6,151 research outputs found
Likelihood Analysis for Mega-Pixel Maps
The derivation of cosmological parameters from astrophysical data sets
routinely involves operations counts which scale as O(N^3) where N is the
number of data points. Currently planned missions, including MAP and Planck,
will generate sky maps with N_d = 10^6 or more pixels. Simple ``brute force''
analysis, applied to such mega-pixel data, would require years of computing
even on the fastest computers. We describe an algorithm which allows estimation
of the likelihood function in the direct pixel basis. The algorithm uses a
conjugate gradient approach to evaluate chi-squared and a geometric
approximation to evaluate the determinant. Monte Carlo simulations provide a
correction to the determinant, yielding an unbiased estimate of the likelihood
surface in an arbitrary region surrounding the likelihood peak. The algorithm
requires O(N_d^{3/2}) operations and O(N_d) storage for each likelihood
evaluation, and allows for significant parallel computation.Comment: 9 pages LaTeX including 2 PostScript figures. Additional discussion
of conjugate gradient chi-squared algorithm. Matches accepted versio
Combined 3D thinning and greedy algorithm to approximate realistic particles with corrected mechanical properties
The shape of irregular particles has significant influence on micro- and
macro-scopic behavior of granular systems. This paper presents a combined 3D
thinning and greedy set-covering algorithm to approximate realistic particles
with a clump of overlapping spheres for discrete element method (DEM)
simulations. First, the particle medial surface (or surface skeleton), from
which all candidate (maximal inscribed) spheres can be generated, is computed
by the topological 3D thinning. Then, the clump generation procedure is
converted into a greedy set-covering (SCP) problem.
To correct the mass distribution due to highly overlapped spheres inside the
clump, linear programming (LP) is used to adjust the density of each component
sphere, such that the aggregate properties mass, center of mass and inertia
tensor are identical or close enough to the prototypical particle. In order to
find the optimal approximation accuracy (volume coverage: ratio of clump's
volume to the original particle's volume), particle flow of 3 different shapes
in a rotating drum are conducted. It was observed that the dynamic angle of
repose starts to converge for all particle shapes at 85% volume coverage
(spheres per clump < 30), which implies the possible optimal resolution to
capture the mechanical behavior of the system.Comment: 34 pages, 13 figure
Interlocking structure design and assembly
Many objects in our life are not manufactured as whole rigid pieces. Instead, smaller components are made to be later assembled into larger structures. Chairs are assembled from wooden pieces, cabins are made of logs, and buildings are constructed from bricks. These components are commonly designed by many iterations of human thinking. In this report, we will look at a few problems related to interlocking components design and assembly. Given an atomic object, how can we design a package that holds the object firmly without a gap in-between? How many pieces should the package be partitioned into? How can we assemble/extract each piece? We will attack this problem by first looking at the lower bound on the number of pieces, then at the upper bound. Afterwards, we will propose a practical algorithm for designing these packages. We also explore a special kind of interlocking structure which has only one or a small number of movable pieces. For example, a burr puzzle. We will design a few blocks with joints whose combination can be assembled into almost any voxelized 3D model. Our blocks require very simple motions to be assembled, enabling robotic assembly. As proof of concept, we also develop a robot system to assemble the blocks. In some extreme conditions where construction components are small, controlling each component individually is impossible. We will discuss an option using global controls. These global controls can be from gravity or magnetic fields. We show that in some special cases where the small units form a rectangular matrix, rearrangement can be done in a small space following a technique similar to bubble sort algorithm
Inertial Douglas-Rachford splitting for monotone inclusion problems
We propose an inertial Douglas-Rachford splitting algorithm for finding the
set of zeros of the sum of two maximally monotone operators in Hilbert spaces
and investigate its convergence properties. To this end we formulate first the
inertial version of the Krasnosel'ski\u{\i}--Mann algorithm for approximating
the set of fixed points of a nonexpansive operator, for which we also provide
an exhaustive convergence analysis. By using a product space approach we employ
these results to the solving of monotone inclusion problems involving linearly
composed and parallel-sum type operators and provide in this way iterative
schemes where each of the maximally monotone mappings is accessed separately
via its resolvent. We consider also the special instance of solving a
primal-dual pair of nonsmooth convex optimization problems and illustrate the
theoretical results via some numerical experiments in clustering and location
theory.Comment: arXiv admin note: text overlap with arXiv:1402.529
Laminated Wave Turbulence: Generic Algorithms II
The model of laminated wave turbulence puts forth a novel computational
problem - construction of fast algorithms for finding exact solutions of
Diophantine equations in integers of order and more. The equations to
be solved in integers are resonant conditions for nonlinearly interacting waves
and their form is defined by the wave dispersion. It is established that for
the most common dispersion as an arbitrary function of a wave-vector length two
different generic algorithms are necessary: (1) one-class-case algorithm for
waves interacting through scales, and (2) two-class-case algorithm for waves
interacting through phases. In our previous paper we described the
one-class-case generic algorithm and in our present paper we present the
two-class-case generic algorithm.Comment: to appear in J. "Communications in Computational Physics" (2006
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