Aplicada 2019

Übergeordnetes Werk:	Enthalten in: SeMA journal - Berlin : Springer, 2010, 77(2019), 1 vom: 31. Juli, Seite 59-72
Übergeordnetes Werk:	volume:77 ; year:2019 ; number:1 ; day:31 ; month:07 ; pages:59-72

Links:	Volltext

DOI / URN:	10.1007/s40324-019-00198-7

Katalog-ID:	SPR037104411

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520			\|a Abstract In a recent paper (Azaïez et al. in SIAM J Math Anal 50(5):5426–5445, 10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace. When the dimension is equal one and making use of a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, the method turns to be a classical progressive proper generalized decomposition. In this contribution we prove the linear convergence of the Power Iterate method applied to compute the modes of the PGD expansion, for both symmetric and non-symmetric problems, when the data are small. We also find a spectral convergence ratio of the PGD expansion in the mean parametric norm, for meaningful parametric elliptic problems.
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allfields	10.1007/s40324-019-00198-7 doi (DE-627)SPR037104411 (SPR)s40324-019-00198-7-e DE-627 ger DE-627 rakwb eng Azaïez, M. verfasserin aut On the computation of Proper Generalized Decomposition modes of parametric elliptic problems 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Sociedad Española de Matemática Aplicada 2019 Abstract In a recent paper (Azaïez et al. in SIAM J Math Anal 50(5):5426–5445, 10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace. When the dimension is equal one and making use of a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, the method turns to be a classical progressive proper generalized decomposition. In this contribution we prove the linear convergence of the Power Iterate method applied to compute the modes of the PGD expansion, for both symmetric and non-symmetric problems, when the data are small. We also find a spectral convergence ratio of the PGD expansion in the mean parametric norm, for meaningful parametric elliptic problems. Proper Generalized Decomposition (dpeaa)DE-He213 Computation of PGD modes (dpeaa)DE-He213 Convergence rate analysis (dpeaa)DE-He213 Chacón Rebollo, T. (orcid)0000-0002-0784-1174 aut Gómez Mármol, M. aut Enthalten in SeMA journal Berlin : Springer, 2010 77(2019), 1 vom: 31. Juli, Seite 59-72 (DE-627)815395299 (DE-600)2805493-3 2254-3902 nnns volume:77 year:2019 number:1 day:31 month:07 pages:59-72 https://dx.doi.org/10.1007/s40324-019-00198-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 77 2019 1 31 07 59-72
spelling	10.1007/s40324-019-00198-7 doi (DE-627)SPR037104411 (SPR)s40324-019-00198-7-e DE-627 ger DE-627 rakwb eng Azaïez, M. verfasserin aut On the computation of Proper Generalized Decomposition modes of parametric elliptic problems 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Sociedad Española de Matemática Aplicada 2019 Abstract In a recent paper (Azaïez et al. in SIAM J Math Anal 50(5):5426–5445, 10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace. When the dimension is equal one and making use of a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, the method turns to be a classical progressive proper generalized decomposition. In this contribution we prove the linear convergence of the Power Iterate method applied to compute the modes of the PGD expansion, for both symmetric and non-symmetric problems, when the data are small. We also find a spectral convergence ratio of the PGD expansion in the mean parametric norm, for meaningful parametric elliptic problems. Proper Generalized Decomposition (dpeaa)DE-He213 Computation of PGD modes (dpeaa)DE-He213 Convergence rate analysis (dpeaa)DE-He213 Chacón Rebollo, T. (orcid)0000-0002-0784-1174 aut Gómez Mármol, M. aut Enthalten in SeMA journal Berlin : Springer, 2010 77(2019), 1 vom: 31. Juli, Seite 59-72 (DE-627)815395299 (DE-600)2805493-3 2254-3902 nnns volume:77 year:2019 number:1 day:31 month:07 pages:59-72 https://dx.doi.org/10.1007/s40324-019-00198-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 77 2019 1 31 07 59-72
