502 research outputs found
Some error estimates for the lumped mass finite element method for a parabolic problem
We study the spatially semidiscrete lumped mass method for the model homogeneous heat equation with homogeneous Dirichlet boundary conditions. Improving earlier results we show that known optimal order smooth initial data error estimates for the standard Galerkin method carry over to the lumped mass method whereas nonsmooth initial data estimates require special assumptions on the triangulation. We also discuss the application to time discretization by the backward Euler and Crank-Nicolson methods
Some error estimates for the finite volume element method for a parabolic problem
We study spatially semidiscrete and fully discrete finite volume element
methods for the homogeneous heat equation with homogeneous Dirichlet boundary
conditions and derive error estimates for smooth and nonsmooth initial data. We
show that the results of our earlier work \cite{clt11} for the lumped mass
method carry over to the present situation. In particular, in order for error
estimates for initial data only in to be of optimal second order for
positive time, a special condition is required, which is satisfied for
symmetric triangulations. Without any such condition, only first order
convergence can be shown, which is illustrated by a counterexample.
Improvements hold for triangulations that are almost symmetric and piecewise
almost symmetric
Precaching of Incomplete Content and Its On Demand Completion
A mechanism to separate a content item provided by a content item service into a first portion that is to be pre-cached by a device of a user and a second portion that is to be initially withheld from the device of the user. The first portion of the content item may correspond to a majority of the content item and may be transmitted to the device. Subsequently, the user may request playback of the content item. In response to the request, the device of the user may retrieve the second portion of the content item that was previously withheld from the user. The second portion may then be inserted into the first portion of the content item so that the complete content item may be played
Describing and Understanding Neighborhood Characteristics through Online Social Media
Geotagged data can be used to describe regions in the world and discover
local themes. However, not all data produced within a region is necessarily
specifically descriptive of that area. To surface the content that is
characteristic for a region, we present the geographical hierarchy model (GHM),
a probabilistic model based on the assumption that data observed in a region is
a random mixture of content that pertains to different levels of a hierarchy.
We apply the GHM to a dataset of 8 million Flickr photos in order to
discriminate between content (i.e., tags) that specifically characterizes a
region (e.g., neighborhood) and content that characterizes surrounding areas or
more general themes. Knowledge of the discriminative and non-discriminative
terms used throughout the hierarchy enables us to quantify the uniqueness of a
given region and to compare similar but distant regions. Our evaluation
demonstrates that our model improves upon traditional Naive Bayes
classification by 47% and hierarchical TF-IDF by 27%. We further highlight the
differences and commonalities with human reasoning about what is locally
characteristic for a neighborhood, distilled from ten interviews and a survey
that covered themes such as time, events, and prior regional knowledgeComment: Accepted in WWW 2015, 2015, Florence, Ital
DeeSIL: Deep-Shallow Incremental Learning
Incremental Learning (IL) is an interesting AI problem when the algorithm is
assumed to work on a budget. This is especially true when IL is modeled using a
deep learning approach, where two com- plex challenges arise due to limited
memory, which induces catastrophic forgetting and delays related to the
retraining needed in order to incorpo- rate new classes. Here we introduce
DeeSIL, an adaptation of a known transfer learning scheme that combines a fixed
deep representation used as feature extractor and learning independent shallow
classifiers to in- crease recognition capacity. This scheme tackles the two
aforementioned challenges since it works well with a limited memory budget and
each new concept can be added within a minute. Moreover, since no deep re-
training is needed when the model is incremented, DeeSIL can integrate larger
amounts of initial data that provide more transferable features. Performance is
evaluated on ImageNet LSVRC 2012 against three state of the art algorithms.
Results show that, at scale, DeeSIL performance is 23 and 33 points higher than
the best baseline when using the same and more initial data respectively
A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem
We analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on m >= 1 intervals of length k/m for the convection part. With h the mesh width in space, this results in an error bound of the form C(0)h(2) + C(m)k for appropriately smooth solutions, where C-m <= C\u27 + C-\u27\u27/m. This work complements the earlier study [V. Thomee and A. S. Vasudeva Murthy, An explicit- implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 (2019), no. 2, 283-293] based on the second-order Strang splitting
Optimal control of number squeezing in trapped Bose-Einstein condensates
We theoretically analyze atom interferometry based on trapped ultracold
atoms, and employ optimal control theory in order to optimize number squeezing
and condensate trapping. In our simulations, we consider a setup where the
confinement potential is transformed from a single to a double well, which
allows to split the condensate. To avoid in the ensuing phase-accumulation
stage of the interferometer dephasing due to the nonlinear atom-atom
interactions, the atom number fluctuations between the two wells should be
sufficiently low. We show that low number fluctuations (high number squeezing)
can be obtained by optimized splitting protocols. Two types of solutions are
found: in the Josephson regime we find an oscillatory tunnel control and a
parametric amplification of number squeezing, while in the Fock regime
squeezing is obtained solely due to the nonlinear coupling, which is
transformed to number squeezing by peaked tunnel pulses. We study splitting and
squeezing within the frameworks of a generic two-mode model, which allows us to
study the basic physical mechanisms, and the multi-configurational time
dependent Hartree for bosons method, which allows for a microscopic modeling of
the splitting dynamics in realistic experiments. Both models give similar
results, thus highlighting the general nature of these two solution schemes. We
finally analyze our results in the context of atom interferometry.Comment: 17 pages, 21 figures, minor correction
The lifecycle of geotagged data
The world is a big place. At any given instant something is happening somewhere, but even when nothing in particular is going on people still find ways to generate data, such as posting on s
An Explicit-Implicit Splitting Method for a Convection-Diffusion Problem
We analyze a second-order accurate finite difference method for a spatially periodic convection-diffusion problem. The method is a time stepping method based on the Strang splitting of the spatially semidiscrete solution, in which the diffusion part uses the Crank-Nicolson method and the convection part the explicit forward Euler approximation on a shorter time interval. When the diffusion coefficient is small, the forward Euler method may be used also for the diffusion term
- …