386,782 research outputs found
Perturbation Analysis and Randomized Algorithms for Large-Scale Total Least Squares Problems
In this paper, we present perturbation analysis and randomized algorithms for
the total least squares (TLS) problems. We derive the perturbation bound and
check its sharpness by numerical experiments. Motivated by the recently popular
probabilistic algorithms for low-rank approximations, we develop randomized
algorithms for the TLS and the truncated total least squares (TTLS) solutions
of large-scale discrete ill-posed problems, which can greatly reduce the
computational time and still keep good accuracy.Comment: 27 pages, 10 figures, 8 table
De-Biasing the Dynamic Mode Decomposition for Applied Koopman Spectral Analysis
The Dynamic Mode Decomposition (DMD)---a popular method for performing
data-driven Koopman spectral analysis---has gained increased adoption as a
technique for extracting dynamically meaningful spatio-temporal descriptions of
fluid flows from snapshot measurements. Often times, DMD descriptions can be
used for predictive purposes as well, which enables informed decision-making
based on DMD model-forecasts. Despite its widespread use and utility, DMD
regularly fails to yield accurate dynamical descriptions when the measured
snapshot data are imprecise due to, e.g., sensor noise. Here, we express DMD as
a two-stage algorithm in order to isolate a source of systematic error. We show
that DMD's first stage, a subspace projection step, systematically introduces
bias errors by processing snapshots asymmetrically. To remove this systematic
error, we propose utilizing an augmented snapshot matrix in a subspace
projection step, as in problems of total least-squares, in order to account for
the error present in all snapshots. The resulting unbiased and noise-aware
total DMD (TDMD) formulation reduces to standard DMD in the absence of snapshot
errors, while the two-stage perspective generalizes the de-biasing framework to
other related methods as well. TDMD's performance is demonstrated in numerical
and experimental fluids examples
Analyzing the Quantum Annealing Approach for Solving Linear Least Squares Problems
With the advent of quantum computers, researchers are exploring if quantum
mechanics can be leveraged to solve important problems in ways that may provide
advantages not possible with conventional or classical methods. A previous work
by O'Malley and Vesselinov in 2016 briefly explored using a quantum annealing
machine for solving linear least squares problems for real numbers. They
suggested that it is best suited for binary and sparse versions of the problem.
In our work, we propose a more compact way to represent variables using two's
and one's complement on a quantum annealer. We then do an in-depth theoretical
analysis of this approach, showing the conditions for which this method may be
able to outperform the traditional classical methods for solving general linear
least squares problems. Finally, based on our analysis and observations, we
discuss potentially promising areas of further research where quantum annealing
can be especially beneficial.Comment: 16 pages, 2 appendice
Efficient Algorithms for Positive Semi-Definite Total Least Squares Problems, Minimum Rank Problem and Correlation Matrix Computation
We have recently presented a method to solve an overdetermined linear system
of equations with multiple right hand side vectors, where the unknown matrix is
to be symmetric and positive definite. The coefficient and the right hand side
matrices are respectively named data and target matrices. A more complicated
problem is encountered when the unknown matrix is to be positive semi-definite.
The problem arises in estimating the compliance matrix to model deformable
structures and approximating correlation and covariance matrices in financial
modeling. Several methods have been proposed for solving such problems assuming
that the data matrix is unrealistically error free. Here, considering error in
measured data and target matrices, we propose a new approach to solve a
positive semi-definite constrained total least squares problem. We first
consider solving the problem when the rank of the unknown matrix is known, by
defining a new error formulation for the positive semi-definite total least
squares problem and use of optimization methods on Stiefel manifolds. We prove
quadratic convergence of our proposed approach. We then describe how to
generalize our proposed method to solve the general positive semi-definite
total least squares problem. We further apply the proposed approach to solve
the minimum rank problem and the problem of computing correlation matrix.
Comparative numerical results show the efficiency of our proposed algorithms.
Finally, the Dolan-More performance profiles are shown to summarize our
comparative study.Comment: 22 pages,16 tables and 4 figure
Condition numbers for the truncated total least squares problem and their estimations
In this paper, we present explicit expressions for the mixed and
componentwise condition numbers of the truncated total least squares (TTLS)
solution of under the genericity
condition, where is a real data matrix and is
a real -vector. Moreover, we reveal that normwise, componentwise and mixed
condition numbers for the TTLS problem can recover the previous corresponding
counterparts for the total least squares (TLS) problem when the truncated level
of for the TTLS problem is . When is a structured matrix, the structured
perturbations for the structured truncated TLS (STTLS) problem are investigated
and the corresponding explicit expressions for the structured normwise,
componentwise and mixed condition numbers for the STTLS problem are obtained.
