Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions

Abstract We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in Beskos et al. (Stoch Process Appl 121(10):2201–2230, 2011). This algorithm can be used as a basis to sample from certain clas...
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Autor*in:	Glatt-Holtz, Nathan E. [verfasserIn] Mondaini, Cecilia F.

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2021

Schlagwörter:	Hamiltonian Monte Carlo (HMC) Infinite dimensional Hamiltonian systems Markov Chain Monte Carlo (MCMC) Statistical sampling Bayesian inversion Advection-diffusion equations Passive scalar transport

Anmerkung:	© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021

Übergeordnetes Werk:	Enthalten in: Stochastics and partial differential equations - New York, NY : Springer, 2013, 10(2021), 4 vom: 18. Sept., Seite 1318-1391
Übergeordnetes Werk:	volume:10 ; year:2021 ; number:4 ; day:18 ; month:09 ; pages:1318-1391

Links:	Volltext

DOI / URN:	10.1007/s40072-021-00211-z

Katalog-ID:	SPR048496987

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520			\|a Abstract We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in Beskos et al. (Stoch Process Appl 121(10):2201–2230, 2011). This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in Beskos et al. (2011), and provides an alternative to a recent proof based on exact coupling techniques given in Bou-Rabee and Eberle (Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019). The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. Stuart (Acta Numer 19:451–559, 2010). Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, Borggaard et al. (SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020).
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allfields	10.1007/s40072-021-00211-z doi (DE-627)SPR048496987 (SPR)s40072-021-00211-z-e DE-627 ger DE-627 rakwb eng Glatt-Holtz, Nathan E. verfasserin aut Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in Beskos et al. (Stoch Process Appl 121(10):2201–2230, 2011). This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in Beskos et al. (2011), and provides an alternative to a recent proof based on exact coupling techniques given in Bou-Rabee and Eberle (Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019). The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. Stuart (Acta Numer 19:451–559, 2010). Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, Borggaard et al. (SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020). Hamiltonian Monte Carlo (HMC) (dpeaa)DE-He213 Infinite dimensional Hamiltonian systems (dpeaa)DE-He213 Markov Chain Monte Carlo (MCMC) (dpeaa)DE-He213 Statistical sampling (dpeaa)DE-He213 Bayesian inversion (dpeaa)DE-He213 Advection-diffusion equations (dpeaa)DE-He213 Passive scalar transport (dpeaa)DE-He213 Mondaini, Cecilia F. (orcid)0000-0002-6880-2814 aut Enthalten in Stochastics and partial differential equations New York, NY : Springer, 2013 10(2021), 4 vom: 18. Sept., Seite 1318-1391 (DE-627)739215078 (DE-600)2708008-0 2194-041X nnns volume:10 year:2021 number:4 day:18 month:09 pages:1318-1391 https://dx.doi.org/10.1007/s40072-021-00211-z 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_152 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_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_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 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_2118 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_4126 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 10 2021 4 18 09 1318-1391
spelling	10.1007/s40072-021-00211-z doi (DE-627)SPR048496987 (SPR)s40072-021-00211-z-e DE-627 ger DE-627 rakwb eng Glatt-Holtz, Nathan E. verfasserin aut Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in Beskos et al. (Stoch Process Appl 121(10):2201–2230, 2011). This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in Beskos et al. (2011), and provides an alternative to a recent proof based on exact coupling techniques given in Bou-Rabee and Eberle (Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019). The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. Stuart (Acta Numer 19:451–559, 2010). Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, Borggaard et al. (SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020). Hamiltonian Monte Carlo (HMC) (dpeaa)DE-He213 Infinite dimensional Hamiltonian systems (dpeaa)DE-He213 Markov Chain Monte Carlo (MCMC) (dpeaa)DE-He213 Statistical sampling (dpeaa)DE-He213 Bayesian inversion (dpeaa)DE-He213 Advection-diffusion equations (dpeaa)DE-He213 Passive scalar transport (dpeaa)DE-He213 Mondaini, Cecilia F. (orcid)0000-0002-6880-2814 aut Enthalten in Stochastics and partial differential equations New York, NY : Springer, 2013 10(2021), 4 vom: 18. Sept., Seite 1318-1391 (DE-627)739215078 (DE-600)2708008-0 2194-041X nnns volume:10 year:2021 number:4 day:18 month:09 pages:1318-1391 https://dx.doi.org/10.1007/s40072-021-00211-z 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_152 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_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_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 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_2118 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_4126 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 10 2021 4 18 09 1318-1391
