392,190 research outputs found
A note on the condition number of the scaled total least squares problem
In this paper, we consider the explicit expressions of the normwise condition
number for the scaled total least squares problem. Some techniques are
introduced to simplify the expression of the condition number, and some new
results are derived. Based on these new results, new expressions of the
condition number for the total least squares problem can be deduced as a
special case. New forms of the condition number enjoy some storage and
computational advantages. We also proposed three different methods to estimate
the condition number. Some numerical experiments are carried out to illustrate
the effectiveness of our results.Comment: 13 pages, 2figure
On the partial condition numbers for the indefinite least squares problem
The condition number of a linear function of the indefinite least squares
solution is called the partial condition number for the indefinite least
squares problem. In this paper, based on a new and very general condition
number which can be called the unified condition number, the expression of the
partial unified condition number is first presented when the data space is
measured by the general weighted product norm. Then, by setting the specific
norms and weight parameters, we obtain the expressions of the partial normwise,
mixed and componentwise condition numbers. Moreover, the corresponding
structured partial condition numbers are also taken into consideration when the
problem is structured, whose expressions are given. Considering the connections
between the indefinite and total least squares problems, we derive the
(structured) partial condition numbers for the latter, which generalize the
ones in the literature. To estimate these condition numbers effectively and
reliably, the probabilistic spectral norm estimator and the small-sample
statistical condition estimation method are applied and three related
algorithms are devised. Finally, the obtained results are illustrated by
numerical experiments.Comment: 22 page
Effectively Subsampled Quadratures For Least Squares Polynomial Approximations
This paper proposes a new deterministic sampling strategy for constructing
polynomial chaos approximations for expensive physics simulation models. The
proposed approach, effectively subsampled quadratures involves sparsely
subsampling an existing tensor grid using QR column pivoting. For polynomial
interpolation using hyperbolic or total order sets, we then solve the following
square least squares problem. For polynomial approximation, we use a column
pruning heuristic that removes columns based on the highest total orders and
then solves the tall least squares problem. While we provide bounds on the
condition number of such tall submatrices, it is difficult to ascertain how
column pruning effects solution accuracy as this is problem specific. We
conclude with numerical experiments on an analytical function and a model
piston problem that show the efficacy of our approach compared with randomized
subsampling. We also show an example where this method fails.Comment: 17 page
Condition numbers of the mixed least squares-total least squares problem: revisited
A new closed formula for the first order perturbation estimate of the mixed
least squares-total least squares (MTLS) solution is presented. It is
mathematically equivalent to the one by Zheng and Yang(Numer. Linear Algebra
Appl. 2019; 26(4):e2239). With this formula, general and structured normwise,
mixed and componentwise condition numbers of the MTLS problem are derived.
Perturbation bounds based on the normwise condition number, and compact forms
for the upper bounds of mixed and componentwise condition numbers are also
given in order for economic storage and efficient computation. It is shown that
the condition numbers and perturbation bound of the TLS problem are unified in
the ones of the MTLS problem.Comment: 20 page
Robust Least Squares for Quantized Data Matrices
In this paper we formulate and solve a robust least squares problem for a
system of linear equations subject to quantization error in the data matrix.
Ordinary least squares fails to consider uncertainty in the operator, modeling
all noise in the observed signal. Total least squares accounts for uncertainty
in the data matrix, but necessarily increases the condition number of the
operator compared to ordinary least squares. Tikhonov regularization or ridge
regression is frequently employed to combat ill-conditioning, but requires
parameter tuning which presents a host of challenges and places strong
assumptions on parameter prior distributions. The proposed method also requires
selection of a parameter, but it can be chosen in a natural way, e.g., a matrix
rounded to the 4th digit uses an uncertainty bounding parameter of 0.5e-4. We
show here that our robust method is theoretically appropriate, tractable, and
performs favorably against ordinary and total least squares.Comment: 10 pages, 5 figure
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
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
A Christoffel function weighted least squares algorithm for collocation approximations
We propose, theoretically investigate, and numerically validate an algorithm
for the Monte Carlo solution of least-squares polynomial approximation problems
in a collocation frame- work. Our method is motivated by generalized Polynomial
Chaos approximation in uncertainty quantification where a polynomial
approximation is formed from a combination of orthogonal polynomials. A
standard Monte Carlo approach would draw samples according to the density of
orthogonality. Our proposed algorithm samples with respect to the equilibrium
measure of the parametric domain, and subsequently solves a weighted
least-squares problem, with weights given by evaluations of the Christoffel
function. We present theoretical analysis to motivate the algorithm, and
numerical results that show our method is superior to standard Monte Carlo
methods in many situations of interest.Comment: 29 pages, 11 figure
Partial condition number for the equality constrained linear least squares problem
In this paper, the normwise condition number of a linear function of the
equality constrained linear least squares solution called the partial condition
number is considered. Its expression and closed formulae are first presented
when the data space and the solution space are measured by the weighted
Frobenius norm and the Euclidean norm, respectively. Then, we investigate the
corresponding structured partial condition number when the problem is
structured. To estimate these condition numbers with high reliability, the
probabilistic spectral norm estimator and the small-sample statistical
condition estimation method are applied and two algorithms are devised. The
obtained results are illustrated by numerical examples.Comment: 17 pages, 2 figure
Discrete least-squares finite element methods
A finite element methodology for large classes of variational boundary value
problems is defined which involves discretizing two linear operators: (1) the
differential operator defining the spatial boundary value problem; and (2) a
Riesz map on the test space. The resulting linear system is overdetermined. Two
different approaches for solving the system are suggested (although others are
discussed): (1) solving the associated normal equation with linear solvers for
symmetric positive-definite systems (e.g. Cholesky factorization); and (2)
solving the overdetermined system with orthogonalization algorithms (e.g. QR
factorization). The finite element assembly algorithm for each of these
approaches is described in detail. The normal equation approach is usually
faster for direct solvers and requires less storage. The second approach
reduces the condition number of the system by a power of two and is less
sensitive to round-off error. The rectangular stiffness matrix of second
approach is demonstrated to have condition number for a
variety of formulations of Poisson's equation. The stiffness matrix from the
normal equation approach is demonstrated to be related to the monolithic
stiffness matrices of least-squares finite element methods and it is proved
that the two are identical in some cases. An example with Poisson's equation
indicates that the solutions of these two different linear systems can be
nearly indistinguishable (if round-off error is not an issue) and rapidly
converge to each other. The orthogonalization approach is suggested to be
beneficial for problems which induce poorly conditioned linear systems.
Experiments with Poisson's equation in single-precision arithmetic as well as
the linear acoustics problem near resonance in double-precision arithmetic
verify this conclusion.Comment: 30 page
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