290 research outputs found
Probabilistic Reduced-Order Modeling for Stochastic Partial Differential Equations
We discuss a Bayesian formulation to coarse-graining (CG) of PDEs where the
coefficients (e.g. material parameters) exhibit random, fine scale variability.
The direct solution to such problems requires grids that are small enough to
resolve this fine scale variability which unavoidably requires the repeated
solution of very large systems of algebraic equations. We establish a
physically inspired, data-driven coarse-grained model which learns a low-
dimensional set of microstructural features that are predictive of the
fine-grained model (FG) response. Once learned, those features provide a sharp
distribution over the coarse scale effec- tive coefficients of the PDE that are
most suitable for prediction of the fine scale model output. This ultimately
allows to replace the computationally expensive FG by a generative proba-
bilistic model based on evaluating the much cheaper CG several times. Sparsity
enforcing pri- ors further increase predictive efficiency and reveal
microstructural features that are important in predicting the FG response.
Moreover, the model yields probabilistic rather than single-point predictions,
which enables the quantification of the unavoidable epistemic uncertainty that
is present due to the information loss that occurs during the coarse-graining
process
Particle-by-Particle Reconstruction of Ultrafiltration Cakes in 3D from Binarized TEM Images
Transmission electron microscopy (TEM) imaging is one of the few techniques available for direct observation of the microstructure of ultrafiltration cakes. TEM images yield local microstructural information in the form of two-dimensional grayscale images of slices a few particle diameters in thickness. This work presents an innovative particle-by-particle reconstruction scheme for simulating ultrafiltration cake microstructure in three dimensions from TEM images. The scheme uses binarized TEM images, thereby permitting use of lesser-quality images. It is able to account for short- and long-range order within ultrafiltration cake structure by matching the morphology of simulated and measured microstructures at a number of resolutions and scales identifiable within the observed microstructure. In the end, simulated microstructures are intended for improving our understanding of the relationships between cake morphology, ultrafiltration performance, and operating conditions
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