@phdthesis{Zillmer2003,
  author    = {Zillmer, R{\"u}diger},
  title     = {Statistical properties and scaling of the Lyapunov exponents in stochastic systems},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0001147},
  school      = {Universit{\"a}t Potsdam},
  year      = {2003},
  abstract  = {Die vorliegende Arbeit umfaßt drei Abhandlungen, welche allgemein mit einer stochastischen Theorie f{\"u}r die Lyapunov-Exponenten befaßt sind. Mit Hilfe dieser Theorie werden universelle Skalengesetze untersucht, die in gekoppelten chaotischen und ungeordneten Systemen auftreten. Zun{\"a}chst werden zwei zeitkontinuierliche stochastische Modelle f{\"u}r schwach gekoppelte chaotische Systeme eingef{\"u}hrt, um die Skalierung der Lyapunov-Exponenten mit der Kopplungsst{\"a}rke ('coupling sensitivity of chaos') zu untersuchen. Mit Hilfe des Fokker-Planck-Formalismus werden Skalengesetze hergeleitet, die von Ergebnissen numerischer Simulationen best{\"a}tigt werden. Anschließend wird gezeigt, daß 'coupling sensitivity' im Fall gekoppelter ungeordneter Ketten auftritt, wobei der Effekt sich durch ein singul{\"a}res Anwachsen der Lokalisierungsl{\"a}nge {\"a}ußert. Numerische Ergebnisse f{\"u}r gekoppelte Anderson-Modelle werden bekr{\"a}ftigt durch analytische Resultate f{\"u}r gekoppelte raumkontinuierliche Schr{\"o}dinger-Gleichungen. Das resultierende Skalengesetz f{\"u}r die Lokalisierungsl{\"a}nge {\"a}hnelt der Skalierung der Lyapunov-Exponenten gekoppelter chaotischer Systeme. Schließlich wird die Statistik der exponentiellen Wachstumsrate des linearen Oszillators mit parametrischem Rauschen studiert. Es wird gezeigt, daß die Verteilung des zeitabh{\"a}ngigen Lyapunov-Exponenten von der Normalverteilung abweicht. Mittels der verallgemeinerten Lyapunov-Exponenten wird der Parameterbereich bestimmt, in welchem die Abweichungen von der Normalverteilung signifikant sind und Multiskalierung wesentlich wird.},
  language  = {en}
}
@article{NardiniRybackiDoehmannetal.2018,
  author    = {Nardini, Livia and Rybacki, Erik and D{\"o}hmann, Maximilian J.E.A. and Morales, Luiz F.G. and Brune, Sascha and Dresen, Georg},
  title     = {High-temperature shear zone formation in Carrara marble},
  series = {Tectonophysics},
  volume    = {749},
  journal   = {Tectonophysics},
  publisher = {Elsevier},
  address   = {Amsterdam [u.a.]},
  issn      = {0040-1951},
  doi       = {10.1016/j.tecto.2018.10.022},
  pages     = {120 -- 139},
  year      = {2018},
  abstract  = {Rock deformation at depths in the Earth's crust is often localized in high temperature shear zones occurring at different scales in a variety of lithologies. The presence of material heterogeneities is known to trigger shear zone development, but the mechanisms controlling initiation and evolution of localization are not fully understood. To investigate the effect of loading conditions on shear zone nucleation along heterogeneities, we performed torsion experiments under constant twist rate (CTR) and constant torque (CT) conditions in a Paterson-type deformation apparatus. The sample assemblage consisted of cylindrical Carrara marble specimens containing a thin plate of Solnhofen limestone perpendicular to the cylinder's longitudinal axis. Under experimental conditions (900 °C, 400 MPa confining pressure), samples were plastically deformed and limestone is about 9 times weaker than marble, acting as a weak inclusion in a strong matrix. CTR experiments were performed at maximum bulk shear strain rates of ~ 2*10-4s-1, yielding peak shear stresses of ~ 20 MPa. CT tests were conducted at shear stresses of ~ 20 MPa resulting in bulk shear strain rates of 1-4*10-4s-1. Experiments were terminated at maximum bulk shear strains of ~ 0.3 and 1.0.Strain was localized within the Carrara marble in front of the inclusion in an area of strongly deformed grains and intense grain size reduction. Locally, evidences for coexisting brittle deformation are also observed regardless of the imposed loading conditions. The local shear strain at the inclusion tipis up to 30 times higher than the strain in the adjacent host rock, rapidly dropping to 5times higher at larger distance from the inclusion. At both investigated bulk strains, the evolution of microstructural and textural parameters is independent of loading conditions. Ourresults suggest that loading conditions do not significantly affect material heterogeneity-induced strain localization during its nucleation and transient stages.},
  language  = {en}
}
@misc{vanLeeuwenKunschNergeretal.2019,
  author    = {van Leeuwen, Peter Jan and Kunsch, Hans R. and Nerger, Lars and Potthast, Roland and Reich, Sebastian},
  title     = {Particle filters for high-dimensional geoscience applications: A review},
