4,617 research outputs found
Numerical Homogenization of Heterogeneous Fractional Laplacians
In this paper, we develop a numerical multiscale method to solve the
fractional Laplacian with a heterogeneous diffusion coefficient. When the
coefficient is heterogeneous, this adds to the computational costs. Moreover,
the fractional Laplacian is a nonlocal operator in its standard form, however
the Caffarelli-Silvestre extension allows for a localization of the equations.
This adds a complexity of an extra spacial dimension and a singular/degenerate
coefficient depending on the fractional order. Using a sub-grid correction
method, we correct the basis functions in a natural weighted Sobolev space and
show that these corrections are able to be truncated to design a
computationally efficient scheme with optimal convergence rates. A key
ingredient of this method is the use of quasi-interpolation operators to
construct the fine scale spaces. Since the solution of the extended problem on
the critical boundary is of main interest, we construct a projective
quasi-interpolation that has both and dimensional averages over
subsets in the spirit of the Scott-Zhang operator. We show that this operator
satisfies local stability and local approximation properties in weighted
Sobolev spaces. We further show that we can obtain a greater rate of
convergence for sufficient smooth forces, and utilizing a global
projection on the critical boundary. We present some numerical examples,
utilizing our projective quasi-interpolation in dimension for analytic
and heterogeneous cases to demonstrate the rates and effectiveness of the
method
Concentration analysis and cocompactness
Loss of compactness that occurs in may significant PDE settings can be
expressed in a well-structured form of profile decomposition for sequences.
Profile decompositions are formulated in relation to a triplet , where
and are Banach spaces, , and is, typically, a
set of surjective isometries on both and . A profile decomposition is a
representation of a bounded sequence in as a sum of elementary
concentrations of the form , , , and a remainder that
vanishes in . A necessary requirement for is, therefore, that any
sequence in that develops no -concentrations has a subsequence
convergent in the norm of . An imbedding with this
property is called -cocompact, a property weaker than, but related to,
compactness. We survey known cocompact imbeddings and their role in profile
decompositions
Asymptotic equivalence for regression under fractional noise
Consider estimation of the regression function based on a model with
equidistant design and measurement errors generated from a fractional Gaussian
noise process. In previous literature, this model has been heuristically linked
to an experiment, where the anti-derivative of the regression function is
continuously observed under additive perturbation by a fractional Brownian
motion. Based on a reformulation of the problem using reproducing kernel
Hilbert spaces, we derive abstract approximation conditions on function spaces
under which asymptotic equivalence between these models can be established and
show that the conditions are satisfied for certain Sobolev balls exceeding some
minimal smoothness. Furthermore, we construct a sequence space representation
and provide necessary conditions for asymptotic equivalence to hold.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1262 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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