173 research outputs found
Local Warming
Using 55 years of daily average temperatures from a local weather station, I
made a least-absolute-deviations (LAD) regression model that accounts for three
effects: seasonal variations, the 11-year solar cycle, and a linear trend. The
model was formulated as a linear programming problem and solved using widely
available optimization software. The solution indicates that temperatures have
gone up by about 2 degrees Fahrenheit over the 55 years covered by the data. It
also correctly identifies the known phase of the solar cycle; i.e., the date of
the last solar minimum. It turns out that the maximum slope of the solar cycle
sinusoid in the regression model is about the same size as the slope produced
by the linear trend. The fact that the solar cycle was correctly extracted by
the model is a strong indicator that effects of this size, in particular the
slope of the linear trend, can be accurately determined from the 55 years of
data analyzed.
The main purpose for doing this analysis is to demonstrate that it is easy to
find and analyze archived temperature data for oneself. In particular, this
problem makes a good class project for upper-level undergraduate courses in
optimization or in statistics.
It is worth noting that a similar least-squares model failed to characterize
the solar cycle correctly and hence even though it too indicates that
temperatures have been rising locally, one can be less confident in this
result.
The paper ends with a section presenting similar results from a few thousand
sites distributed world-wide, some results from a modification of the model
that includes both temperature and humidity, as well as a number of suggestions
for future work and/or ideas for enhancements that could be used as classroom
projects.Comment: 12 pages, 5 figures, to appear in SIAM Revie
A Parametric Simplex Algorithm for Linear Vector Optimization Problems
In this paper, a parametric simplex algorithm for solving linear vector
optimization problems (LVOPs) is presented. This algorithm can be seen as a
variant of the multi-objective simplex (Evans-Steuer) algorithm [12]. Different
from it, the proposed algorithm works in the parameter space and does not aim
to find the set of all efficient solutions. Instead, it finds a solution in the
sense of Loehne [16], that is, it finds a subset of efficient solutions that
allows to generate the whole frontier. In that sense, it can also be seen as a
generalization of the parametric self-dual simplex algorithm, which originally
is designed for solving single objective linear optimization problems, and is
modified to solve two objective bounded LVOPs with the positive orthant as the
ordering cone in Ruszczynski and Vanderbei [21]. The algorithm proposed here
works for any dimension, any solid pointed polyhedral ordering cone C and for
bounded as well as unbounded problems. Numerical results are provided to
compare the proposed algorithm with an objective space based LVOP algorithm
(Benson algorithm in [13]), that also provides a solution in the sense of [16],
and with Evans-Steuer algorithm [12]. The results show that for non-degenerate
problems the proposed algorithm outperforms Benson algorithm and is on par with
Evan-Steuer algorithm. For highly degenerate problems Benson's algorithm [13]
excels the simplex-type algorithms; however, the parametric simplex algorithm
is for these problems computationally much more efficient than Evans-Steuer
algorithm.Comment: 27 pages, 4 figures, 5 table
Splitting dense columns in sparse linear systems
AbstractWe consider systems of equations of the form AATx = b, where A is a sparse matrix having a small number of columns which are much denser than the other columns. These dense columns in A cause AAT to be very (or even completely) dense, which greatly limits the effectiveness of sparse-matrix techniques for directly solving the above system of equations. In the literature on interior-point methods for linear programming, the usual technique for dealing with this problem is to split A into a sparse part S and a dense part D, A = [S D], and to solve systems involving AAT in terms of the solution of systems involving SST using either conjugate-gradient techniques or the Sherman-Morrison-Woodbury formula. This approach has the difficulty that SST is often rank-deficient even when AAT has full rank. In this paper we propose an alternative method which avoids the rank-deficiency problem and at the same time allows for the effective use of sparse-matrix techniques. The resulting algorithm is both efficient and robust
Two-Mirror Apodization for High-Contrast Imaging
Direct detection of extrasolar planets will require imaging systems capable
of unprecedented contrast. Apodized pupils provide an attractive way to achieve
such contrast but they are difficult, perhaps impossible, to manufacture to the
required tolerance and they absorb about 90% of the light in order to create
the apodization, which of course lengthens the exposure times needed for planet
detection. A recently proposed alternative is to use two mirrors to accomplish
the apodization. With such a system, no light is lost. In this paper, we
provide a careful mathematical analysis, using one dimensional mirrors, of the
on-axis and off-axis performance of such a two-mirror apodization system. There
appear to be advantages and disadvantages to this approach. In addition to not
losing any light, we show that the nonuniformity of the apodization implies an
extra magnification of off-axis sources and thereby makes it possible to build
a real system with about half the aperture that one would otherwise require or,
equivalently, resolve planets at about half the angular separation as one can
achieve with standard apodization. More specifically, ignoring pointing error
and stellar disk size, a planet at ought to be at the edge of
detectability. However, we show that the non-zero size of a stellar disk pushes
the threshold for high-contrast so that a planet must be at least from its star to be detectable. The off-axis analysis of
two-dimensional mirrors is left for future study.Comment: 21 pages, 7 figures. For author's webpage version see
http://www.orfe.princeton.edu/~rvdb/tex/piaa/ms.pdf This version has improved
figures and addresses comments of a refere
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