  series = {Quarterly journal of the Royal Meteorological Society},
  volume    = {145},
  journal   = {Quarterly journal of the Royal Meteorological Society},
  number    = {723},
  publisher = {Wiley},
  address   = {Hoboken},
  issn      = {0035-9009},
  doi       = {10.1002/qj.3551},
  pages     = {2335 -- 2365},
  year      = {2019},
  abstract  = {Particle filters contain the promise of fully nonlinear data assimilation. They have been applied in numerous science areas, including the geosciences, but their application to high-dimensional geoscience systems has been limited due to their inefficiency in high-dimensional systems in standard settings. However, huge progress has been made, and this limitation is disappearing fast due to recent developments in proposal densities, the use of ideas from (optimal) transportation, the use of localization and intelligent adaptive resampling strategies. Furthermore, powerful hybrids between particle filters and ensemble Kalman filters and variational methods have been developed. We present a state-of-the-art discussion of present efforts of developing particle filters for high-dimensional nonlinear geoscience state-estimation problems, with an emphasis on atmospheric and oceanic applications, including many new ideas, derivations and unifications, highlighting hidden connections, including pseudo-code, and generating a valuable tool and guide for the community. Initial experiments show that particle filters can be competitive with present-day methods for numerical weather prediction, suggesting that they will become mainstream soon.},
  language  = {en}
}
@article{deWiljesPathirajaReich2020,
  author    = {de Wiljes, Jana and Pathiraja, Sahani Darschika and Reich, Sebastian},
  title     = {Ensemble transform algorithms for nonlinear smoothing problems},
  series = {SIAM journal on scientific computing},
  volume    = {42},
  journal   = {SIAM journal on scientific computing},
  number    = {1},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1064-8275},
  doi       = {10.1137/19M1239544},
  pages     = {A87 -- A114},
  year      = {2020},
  abstract  = {Several numerical tools designed to overcome the challenges of smoothing in a non-linear and non-Gaussian setting are investigated for a class of particle smoothers. The considered family of smoothers is induced by the class of linear ensemble transform filters which contains classical filters such as the stochastic ensemble Kalman filter, the ensemble square root filter, and the recently introduced nonlinear ensemble transform filter. Further the ensemble transform particle smoother is introduced and particularly highlighted as it is consistent in the particle limit and does not require assumptions with respect to the family of the posterior distribution. The linear update pattern of the considered class of linear ensemble transform smoothers allows one to implement important supplementary techniques such as adaptive spread corrections, hybrid formulations, and localization in order to facilitate their application to complex estimation problems. These additional features are derived and numerically investigated for a sequence of increasingly challenging test problems.},
  language  = {en}
}
@article{ReichWeissmann2021,
  author    = {Reich, Sebastian and Weissmann, Simon},
  title     = {Fokker-Planck particle systems for Bayesian inference: computational approaches},
  series = {SIAM ASA journal on uncertainty quantification},
  volume    = {9},
  journal   = {SIAM ASA journal on uncertainty quantification},
  number    = {2},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {2166-2525},
  doi       = {10.1137/19M1303162},
  pages     = {446 -- 482},
  year      = {2021},
  abstract  = {Bayesian inference can be embedded into an appropriately defined dynamics in the space of probability measures. In this paper, we take Brownian motion and its associated Fokker-Planck equation as a starting point for such embeddings and explore several interacting particle approximations. More specifically, we consider both deterministic and stochastic interacting particle systems and combine them with the idea of preconditioning by the empirical covariance matrix. In addition to leading to affine invariant formulations which asymptotically speed up convergence, preconditioning allows for gradient-free implementations in the spirit of the ensemble Kalman filter. While such gradient-free implementations have been demonstrated to work well for posterior measures that are nearly Gaussian, we extend their scope of applicability to multimodal measures by introducing localized gradient-free approximations. Numerical results demonstrate the effectiveness of the considered methodologies.},
  language  = {en}
}