allfields_unstemmed	10.1007/s40324-019-00198-7 doi (DE-627)SPR037104411 (SPR)s40324-019-00198-7-e DE-627 ger DE-627 rakwb eng Azaïez, M. verfasserin aut On the computation of Proper Generalized Decomposition modes of parametric elliptic problems 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Sociedad Española de Matemática Aplicada 2019 Abstract In a recent paper (Azaïez et al. in SIAM J Math Anal 50(5):5426–5445, 10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace. When the dimension is equal one and making use of a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, the method turns to be a classical progressive proper generalized decomposition. In this contribution we prove the linear convergence of the Power Iterate method applied to compute the modes of the PGD expansion, for both symmetric and non-symmetric problems, when the data are small. We also find a spectral convergence ratio of the PGD expansion in the mean parametric norm, for meaningful parametric elliptic problems. Proper Generalized Decomposition (dpeaa)DE-He213 Computation of PGD modes (dpeaa)DE-He213 Convergence rate analysis (dpeaa)DE-He213 Chacón Rebollo, T. (orcid)0000-0002-0784-1174 aut Gómez Mármol, M. aut Enthalten in SeMA journal Berlin : Springer, 2010 77(2019), 1 vom: 31. Juli, Seite 59-72 (DE-627)815395299 (DE-600)2805493-3 2254-3902 nnns volume:77 year:2019 number:1 day:31 month:07 pages:59-72 https://dx.doi.org/10.1007/s40324-019-00198-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 77 2019 1 31 07 59-72
allfieldsGer	10.1007/s40324-019-00198-7 doi (DE-627)SPR037104411 (SPR)s40324-019-00198-7-e DE-627 ger DE-627 rakwb eng Azaïez, M. verfasserin aut On the computation of Proper Generalized Decomposition modes of parametric elliptic problems 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Sociedad Española de Matemática Aplicada 2019 Abstract In a recent paper (Azaïez et al. in SIAM J Math Anal 50(5):5426–5445, 10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace. When the dimension is equal one and making use of a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, the method turns to be a classical progressive proper generalized decomposition. In this contribution we prove the linear convergence of the Power Iterate method applied to compute the modes of the PGD expansion, for both symmetric and non-symmetric problems, when the data are small. We also find a spectral convergence ratio of the PGD expansion in the mean parametric norm, for meaningful parametric elliptic problems. Proper Generalized Decomposition (dpeaa)DE-He213 Computation of PGD modes (dpeaa)DE-He213 Convergence rate analysis (dpeaa)DE-He213 Chacón Rebollo, T. (orcid)0000-0002-0784-1174 aut Gómez Mármol, M. aut Enthalten in SeMA journal Berlin : Springer, 2010 77(2019), 1 vom: 31. Juli, Seite 59-72 (DE-627)815395299 (DE-600)2805493-3 2254-3902 nnns volume:77 year:2019 number:1 day:31 month:07 pages:59-72 https://dx.doi.org/10.1007/s40324-019-00198-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 77 2019 1 31 07 59-72
allfieldsSound	10.1007/s40324-019-00198-7 doi (DE-627)SPR037104411 (SPR)s40324-019-00198-7-e DE-627 ger DE-627 rakwb eng Azaïez, M. verfasserin aut On the computation of Proper Generalized Decomposition modes of parametric elliptic problems 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Sociedad Española de Matemática Aplicada 2019 Abstract In a recent paper (Azaïez et al. in SIAM J Math Anal 50(5):5426–5445, 10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace. When the dimension is equal one and making use of a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, the method turns to be a classical progressive proper generalized decomposition. In this contribution we prove the linear convergence of the Power Iterate method applied to compute the modes of the PGD expansion, for both symmetric and non-symmetric problems, when the data are small. We also find a spectral convergence ratio of the PGD expansion in the mean parametric norm, for meaningful parametric elliptic problems. Proper Generalized Decomposition (dpeaa)DE-He213 Computation of PGD modes (dpeaa)DE-He213 Convergence rate analysis (dpeaa)DE-He213 Chacón Rebollo, T. (orcid)0000-0002-0784-1174 aut Gómez Mármol, M. aut Enthalten in SeMA journal Berlin : Springer, 2010 77(2019), 1 vom: 31. Juli, Seite 59-72 (DE-627)815395299 (DE-600)2805493-3 2254-3902 nnns volume:77 year:2019 number:1 day:31 month:07 pages:59-72 https://dx.doi.org/10.1007/s40324-019-00198-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 77 2019 1 31 07 59-72
language	English
source	Enthalten in SeMA journal 77(2019), 1 vom: 31. Juli, Seite 59-72 volume:77 year:2019 number:1 day:31 month:07 pages:59-72
sourceStr	Enthalten in SeMA journal 77(2019), 1 vom: 31. Juli, Seite 59-72 volume:77 year:2019 number:1 day:31 month:07 pages:59-72
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Proper Generalized Decomposition Computation of PGD modes Convergence rate analysis
isfreeaccess_bool	false
container_title	SeMA journal
