15,548 research outputs found
A Finite Element Method With Singularity Reconstruction for Fractional Boundary Value Problems
We consider a two-point boundary value problem involving a Riemann-Liouville
fractional derivative of order \al\in (1,2) in the leading term on the unit
interval . Generally the standard Galerkin finite element method can
only give a low-order convergence even if the source term is very smooth due to
the presence of the singularity term x^{\al-1} in the solution
representation. In order to enhance the convergence, we develop a simple
singularity reconstruction strategy by splitting the solution into a singular
part and a regular part, where the former captures explicitly the singularity.
We derive a new variational formulation for the regular part, and establish
that the Galerkin approximation of the regular part can achieve a better
convergence order in the , H^{\al/2}(0,1) and -norms
than the standard Galerkin approach, with a convergence rate for the recovered
singularity strength identical with the error estimate. The
reconstruction approach is very flexible in handling explicit singularity, and
it is further extended to the case of a Neumann type boundary condition on the
left end point, which involves a strong singularity x^{\al-2}. Extensive
numerical results confirm the theoretical study and efficiency of the proposed
approach.Comment: 23 pp. ESAIM: Math. Model. Numer. Anal., to appea
An Analysis of Galerkin Proper Orthogonal Decomposition for Subdiffusion
In this work, we develop a novel Galerkin-L1-POD scheme for the subdiffusion
model with a Caputo fractional derivative of order in time,
which is often used to describe anomalous diffusion processes in heterogeneous
media. The nonlocality of the fractional derivative requires storing all the
solutions from time zero. The proposed scheme is based on continuous piecewise
linear finite elements, L1 time stepping, and proper orthogonal decomposition
(POD). By constructing an effective reduced-order scheme using problem-adapted
basis functions, it can significantly reduce the computational complexity and
storage requirement. We shall provide a complete error analysis of the scheme
under realistic regularity assumptions by means of a novel energy argument.
Extensive numerical experiments are presented to verify the convergence
analysis and the efficiency of the proposed scheme.Comment: 25 pp, 5 figure
Microstructural Characterization of Shrouded Plasma-Sprayed Titanium Coatings
Titanium and its alloys are often used for corrosion protection because they are able to
offer high chemical resistance against various corrosive media. In this paper, shrouded plasma spray
technology was applied to produce titanium coatings. A solid shroud with an external shrouding
gas was used to plasma spray titanium powder feedstock with aim of reducing the oxide content
in the as-sprayed coatings. The titanium coatings were assessed by optical microscope, scanning
electron microscopy, X-ray diffraction, LECO combustion method and Vickers microhardness testing.
The results showed that the presence of the shroud and the external shrouding gas led to a dense
microstructure with a low porosity in the plasma-sprayed titanium coatings. The oxygen and nitrogen
contents in the titanium coating were kept at a low level due to the shielding effect of the shroud
attachment and the external shrouding gas. The dominant phase in the shrouded titanium coatings
was mainly composed of α-Ti phase, which was very similar to the titanium feedstock powders.
The shrouded plasma-sprayed titanium coatings had a Vickers microhardness of 404.2 ± 103.2 HV
Multi-View Active Learning in the Non-Realizable Case
The sample complexity of active learning under the realizability assumption
has been well-studied. The realizability assumption, however, rarely holds in
practice. In this paper, we theoretically characterize the sample complexity of
active learning in the non-realizable case under multi-view setting. We prove
that, with unbounded Tsybakov noise, the sample complexity of multi-view active
learning can be , contrasting to
single-view setting where the polynomial improvement is the best possible
achievement. We also prove that in general multi-view setting the sample
complexity of active learning with unbounded Tsybakov noise is
, where the order of is
independent of the parameter in Tsybakov noise, contrasting to previous
polynomial bounds where the order of is related to the parameter
in Tsybakov noise.Comment: 22 pages, 1 figur
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