537 research outputs found
RBF multiscale collocation for second order elliptic boundary value problems
In this paper, we discuss multiscale radial basis function collocation methods for solving elliptic partial differential equations on bounded domains. The approximate solution is constructed in a multi-level fashion, each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh. On each level, standard symmetric collocation is employed. A convergence theory is given, which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions. We are able to show that the convergence is linear in the number of levels. We also discuss the condition numbers of the arising systems and the effect of simple, diagonal preconditioners, now proving rigorously previous numerical observations
Preconditioning of weighted H(div)-norm and applications to numerical simulation of highly heterogeneous media
In this paper we propose and analyze a preconditioner for a system arising
from a finite element approximation of second order elliptic problems
describing processes in highly het- erogeneous media. Our approach uses the
technique of multilevel methods and the recently proposed preconditioner based
on additive Schur complement approximation by J. Kraus (see [8]). The main
results are the design and a theoretical and numerical justification of an
iterative method for such problems that is robust with respect to the contrast
of the media, defined as the ratio between the maximum and minimum values of
the coefficient (related to the permeability/conductivity).Comment: 28 page
Well-posedness and Robust Preconditioners for the Discretized Fluid-Structure Interaction Systems
In this paper we develop a family of preconditioners for the linear algebraic
systems arising from the arbitrary Lagrangian-Eulerian discretization of some
fluid-structure interaction models. After the time discretization, we formulate
the fluid-structure interaction equations as saddle point problems and prove
the uniform well-posedness. Then we discretize the space dimension by finite
element methods and prove their uniform well-posedness by two different
approaches under appropriate assumptions. The uniform well-posedness makes it
possible to design robust preconditioners for the discretized fluid-structure
interaction systems. Numerical examples are presented to show the robustness
and efficiency of these preconditioners.Comment: 1. Added two preconditioners into the analysis and implementation 2.
Rerun all the numerical tests 3. changed title, abstract and corrected lots
of typos and inconsistencies 4. added reference
Empirical Bayes selection of wavelet thresholds
This paper explores a class of empirical Bayes methods for level-dependent
threshold selection in wavelet shrinkage. The prior considered for each wavelet
coefficient is a mixture of an atom of probability at zero and a heavy-tailed
density. The mixing weight, or sparsity parameter, for each level of the
transform is chosen by marginal maximum likelihood. If estimation is carried
out using the posterior median, this is a random thresholding procedure; the
estimation can also be carried out using other thresholding rules with the same
threshold. Details of the calculations needed for implementing the procedure
are included. In practice, the estimates are quick to compute and there is
software available. Simulations on the standard model functions show excellent
performance, and applications to data drawn from various fields of application
are used to explore the practical performance of the approach. By using a
general result on the risk of the corresponding marginal maximum likelihood
approach for a single sequence, overall bounds on the risk of the method are
found subject to membership of the unknown function in one of a wide range of
Besov classes, covering also the case of f of bounded variation. The rates
obtained are optimal for any value of the parameter p in (0,\infty],
simultaneously for a wide range of loss functions, each dominating the L_q norm
of the \sigmath derivative, with \sigma\ge0 and 0<q\le2.Comment: Published at http://dx.doi.org/10.1214/009053605000000345 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Quantum algorithms and the finite element method
The finite element method is used to approximately solve boundary value
problems for differential equations. The method discretises the parameter space
and finds an approximate solution by solving a large system of linear
equations. Here we investigate the extent to which the finite element method
can be accelerated using an efficient quantum algorithm for solving linear
equations. We consider the representative general question of approximately
computing a linear functional of the solution to a boundary value problem, and
compare the quantum algorithm's theoretical performance with that of a standard
classical algorithm -- the conjugate gradient method. Prior work had claimed
that the quantum algorithm could be exponentially faster, but did not determine
the overall classical and quantum runtimes required to achieve a predetermined
solution accuracy. Taking this into account, we find that the quantum algorithm
can achieve a polynomial speedup, the extent of which grows with the dimension
of the partial differential equation. In addition, we give evidence that no
improvement of the quantum algorithm could lead to a super-polynomial speedup
when the dimension is fixed and the solution satisfies certain smoothness
properties.Comment: 16 pages, 2 figure
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