207 research outputs found
A reduced basis localized orthogonal decomposition
In this work we combine the framework of the Reduced Basis method (RB) with
the framework of the Localized Orthogonal Decomposition (LOD) in order to solve
parametrized elliptic multiscale problems. The idea of the LOD is to split a
high dimensional Finite Element space into a low dimensional space with
comparably good approximation properties and a remainder space with negligible
information. The low dimensional space is spanned by locally supported basis
functions associated with the node of a coarse mesh obtained by solving
decoupled local problems. However, for parameter dependent multiscale problems,
the local basis has to be computed repeatedly for each choice of the parameter.
To overcome this issue, we propose an RB approach to compute in an "offline"
stage LOD for suitable representative parameters. The online solution of the
multiscale problems can then be obtained in a coarse space (thanks to the LOD
decomposition) and for an arbitrary value of the parameters (thanks to a
suitable "interpolation" of the selected RB). The online RB-LOD has a basis
with local support and leads to sparse systems. Applications of the strategy to
both linear and nonlinear problems are given
Localized orthogonal decomposition method for the wave equation with a continuum of scales
This paper is devoted to numerical approximations for the wave equation with
a multiscale character. Our approach is formulated in the framework of the
Localized Orthogonal Decomposition (LOD) interpreted as a numerical
homogenization with an -projection. We derive explicit convergence rates
of the method in the -, - and
-norms without any assumptions on higher order space
regularity or scale-separation. The order of the convergence rates depends on
further graded assumptions on the initial data. We also prove the convergence
of the method in the framework of G-convergence without any structural
assumptions on the initial data, i.e. without assuming that it is
well-prepared. This rigorously justifies the method. Finally, the performance
of the method is demonstrated in numerical experiments
A Bayesian numerical homogenization method for elliptic multiscale inverse problems
A new strategy based on numerical homogenization and Bayesian techniques for
solving multiscale inverse problems is introduced. We consider a class of
elliptic problems which vary at a microscopic scale, and we aim at recovering
the highly oscillatory tensor from measurements of the fine scale solution at
the boundary, using a coarse model based on numerical homogenization and model
order reduction. We provide a rigorous Bayesian formulation of the problem,
taking into account different possibilities for the choice of the prior
measure. We prove well-posedness of the effective posterior measure and, by
means of G-convergence, we establish a link between the effective posterior and
the fine scale model. Several numerical experiments illustrate the efficiency
of the proposed scheme and confirm the theoretical findings
Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems
This paper presents two new approaches for finding the homogenized
coefficients of multiscale elliptic PDEs. Standard approaches for computing the
homogenized coefficients suffer from the so-called resonance error, originating
from a mismatch between the true and the computational boundary conditions. Our
new methods, based on solutions of parabolic and elliptic cell-problems, result
in an exponential decay of the resonance error
A priori and a posteriori error analysis of a QC method for complex lattices
In this paper we prove a priori and a posteriori error estimates for a
multiscale numerical method for computing equilibria of multilattices under an
external force. The error estimates are derived in a norm in one
space dimension. One of the features of our analysis is that we establish an
equivalent way of formulating the coarse-grained problem which greatly
simplifies derivation of the error bounds (both, a priori and a posteriori). We
illustrate our error estimates with numerical experiments.Comment: 23 page
A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems
The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L 2 and the H 1 norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rate
A probabilistic finite element method based on random meshes: Error estimators and Bayesian inverse problems
We present a novel probabilistic finite element method (FEM) for the solution
and uncertainty quantification of elliptic partial differential equations based
on random meshes, which we call random mesh FEM (RM-FEM). Our methodology
allows to introduce a probability measure on standard piecewise linear FEM. We
present a posteriori error estimators based uniquely on probabilistic
information. A series of numerical experiments illustrates the potential of the
RM-FEM for error estimation and validates our analysis. We furthermore
demonstrate how employing the RM-FEM enhances the quality of the solution of
Bayesian inverse problems, thus allowing a better quantification of numerical
errors in pipelines of computations
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