authorswithroles_txt_mv	Azaïez, M. @@aut@@ Chacón Rebollo, T. @@aut@@ Gómez Mármol, M. @@aut@@
publishDateDaySort_date	2019-07-31T00:00:00Z
hierarchy_top_id	815395299
id	SPR037104411
language_de	englisch
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author	Azaïez, M.
spellingShingle	Azaïez, M. misc Proper Generalized Decomposition misc Computation of PGD modes misc Convergence rate analysis On the computation of Proper Generalized Decomposition modes of parametric elliptic problems
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topic_title	On the computation of Proper Generalized Decomposition modes of parametric elliptic problems Proper Generalized Decomposition (dpeaa)DE-He213 Computation of PGD modes (dpeaa)DE-He213 Convergence rate analysis (dpeaa)DE-He213
topic	misc Proper Generalized Decomposition misc Computation of PGD modes misc Convergence rate analysis
topic_unstemmed	misc Proper Generalized Decomposition misc Computation of PGD modes misc Convergence rate analysis
topic_browse	misc Proper Generalized Decomposition misc Computation of PGD modes misc Convergence rate analysis
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title	On the computation of Proper Generalized Decomposition modes of parametric elliptic problems
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title_full	On the computation of Proper Generalized Decomposition modes of parametric elliptic problems
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author_browse	Azaïez, M. Chacón Rebollo, T. Gómez Mármol, M.
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title_sort	on the computation of proper generalized decomposition modes of parametric elliptic problems
title_auth	On the computation of Proper Generalized Decomposition modes of parametric elliptic problems
abstract	Abstract In a recent paper (Azaïez et al. in SIAM J Math Anal 50(5):5426–5445, 10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace. When the dimension is equal one and making use of a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, the method turns to be a classical progressive proper generalized decomposition. In this contribution we prove the linear convergence of the Power Iterate method applied to compute the modes of the PGD expansion, for both symmetric and non-symmetric problems, when the data are small. We also find a spectral convergence ratio of the PGD expansion in the mean parametric norm, for meaningful parametric elliptic problems. © Sociedad Española de Matemática Aplicada 2019
abstractGer	Abstract In a recent paper (Azaïez et al. in SIAM J Math Anal 50(5):5426–5445, 10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace. When the dimension is equal one and making use of a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, the method turns to be a classical progressive proper generalized decomposition. In this contribution we prove the linear convergence of the Power Iterate method applied to compute the modes of the PGD expansion, for both symmetric and non-symmetric problems, when the data are small. We also find a spectral convergence ratio of the PGD expansion in the mean parametric norm, for meaningful parametric elliptic problems. © Sociedad Española de Matemática Aplicada 2019
abstract_unstemmed	Abstract In a recent paper (Azaïez et al. in SIAM J Math Anal 50(5):5426–5445, 10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace. When the dimension is equal one and making use of a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, the method turns to be a classical progressive proper generalized decomposition. In this contribution we prove the linear convergence of the Power Iterate method applied to compute the modes of the PGD expansion, for both symmetric and non-symmetric problems, when the data are small. We also find a spectral convergence ratio of the PGD expansion in the mean parametric norm, for meaningful parametric elliptic problems. © Sociedad Española de Matemática Aplicada 2019
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title_short	On the computation of Proper Generalized Decomposition modes of parametric elliptic problems
url	https://dx.doi.org/10.1007/s40324-019-00198-7
remote_bool	true
author2	Chacón Rebollo, T. Gómez Mármol, M.
author2Str	Chacón Rebollo, T. Gómez Mármol, M.
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doi_str	10.1007/s40324-019-00198-7
up_date	2024-07-03T21:09:05.532Z
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On the computation of Proper Generalized Decomposition modes of parametric elliptic problems

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