227,757 research outputs found
Sparse Generalized Multiscale Finite Element Methods and their applications
In a number of previous papers, local (coarse grid) multiscale model
reduction techniques are developed using a Generalized Multiscale Finite
Element Method. In these approaches, multiscale basis functions are constructed
using local snapshot spaces, where a snapshot space is a large space that
represents the solution behavior in a coarse block. In a number of applications
(e.g., those discussed in the paper), one may have a sparsity in the snapshot
space for an appropriate choice of a snapshot space. More precisely, the
solution may only involve a portion of the snapshot space. In this case, one
can use sparsity techniques to identify multiscale basis functions. In this
paper, we consider two such sparse local multiscale model reduction approaches.
In the first approach (which is used for parameter-dependent multiscale
PDEs), we use local minimization techniques, such as sparse POD, to identify
multiscale basis functions, which are sparse in the snapshot space. These
minimization techniques use minimization to find local multiscale basis
functions, which are further used for finding the solution. In the second
approach (which is used for the Helmholtz equation), we directly apply
minimization techniques to solve the underlying PDEs. This approach is more
expensive as it involves a large snapshot space; however, in this example, we
can not identify a local minimization principle, such as local generalized SVD
Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
In this paper, we discuss a general multiscale model reduction framework
based on multiscale finite element methods. We give a brief overview of related
multiscale methods. Due to page limitations, the overview focuses on a few
related methods and is not intended to be comprehensive. We present a general
adaptive multiscale model reduction framework, the Generalized Multiscale
Finite Element Method. Besides the method's basic outline, we discuss some
important ingredients needed for the method's success. We also discuss several
applications. The proposed method allows performing local model reduction in
the presence of high contrast and no scale separation
Local Improvements to Reduced-Order Approximations of Optimal Control Problems Governed by Diffusion-Convection-Reaction Equation
We consider the optimal control problem governed by diffusion convection
reaction equation without control constraints. The proper orthogonal
decomposition(POD) method is used to reduce the dimension of the problem. The
POD method may be lack of accuracy if the POD basis depending on a set of
parameters is used to approximate the problem depending on a different set of
parameters. We are interested in the perturbation of diffusion term. To
increase the accuracy and robustness of the basis, we compute three bases
additional to the baseline POD. The first two of them use the sensitivity
information to extrapolate and expand the POD basis. The other one is based on
the subspace angle interpolation method. We compare these different bases in
terms of accuracy and complexity and investigate the advantages and main
drawbacks of them.Comment: 19 pages, 5 figures, 2 table
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