38,041 research outputs found
Highly accurate special quadrature methods for Stokesian particle suspensions in confined geometries
Boundary integral methods are highly suited for problems with complicated
geometries, but require special quadrature methods to accurately compute the
singular and nearly singular layer potentials that appear in them. This paper
presents a boundary integral method that can be used to study the motion of
rigid particles in three-dimensional periodic Stokes flow with confining walls.
A centrepiece of our method is the highly accurate special quadrature method,
which is based on a combination of upsampled quadrature and quadrature by
expansion (QBX), accelerated using a precomputation scheme. The method is
demonstrated for rodlike and spheroidal particles, with the confining geometry
given by a pipe or a pair of flat walls. A parameter selection strategy for the
special quadrature method is presented and tested. Periodic interactions are
computed using the Spectral Ewald (SE) fast summation method, which allows our
method to run in O(n log n) time for n grid points, assuming the number of
geometrical objects grows while the grid point concentration is kept fixed.Comment: 46 pages, 41 figure
Adaptive probability scheme for behaviour monitoring of the elderly using a specialised ambient device
A Hidden Markov Model (HMM) modified to work in combination with a Fuzzy System is utilised to determine the current behavioural state of the user from information obtained with specialised hardware. Due to the high dimensionality and not-linearly-separable nature of the Fuzzy System and the sensor data obtained with the hardware which informs the state decision, a new method is devised to update the HMM and replace the initial Fuzzy System such that subsequent state decisions are based on the most recent information. The resultant system first reduces the dimensionality of the original information by using a manifold representation in the high dimension which is unfolded in the lower dimension. The data is then linearly separable in the lower dimension where a simple linear classifier, such as the perceptron used here, is applied to determine the probability of the observations belonging to a state. Experiments using the new system verify its applicability in a real scenario
High-dimensional estimation with geometric constraints
Consider measuring an n-dimensional vector x through the inner product with
several measurement vectors, a_1, a_2, ..., a_m. It is common in both signal
processing and statistics to assume the linear response model y_i = +
e_i, where e_i is a noise term. However, in practice the precise relationship
between the signal x and the observations y_i may not follow the linear model,
and in some cases it may not even be known. To address this challenge, in this
paper we propose a general model where it is only assumed that each observation
y_i may depend on a_i only through . We do not assume that the
dependence is known. This is a form of the semiparametric single index model,
and it includes the linear model as well as many forms of the generalized
linear model as special cases. We further assume that the signal x has some
structure, and we formulate this as a general assumption that x belongs to some
known (but arbitrary) feasible set K. We carefully detail the benefit of using
the signal structure to improve estimation. The theory is based on the mean
width of K, a geometric parameter which can be used to understand its effective
dimension in estimation problems. We determine a simple, efficient two-step
procedure for estimating the signal based on this model -- a linear estimation
followed by metric projection onto K. We give general conditions under which
the estimator is minimax optimal up to a constant. This leads to the intriguing
conclusion that in the high noise regime, an unknown non-linearity in the
observations does not significantly reduce one's ability to determine the
signal, even when the non-linearity may be non-invertible. Our results may be
specialized to understand the effect of non-linearities in compressed sensing.Comment: This version incorporates minor revisions suggested by referee
Far-Field Compression for Fast Kernel Summation Methods in High Dimensions
We consider fast kernel summations in high dimensions: given a large set of
points in dimensions (with ) and a pair-potential function (the
{\em kernel} function), we compute a weighted sum of all pairwise kernel
interactions for each point in the set. Direct summation is equivalent to a
(dense) matrix-vector multiplication and scales quadratically with the number
of points. Fast kernel summation algorithms reduce this cost to log-linear or
linear complexity.
Treecodes and Fast Multipole Methods (FMMs) deliver tremendous speedups by
constructing approximate representations of interactions of points that are far
from each other. In algebraic terms, these representations correspond to
low-rank approximations of blocks of the overall interaction matrix. Existing
approaches require an excessive number of kernel evaluations with increasing
and number of points in the dataset.
To address this issue, we use a randomized algebraic approach in which we
first sample the rows of a block and then construct its approximate, low-rank
interpolative decomposition. We examine the feasibility of this approach
theoretically and experimentally. We provide a new theoretical result showing a
tighter bound on the reconstruction error from uniformly sampling rows than the
existing state-of-the-art. We demonstrate that our sampling approach is
competitive with existing (but prohibitively expensive) methods from the
literature. We also construct kernel matrices for the Laplacian, Gaussian, and
polynomial kernels -- all commonly used in physics and data analysis. We
explore the numerical properties of blocks of these matrices, and show that
they are amenable to our approach. Depending on the data set, our randomized
algorithm can successfully compute low rank approximations in high dimensions.
We report results for data sets with ambient dimensions from four to 1,000.Comment: 43 pages, 21 figure
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