allfields_unstemmed	10.1007/s40072-021-00211-z doi (DE-627)SPR048496987 (SPR)s40072-021-00211-z-e DE-627 ger DE-627 rakwb eng Glatt-Holtz, Nathan E. verfasserin aut Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in Beskos et al. (Stoch Process Appl 121(10):2201–2230, 2011). This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in Beskos et al. (2011), and provides an alternative to a recent proof based on exact coupling techniques given in Bou-Rabee and Eberle (Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019). The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. Stuart (Acta Numer 19:451–559, 2010). Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, Borggaard et al. (SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020). Hamiltonian Monte Carlo (HMC) (dpeaa)DE-He213 Infinite dimensional Hamiltonian systems (dpeaa)DE-He213 Markov Chain Monte Carlo (MCMC) (dpeaa)DE-He213 Statistical sampling (dpeaa)DE-He213 Bayesian inversion (dpeaa)DE-He213 Advection-diffusion equations (dpeaa)DE-He213 Passive scalar transport (dpeaa)DE-He213 Mondaini, Cecilia F. (orcid)0000-0002-6880-2814 aut Enthalten in Stochastics and partial differential equations New York, NY : Springer, 2013 10(2021), 4 vom: 18. Sept., Seite 1318-1391 (DE-627)739215078 (DE-600)2708008-0 2194-041X nnns volume:10 year:2021 number:4 day:18 month:09 pages:1318-1391 https://dx.doi.org/10.1007/s40072-021-00211-z 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_152 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_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_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 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_2118 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_4126 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 10 2021 4 18 09 1318-1391
allfieldsGer	10.1007/s40072-021-00211-z doi (DE-627)SPR048496987 (SPR)s40072-021-00211-z-e DE-627 ger DE-627 rakwb eng Glatt-Holtz, Nathan E. verfasserin aut Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in Beskos et al. (Stoch Process Appl 121(10):2201–2230, 2011). This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in Beskos et al. (2011), and provides an alternative to a recent proof based on exact coupling techniques given in Bou-Rabee and Eberle (Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019). The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. Stuart (Acta Numer 19:451–559, 2010). Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, Borggaard et al. (SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020). Hamiltonian Monte Carlo (HMC) (dpeaa)DE-He213 Infinite dimensional Hamiltonian systems (dpeaa)DE-He213 Markov Chain Monte Carlo (MCMC) (dpeaa)DE-He213 Statistical sampling (dpeaa)DE-He213 Bayesian inversion (dpeaa)DE-He213 Advection-diffusion equations (dpeaa)DE-He213 Passive scalar transport (dpeaa)DE-He213 Mondaini, Cecilia F. (orcid)0000-0002-6880-2814 aut Enthalten in Stochastics and partial differential equations New York, NY : Springer, 2013 10(2021), 4 vom: 18. Sept., Seite 1318-1391 (DE-627)739215078 (DE-600)2708008-0 2194-041X nnns volume:10 year:2021 number:4 day:18 month:09 pages:1318-1391 https://dx.doi.org/10.1007/s40072-021-00211-z 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_152 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_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_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 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_2118 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_4126 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 10 2021 4 18 09 1318-1391
allfieldsSound	10.1007/s40072-021-00211-z doi (DE-627)SPR048496987 (SPR)s40072-021-00211-z-e DE-627 ger DE-627 rakwb eng Glatt-Holtz, Nathan E. verfasserin aut Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in Beskos et al. (Stoch Process Appl 121(10):2201–2230, 2011). This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in Beskos et al. (2011), and provides an alternative to a recent proof based on exact coupling techniques given in Bou-Rabee and Eberle (Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019). The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. Stuart (Acta Numer 19:451–559, 2010). Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, Borggaard et al. (SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020). Hamiltonian Monte Carlo (HMC) (dpeaa)DE-He213 Infinite dimensional Hamiltonian systems (dpeaa)DE-He213 Markov Chain Monte Carlo (MCMC) (dpeaa)DE-He213 Statistical sampling (dpeaa)DE-He213 Bayesian inversion (dpeaa)DE-He213 Advection-diffusion equations (dpeaa)DE-He213 Passive scalar transport (dpeaa)DE-He213 Mondaini, Cecilia F. (orcid)0000-0002-6880-2814 aut Enthalten in Stochastics and partial differential equations New York, NY : Springer, 2013 10(2021), 4 vom: 18. Sept., Seite 1318-1391 (DE-627)739215078 (DE-600)2708008-0 2194-041X nnns volume:10 year:2021 number:4 day:18 month:09 pages:1318-1391 https://dx.doi.org/10.1007/s40072-021-00211-z 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_152 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_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_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 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_2118 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_4126 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 10 2021 4 18 09 1318-1391
language	English
source	Enthalten in Stochastics and partial differential equations 10(2021), 4 vom: 18. Sept., Seite 1318-1391 volume:10 year:2021 number:4 day:18 month:09 pages:1318-1391
sourceStr	Enthalten in Stochastics and partial differential equations 10(2021), 4 vom: 18. Sept., Seite 1318-1391 volume:10 year:2021 number:4 day:18 month:09 pages:1318-1391
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Hamiltonian Monte Carlo (HMC) Infinite dimensional Hamiltonian systems Markov Chain Monte Carlo (MCMC) Statistical sampling Bayesian inversion Advection-diffusion equations Passive scalar transport
