3 research outputs found
Cluster-based Generalized Multiscale Finite Element Method for elliptic PDEs with random coefficients
We propose a generalized multiscale finite element method (GMsFEM) based on
clustering algorithm to study the elliptic PDEs with random coefficients in the
multi-query setting. Our method consists of offline and online stages. In the
offline stage, we construct a small number of reduced basis functions within
each coarse grid block, which can then be used to approximate the multiscale
finite element basis functions. In addition, we coarsen the corresponding
random space through a clustering algorithm. In the online stage, we can obtain
the multiscale finite element basis very efficiently on a coarse grid by using
the pre-computed multiscale basis. The new GMsFEM can be applied to multiscale
SPDE starting with a relatively coarse grid, without requiring the coarsest
grid to resolve the smallest-scale of the solution. The new method offers
considerable savings in solving multiscale SPDEs. Numerical results are
presented to demonstrate the accuracy and efficiency of the proposed method for
several multiscale stochastic problems without scale separation
Learning Algorithms for Coarsening Uncertainty Space and Applications to Multiscale Simulations
In this paper, we investigate and design multiscale simulations for
stochastic multiscale PDEs. As for the space, we consider a coarse grid and a
known multiscale method, the Generalized Multiscale Finite Element Method
(GMsFEM). In order to obtain a small dimensional representation of the solution
in each coarse block, the uncertainty space needs to be partitioned
(coarsened). This coarsening collects realizations that provide similar
multiscale features as outlined in GMsFEM (or other method of choice). This
step is known to be computationally demanding as it requires many local solves
and clustering based on them. In this paper, we take a different approach and
learn coarsening the uncertainty space. Our methods use deep learning
techniques in identifying clusters(coarsening) in the uncertainty space. We use
convolutional neural networks combined with some techniques in adversary neural
networks. We define appropriate loss functions in the proposed neural networks,
where the loss function is composed of several parts that includes terms
related to clusters and reconstruction of basis functions. We present numerical
results for channelized permeability fields in the examples of flows in porous
media
A data-driven approach for multiscale elliptic PDEs with random coefficients based on intrinsic dimension reduction
We propose a data-driven approach to solve multiscale elliptic PDEs with
random coefficients based on the intrinsic low dimension structure of the
underlying elliptic differential operators. Our method consists of offline and
online stages. At the offline stage, a low dimension space and its basis are
extracted from the data to achieve significant dimension reduction in the
solution space. At the online stage, the extracted basis will be used to solve
a new multiscale elliptic PDE efficiently. The existence of low dimension
structure is established by showing the high separability of the underlying
Green's functions. Different online construction methods are proposed depending
on the problem setup. We provide error analysis based on the sampling error and
the truncation threshold in building the data-driven basis. Finally, we present
numerical examples to demonstrate the accuracy and efficiency of the proposed
method