2,993 research outputs found
Port reduction in parametrized component static condensation: approximation and a posteriori error estimation
We introduce a port (interface) approximation and a posteriori error bound framework for a general component-based static condensation method in the context of parameter-dependent linear elliptic partial differential equations. The key ingredients are as follows: (i) efficient empirical port approximation spaces—the dimensions of these spaces may be chosen small to reduce the computational cost associated with formation and solution of the static condensation system; and (ii) a computationally tractable a posteriori error bound realized through a non-conforming approximation and associated conditioner—the error in the global system approximation, or in a scalar output quantity, may be bounded relatively sharply with respect to the underlying finite element discretization.
Our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element (SCRBE) method. We provide several numerical examples within the SCRBE context, which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds.Research Council of NorwayUnited States. Office of Naval Research (Grant N00014-11-0713
Accurate and efficient evaluation of the a posteriori error estimator in the reduced basis method
The reduced basis method is a model reduction technique yielding substantial
savings of computational time when a solution to a parametrized equation has to
be computed for many values of the parameter. Certification of the
approximation is possible by means of an a posteriori error bound. Under
appropriate assumptions, this error bound is computed with an algorithm of
complexity independent of the size of the full problem. In practice, the
evaluation of the error bound can become very sensitive to round-off errors. We
propose herein an explanation of this fact. A first remedy has been proposed in
[F. Casenave, Accurate \textit{a posteriori} error evaluation in the reduced
basis method. \textit{C. R. Math. Acad. Sci. Paris} \textbf{350} (2012)
539--542.]. Herein, we improve this remedy by proposing a new approximation of
the error bound using the Empirical Interpolation Method (EIM). This method
achieves higher levels of accuracy and requires potentially less
precomputations than the usual formula. A version of the EIM stabilized with
respect to round-off errors is also derived. The method is illustrated on a
simple one-dimensional diffusion problem and a three-dimensional acoustic
scattering problem solved by a boundary element method.Comment: 26 pages, 10 figures. ESAIM: Mathematical Modelling and Numerical
Analysis, 201
Randomized Local Model Order Reduction
In this paper we propose local approximation spaces for localized model order
reduction procedures such as domain decomposition and multiscale methods. Those
spaces are constructed from local solutions of the partial differential
equation (PDE) with random boundary conditions, yield an approximation that
converges provably at a nearly optimal rate, and can be generated at close to
optimal computational complexity. In many localized model order reduction
approaches like the generalized finite element method, static condensation
procedures, and the multiscale finite element method local approximation spaces
can be constructed by approximating the range of a suitably defined transfer
operator that acts on the space of local solutions of the PDE. Optimal local
approximation spaces that yield in general an exponentially convergent
approximation are given by the left singular vectors of this transfer operator
[I. Babu\v{s}ka and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However,
the direct calculation of these singular vectors is computationally very
expensive. In this paper, we propose an adaptive randomized algorithm based on
methods from randomized linear algebra [N. Halko et al. 2011], which constructs
a local reduced space approximating the range of the transfer operator and thus
the optimal local approximation spaces. The adaptive algorithm relies on a
probabilistic a posteriori error estimator for which we prove that it is both
efficient and reliable with high probability. Several numerical experiments
confirm the theoretical findings.Comment: 31 pages, 14 figures, 1 table, 1 algorith
Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings
In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinley parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth "parametric manifold" - dimension reduction, an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations - rapid convergence, an a posteriori error estimation procedures - rigorous and sharp bounds for the functional outputs related with the underlying solution or related quantities of interest, like stress intensity factor, and Offline-Online computational decomposition strategies - minimum marginal cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present several illustrative results for linear elasticity problem in parametrized geometries representing 2D Cartesian or 3D axisymmetric configurations like an arc-cantilever beam, a center crack problem, a composite unit cell or a woven composite beam, a multi-material plate, and a closed vessel. We consider different parametrization for the systems: either physical quantities - to model the materials and loads - and geometrical parameters - to model different geometrical configurations - with isotropic and orthotropic materials working in plane stress and plane strain approximation. We would like to underline the versatility of the methodology in very different problems. As last example we provide a nonlinear setting with increased complexity
Parametric and nonparametric inference in equilibrium job search models
Equilibrium job search models allow for labor markets with homogeneous workers and firms to yield nondegenerate wage densities. However, the resulting wage densities do not accord well with empirical regularities. Accordingly, many extensions to the basic equilibrium search model have been considered (e.g., heterogeneity in productivity, heterogeneity in the value of leisure, etc.). It is increasingly common to use nonparametric forms for these extensions and, hence, researchers can obtain a perfect fit (in a kernel smoothed sense) between theoretical and empirical wage densities. This makes it difficult to carry out model comparison of different model extensions. In this paper, we first develop Bayesian parametric and nonparametric methods which are comparable to the existing non-Bayesian literature. We then show how Bayesian methods can be used to compare various nonparametric equilibrium search models in a statistically rigorous sense
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