isfreeaccess_bool	false
container_title	Stochastics and partial differential equations
authorswithroles_txt_mv	Glatt-Holtz, Nathan E. @@aut@@ Mondaini, Cecilia F. @@aut@@
publishDateDaySort_date	2021-09-18T00:00:00Z
hierarchy_top_id	739215078
id	SPR048496987
language_de	englisch
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author	Glatt-Holtz, Nathan E.
spellingShingle	Glatt-Holtz, Nathan E. misc Hamiltonian Monte Carlo (HMC) misc Infinite dimensional Hamiltonian systems misc Markov Chain Monte Carlo (MCMC) misc Statistical sampling misc Bayesian inversion misc Advection-diffusion equations misc Passive scalar transport Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions
authorStr	Glatt-Holtz, Nathan E.
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topic_title	Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions Hamiltonian Monte Carlo (HMC) (dpeaa)DE-He213 Infinite dimensional Hamiltonian systems (dpeaa)DE-He213 Markov Chain Monte Carlo (MCMC) (dpeaa)DE-He213 Statistical sampling (dpeaa)DE-He213 Bayesian inversion (dpeaa)DE-He213 Advection-diffusion equations (dpeaa)DE-He213 Passive scalar transport (dpeaa)DE-He213
topic	misc Hamiltonian Monte Carlo (HMC) misc Infinite dimensional Hamiltonian systems misc Markov Chain Monte Carlo (MCMC) misc Statistical sampling misc Bayesian inversion misc Advection-diffusion equations misc Passive scalar transport
topic_unstemmed	misc Hamiltonian Monte Carlo (HMC) misc Infinite dimensional Hamiltonian systems misc Markov Chain Monte Carlo (MCMC) misc Statistical sampling misc Bayesian inversion misc Advection-diffusion equations misc Passive scalar transport
topic_browse	misc Hamiltonian Monte Carlo (HMC) misc Infinite dimensional Hamiltonian systems misc Markov Chain Monte Carlo (MCMC) misc Statistical sampling misc Bayesian inversion misc Advection-diffusion equations misc Passive scalar transport
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title	Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions
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title_full	Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions
author_sort	Glatt-Holtz, Nathan E.
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author_browse	Glatt-Holtz, Nathan E. Mondaini, Cecilia F.
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author-letter	Glatt-Holtz, Nathan E.
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title_sort	mixing rates for hamiltonian monte carlo algorithms in finite and infinite dimensions
title_auth	Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions
abstract	Abstract We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in Beskos et al. (Stoch Process Appl 121(10):2201–2230, 2011). This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in Beskos et al. (2011), and provides an alternative to a recent proof based on exact coupling techniques given in Bou-Rabee and Eberle (Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019). The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. Stuart (Acta Numer 19:451–559, 2010). Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, Borggaard et al. (SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020). © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021
abstractGer	Abstract We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in Beskos et al. (Stoch Process Appl 121(10):2201–2230, 2011). This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in Beskos et al. (2011), and provides an alternative to a recent proof based on exact coupling techniques given in Bou-Rabee and Eberle (Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019). The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. Stuart (Acta Numer 19:451–559, 2010). Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, Borggaard et al. (SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020). © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021
abstract_unstemmed	Abstract We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in Beskos et al. (Stoch Process Appl 121(10):2201–2230, 2011). This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in Beskos et al. (2011), and provides an alternative to a recent proof based on exact coupling techniques given in Bou-Rabee and Eberle (Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019). The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. Stuart (Acta Numer 19:451–559, 2010). Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, Borggaard et al. (SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020). © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021
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title_short	Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions
url	https://dx.doi.org/10.1007/s40072-021-00211-z
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author2	Mondaini, Cecilia F.
author2Str	Mondaini, Cecilia F.
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doi_str	10.1007/s40072-021-00211-z
up_date	2024-07-03T19:35:14.120Z
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(Stoch Process Appl 121(10):2201–2230, 2011). This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in Beskos et al. (2011), and provides an alternative to a recent proof based on exact coupling techniques given in Bou-Rabee and Eberle (Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019). The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. Stuart (Acta Numer 19:451–559, 2010). Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, Borggaard et al. 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