527 research outputs found
Ensemble-based implicit sampling for Bayesian inverse problems with non-Gaussian priors
In the paper, we develop an ensemble-based implicit sampling method for
Bayesian inverse problems. For Bayesian inference, the iterative ensemble
smoother (IES) and implicit sampling are integrated to obtain importance
ensemble samples, which build an importance density. The proposed method shares
a similar idea to importance sampling. IES is used to approximate mean and
covariance of a posterior distribution. This provides the MAP point and the
inverse of Hessian matrix, which are necessary to construct the implicit map in
implicit sampling. The importance samples are generated by the implicit map and
the corresponding weights are the ratio between the importance density and
posterior density. In the proposed method, we use the ensemble samples of IES
to find the optimization solution of likelihood function and the inverse of
Hessian matrix. This approach avoids the explicit computation for Jacobian
matrix and Hessian matrix, which are very computationally expensive in high
dimension spaces. To treat non-Gaussian models, discrete cosine transform and
Gaussian mixture model are used to characterize the non-Gaussian priors. The
ensemble-based implicit sampling method is extended to the non-Gaussian priors
for exploring the posterior of unknowns in inverse problems. The proposed
method is used for each individual Gaussian model in the Gaussian mixture
model. The proposed approach substantially improves the applicability of
implicit sampling method. A few numerical examples are presented to demonstrate
the efficacy of the proposed method with applications of inverse problems for
subsurface flow problems and anomalous diffusion models in porous media.Comment: 27 page
Neural Ideal Large Eddy Simulation: Modeling Turbulence with Neural Stochastic Differential Equations
We introduce a data-driven learning framework that assimilates two powerful
ideas: ideal large eddy simulation (LES) from turbulence closure modeling and
neural stochastic differential equations (SDE) for stochastic modeling. The
ideal LES models the LES flow by treating each full-order trajectory as a
random realization of the underlying dynamics, as such, the effect of
small-scales is marginalized to obtain the deterministic evolution of the LES
state. However, ideal LES is analytically intractable. In our work, we use a
latent neural SDE to model the evolution of the stochastic process and an
encoder-decoder pair for transforming between the latent space and the desired
ideal flow field. This stands in sharp contrast to other types of neural
parameterization of closure models where each trajectory is treated as a
deterministic realization of the dynamics. We show the effectiveness of our
approach (niLES - neural ideal LES) on a challenging chaotic dynamical system:
Kolmogorov flow at a Reynolds number of 20,000. Compared to competing methods,
our method can handle non-uniform geometries using unstructured meshes
seamlessly. In particular, niLES leads to trajectories with more accurate
statistics and enhances stability, particularly for long-horizon rollouts.Comment: 18 page
Multiscale modeling for fluid transport in nanosystems.
Atomistic-scale behavior drives performance in many micro- and nano-fluidic systems, such as mircrofludic mixers and electrical energy storage devices. Bringing this information into the traditionally continuum models used for engineering analysis has proved challenging. This work describes one such approach to address this issue by developing atomistic-to-continuum multi scale and multi physics methods to enable molecular dynamics (MD) representations of atoms to incorporated into continuum simulations. Coupling is achieved by imposing constraints based on fluxes of conserved quantities between the two regions described by one of these models. The impact of electric fields and surface charges are also critical, hence, methodologies to extend finite-element (FE) MD electric field solvers have been derived to account for these effects. Finally, the continuum description can have inconsistencies with the coarse-grained MD dynamics, so FE equations based on MD statistics were derived to facilitate the multi scale coupling. Examples are shown relevant to nanofluidic systems, such as pore flow, Couette flow, and electric double layer
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