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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.
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.
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.
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.
This work incorporates three treatises which are commonly concerned with a stochastic theory of the Lyapunov exponents. With the help of this theory universal scaling laws are investigated which appear in coupled chaotic and disordered systems. First, two continuous-time stochastic models for weakly coupled chaotic systems are introduced to study the scaling of the Lyapunov exponents with the coupling strength (coupling sensitivity of chaos). By means of the the Fokker-Planck formalism scaling relations are derived, which are confirmed by results of numerical simulations. Next, coupling sensitivity is shown to exist for coupled disordered chains, where it appears as a singular increase of the localization length. Numerical findings for coupled Anderson models are confirmed by analytic results for coupled continuous-space Schrödinger equations. The resulting scaling relation of the localization length resembles the scaling of the Lyapunov exponent of coupled chaotic systems. Finally, the statistics of the exponential growth rate of the linear oscillator with parametric noise are studied. It is shown that the distribution of the finite-time Lyapunov exponent deviates from a Gaussian one. By means of the generalized Lyapunov exponents the parameter range is determined where the non-Gaussian part of the distribution is significant and multiscaling becomes essential.