28 research outputs found
Bayesian Numerical Homogenization
Numerical homogenization, i.e. the finite-dimensional approximation of
solution spaces of PDEs with arbitrary rough coefficients, requires the
identification of accurate basis elements. These basis elements are oftentimes
found after a laborious process of scientific investigation and plain
guesswork. Can this identification problem be facilitated? Is there a general
recipe/decision framework for guiding the design of basis elements? We suggest
that the answer to the above questions could be positive based on the
reformulation of numerical homogenization as a Bayesian Inference problem in
which a given PDE with rough coefficients (or multi-scale operator) is excited
with noise (random right hand side/source term) and one tries to estimate the
value of the solution at a given point based on a finite number of
observations. We apply this reformulation to the identification of bases for
the numerical homogenization of arbitrary integro-differential equations and
show that these bases have optimal recovery properties. In particular we show
how Rough Polyharmonic Splines can be re-discovered as the optimal solution of
a Gaussian filtering problem.Comment: 22 pages. To appear in SIAM Multiscale Modeling and Simulatio
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Mini-Workshop: Numerical Upscaling for Flow Problems: Theory and Applications
The objective of this workshop was to bring together researchers working in multiscale simulations with emphasis on multigrid methods and multiscale ïŹnite element methods, aiming at chieving of better understanding and synergy between these methods. The goal of multiscale ïŹnite element methods, as upscaling methods, is to compute coarse scale solutions of the underlying equations as accurately as possible. On the other hand, multigrid methods attempt to solve ïŹne-scale equations rapidly using a hierarchy of coarse spaces. Multigrid methods need âgoodâ coarse scale spaces for their eïŹciency. The discussions of this workshop partly focused on approximation properties of coarse scale spaces and multigrid convergence. Some other presentations were on upscaling, domain decomposition methods and nonlinear multiscale methods. Some researchers discussed applications of these methods to reservoir simulations, as well as to simulations of ïŹltration, insulating materials, and turbulence
Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast
We construct finite-dimensional approximations of solution spaces of
divergence form operators with -coefficients. Our method does not
rely on concepts of ergodicity or scale-separation, but on the property that
the solution space of these operators is compactly embedded in if source
terms are in the unit ball of instead of the unit ball of .
Approximation spaces are generated by solving elliptic PDEs on localized
sub-domains with source terms corresponding to approximation bases for .
The -error estimates show that -dimensional spaces
with basis elements localized to sub-domains of diameter (with ) result in an
accuracy for elliptic, parabolic and hyperbolic
problems. For high-contrast media, the accuracy of the method is preserved
provided that localized sub-domains contain buffer zones of width
where the contrast of the medium
remains bounded. The proposed method can naturally be generalized to vectorial
equations (such as elasto-dynamics).Comment: Accepted for publication in SIAM MM
Expanded mixed multiscale finite element methods and their applications for flows in porous media
We develop a family of expanded mixed Multiscale Finite Element Methods
(MsFEMs) and their hybridizations for second-order elliptic equations. This
formulation expands the standard mixed Multiscale Finite Element formulation in
the sense that four unknowns (hybrid formulation) are solved simultaneously:
pressure, gradient of pressure, velocity and Lagrange multipliers. We use
multiscale basis functions for the both velocity and gradient of pressure. In
the expanded mixed MsFEM framework, we consider both cases of separable-scale
and non-separable spatial scales. We specifically analyze the methods in three
categories: periodic separable scales, - convergence separable scales, and
continuum scales. When there is no scale separation, using some global
information can improve accuracy for the expanded mixed MsFEMs. We present
rigorous convergence analysis for expanded mixed MsFEMs. The analysis includes
both conforming and nonconforming expanded mixed MsFEM. Numerical results are
presented for various multiscale models and flows in porous media with shales
to illustrate the efficiency of the expanded mixed MsFEMs.Comment: 33 page
Local multiscale model reduction using discontinuous Galerkin coupling for elasticity problems
In this paper, we consider the constrained energy minimizing generalized
multiscale finite element method (CEM-GMsFEM) with discontinuous Galerkin (DG)
coupling for the linear elasticity equations in highly heterogeneous and high
contrast media. We will introduce the construction of a DG version of the
CEM-GMsFEM, such as auxiliary basis functions and offline basis functions. The
DG version of the method offers some advantages such as flexibility in coarse
grid construction and sparsity of resulting discrete systems. Moreover, to our
best knowledge, this is the first time where the proof of the convergence of
the CEM-GMsFEM in the DG form is given. Some numerical examples will be
presented to illustrate the performance of the method