320 research outputs found
Design of engineering systems in Polish mines in the third quarter of the 20th century
Participation of mathematicians in the implementation of economic projects in
Poland, in which mathematics-based methods played an important role, happened
sporadically in the past. Usually methods known from publications and verified
were adapted to solving related problems. The subject of this paper is the
cooperation between mathematicians and engineers in Wroc{\l}aw in the second
half of the twentieth century established in the form of an analysis of the
effectiveness of engineering systems used in mining. The results of this
cooperation showed that at the design stage of technical systems it is
necessary to take into account factors that could not have been rationally
controlled before. The need to explain various aspects of future exploitation
was a strong motivation for the development of mathematical modeling methods.
These methods also opened research topics in the theory of stochastic processes
and graph theory. The social aspects of this cooperation are also interesting.Comment: 45 pages, 11 figures, 116 reference
The random component-wise power method
This paper considers a random component-wise variant of the unnormalized power method, which is similar to the regular power iteration except that only a random subset of indices is updated in each iteration. For the case of normal matrices, it was previously shown that random component-wise updates converge in the mean-squared sense to an eigenvector of eigenvalue 1 of the underlying matrix even in the case of the matrix having spectral radius larger than unity. In addition to the enlarged convergence regions, this study shows that the eigenvalue gap does not directly affect the convergence rate of the randomized updates unlike the regular power method. In particular, it is shown that the rate of convergence is affected by the phase of the eigenvalues in the case of random component-wise updates, and the randomized updates favor negative eigenvalues over positive ones. As an application, this study considers a reformulation of the component-wise updates revealing a randomized algorithm that is proven to converge to the dominant left and right singular vectors of a normalized data matrix. The algorithm is also extended to handle large-scale distributed data when computing an arbitrary rank approximation of an arbitrary data matrix. Numerical simulations verify the convergence of the proposed algorithms under different parameter settings
Randomized Extended Kaczmarz for Solving Least-Squares
We present a randomized iterative algorithm that exponentially converges in
expectation to the minimum Euclidean norm least squares solution of a given
linear system of equations. The expected number of arithmetic operations
required to obtain an estimate of given accuracy is proportional to the square
condition number of the system multiplied by the number of non-zeros entries of
the input matrix. The proposed algorithm is an extension of the randomized
Kaczmarz method that was analyzed by Strohmer and Vershynin.Comment: 19 Pages, 5 figures; code is available at
https://github.com/zouzias/RE
Sparse Randomized Kaczmarz for Support Recovery of Jointly Sparse Corrupted Multiple Measurement Vectors
While single measurement vector (SMV) models have been widely studied in
signal processing, there is a surging interest in addressing the multiple
measurement vectors (MMV) problem. In the MMV setting, more than one
measurement vector is available and the multiple signals to be recovered share
some commonalities such as a common support. Applications in which MMV is a
naturally occurring phenomenon include online streaming, medical imaging, and
video recovery. This work presents a stochastic iterative algorithm for the
support recovery of jointly sparse corrupted MMV. We present a variant of the
Sparse Randomized Kaczmarz algorithm for corrupted MMV and compare our proposed
method with an existing Kaczmarz type algorithm for MMV problems. We also
showcase the usefulness of our approach in the online (streaming) setting and
provide empirical evidence that suggests the robustness of the proposed method
to the distribution of the corruption and the number of corruptions occurring.Comment: 13 pages, 6 figure
Preasymptotic Convergence of Randomized Kaczmarz Method
Kaczmarz method is one popular iterative method for solving inverse problems,
especially in computed tomography. Recently, it was established that a
randomized version of the method enjoys an exponential convergence for
well-posed problems, and the convergence rate is determined by a variant of the
condition number. In this work, we analyze the preasymptotic convergence
behavior of the randomized Kaczmarz method, and show that the low-frequency
error (with respect to the right singular vectors) decays faster during first
iterations than the high-frequency error. Under the assumption that the inverse
solution is smooth (e.g., sourcewise representation), the result explains the
fast empirical convergence behavior, thereby shedding new insights into the
excellent performance of the randomized Kaczmarz method in practice. Further,
we propose a simple strategy to stabilize the asymptotic convergence of the
iteration by means of variance reduction. We provide extensive numerical
experiments to confirm the analysis and to elucidate the behavior of the
algorithms.Comment: 20 page
The random component-wise power method
This paper considers a random component-wise variant of the unnormalized power method, which is similar to the regular power iteration except that only a random subset of indices is updated in each iteration. For the case of normal matrices, it was previously shown that random component-wise updates converge in the mean-squared sense to an eigenvector of eigenvalue 1 of the underlying matrix even in the case of the matrix having spectral radius larger than unity. In addition to the enlarged convergence regions, this study shows that the eigenvalue gap does not directly affect the convergence rate of the randomized updates unlike the regular power method. In particular, it is shown that the rate of convergence is affected by the phase of the eigenvalues in the case of random component-wise updates, and the randomized updates favor negative eigenvalues over positive ones. As an application, this study considers a reformulation of the component-wise updates revealing a randomized algorithm that is proven to converge to the dominant left and right singular vectors of a normalized data matrix. The algorithm is also extended to handle large-scale distributed data when computing an arbitrary rank approximation of an arbitrary data matrix. Numerical simulations verify the convergence of the proposed algorithms under different parameter settings
Real-time Image Generation for Compressive Light Field Displays
With the invention of integral imaging and parallax barriers in the beginning of the 20th century, glasses-free 3D displays have become feasible. Only today—more than a century later—glasses-free 3D displays are finally emerging in the consumer market. The technologies being employed in current-generation devices, however, are fundamentally the same as what was invented 100 years ago. With rapid advances in optical fabrication, digital processing power, and computational perception, a new generation of display technology is emerging: compressive displays exploring the co-design of optical elements and computational processing while taking particular characteristics of the human visual system into account. In this paper, we discuss real-time implementation strategies for emerging compressive light field displays. We consider displays composed of multiple stacked layers of light-attenuating or polarization-rotating layers, such as LCDs. The involved image generation requires iterative tomographic image synthesis. We demonstrate that, for the case of light field display, computed tomographic light field synthesis maps well to operations included in the standard graphics pipeline, facilitating efficient GPU-based implementations with real-time framerates.United States. Defense Advanced Research Projects Agency. Soldier Centric Imaging via Computational CamerasNational Science Foundation (U.S.) (Grant IIS-1116452)United States. Defense Advanced Research Projects Agency. Maximally scalable Optical Sensor Array Imaging with Computation ProgramAlfred P. Sloan Foundation (Research Fellowship)United States. Defense Advanced Research Projects Agency (Young Faculty Award
Randomized Kaczmarz solver for noisy linear systems
The Kaczmarz method is an iterative algorithm for solving systems of linear
equations Ax=b. Theoretical convergence rates for this algorithm were largely
unknown until recently when work was done on a randomized version of the
algorithm. It was proved that for overdetermined systems, the randomized
Kaczmarz method converges with expected exponential rate, independent of the
number of equations in the system. Here we analyze the case where the system
Ax=b is corrupted by noise, so we consider the system where Ax is approximately
b + r where r is an arbitrary error vector. We prove that in this noisy
version, the randomized method reaches an error threshold dependent on the
matrix A with the same rate as in the error-free case. We provide examples
showing our results are sharp in the general context
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