Furthermore, the relationships between the structured and unstructured
normwise, componentwise and mixed condition numbers for the STTLS problem are
studied. Based on small sample statistical condition estimation (SCE), reliable
condition estimation algorithms for both unstructured and structured normwise,
mixed and componentwise are devised, which utilize the SVD of the augmented
matrix . The efficient proposed condition estimation
algorithms can be integrated into the SVD-based direct solver for the small and
medium size TTLS problem to give the error estimation for the numerical TTLS
solution. Numerical experiments are reported to illustrate the reliability of
the proposed estimation algorithms, which coincide with our theoretical
results
What can Lattice QCD theorists learn from NMR spectroscopists?
Euclidean-time hadron correlation functions computed in Lattice QCD (LQCD)
are modeled by a sum of decaying exponentials, reminiscent of the exponentially
damped sinusoid models of free induction decay (FID) in Nuclear Magnetic
Resonance (NMR) spectroscopy. We present our initial progress in studying how
data modeling techniques commonly used in NMR perform when applied to LQCD
data.Comment: 11 pages, svmult.cls. Minor changes in response to reviewers'
comments. To appear in the Proceedings of the Third International Workshop on
Numerical Analysis and Lattice QCD, Edinburgh, Scotland, 30 Jun - 04 Jul 200
Runtime Guarantees for Regression Problems
We study theoretical runtime guarantees for a class of optimization problems
that occur in a wide variety of inference problems. these problems are
motivated by the lasso framework and have applications in machine learning and
computer vision.
Our work shows a close connection between these problems and core questions
in algorithmic graph theory. While this connection demonstrates the
difficulties of obtaining runtime guarantees, it also suggests an approach of
using techniques originally developed for graph algorithms.
We then show that most of these problems can be formulated as a grouped least
squares problem, and give efficient algorithms for this formulation. Our
algorithms rely on routines for solving quadratic minimization problems, which
in turn are equivalent to solving linear systems. Finally we present some
experimental results on applying our approximation algorithm to image
processing problems
Isogeometric Least-squares Collocation Method with Consistency and Convergence Analysis
In this paper, we present the isogeometric least-squares collocation (IGA-L)
method, which determines the numerical solution by making the approximate
differential operator fit the real differential operator in a least-squares
sense. The number of collocation points employed in IGA-L can be larger than
that of the unknowns. Theoretical analysis and numerical examples presented in
this paper show the superiority of IGA-L over state-of-the-art collocation
methods. First, a small increase in the number of collocation points in IGA-L
leads to a large improvement in the accuracy of its numerical solution. Second,
IGA-L method is more flexible and more stable, because the number of
collocation points in IGA-L is variable. Third, IGA-L is convergent in some
cases of singular parameterization. Moreover, the consistency and convergence
analysis are also developed in this paper
Distributed Least-Squares Iterative Methods in Networks: A Survey
Many science and engineering applications involve solving a linear
least-squares system formed from some field measurements. In the distributed
cyber-physical systems (CPS), often each sensor node used for measurement only
knows partial independent rows of the least-squares system. To compute the
least-squares solution they need to gather all these measurement at a
centralized location and then compute the solution. These data collection and
computation are inefficient because of bandwidth and time constraints and
sometimes are infeasible because of data privacy concerns. Thus distributed
computations are strongly preferred or demanded in many of the real world
applications e.g.: smart-grid, target tracking etc. To compute least squares
for the large sparse system of linear equation iterative methods are natural
candidates and there are a lot of studies regarding this, however, most of them
are related to the efficiency of centralized/parallel computations while and
only a few are explicitly about distributed computation or have the potential
to apply in distributed networks. This paper surveys the representative
iterative methods from several research communities. Some of them were not
originally designed for this need, so we slightly modified them to suit our
requirement and maintain the consistency. In this survey, we sketch the
skeleton of the algorithm first and then analyze its time-to-completion and
communication cost. To our best knowledge, this is the first survey of
distributed least-squares in distributed networks
A New Error in Variables Model for Solving Positive Definite Linear System Using Orthogonal Matrix Decompositions
The need to estimate a positive definite solution to an overdetermined linear
system of equations with multiple right hand side vectors arises in several
process control contexts. The coefficient and the right hand side matrices are
respectively named data and target matrices. A number of optimization methods
were proposed for solving such problems, in which the data matrix is
unrealistically assumed to be error free. Here, considering error in measured
data and target matrices, we present an approach to solve a positive definite
constrained linear system of equations based on the use of a newly defined
error function. To minimize the defined error function, we derive necessary and
sufficient optimality conditions and outline a direct algorithm to compute the
solution. We provide a comparison of our proposed approach and two existing
methods, the interior point method and a method based on quadratic programming.
Two important characteristics of our proposed method as compared to the
existing methods are computing the solution directly and considering error both
in data and target matrices. Moreover, numerical test results show that the new
approach leads to smaller standard deviations of error entries and smaller
effective rank as desired by control problems. Furthermore, in a comparative
study, using the Dolan-Mor\'{e} performance profiles, we show the approach to
be more efficient.Comment: 22 pages, 10 figures, 10 table
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