Small Sample Statistical Condition Estimation Research Articles

Conditioning Theory for Generalized Inverse CA‡ and Their Estimations

The conditioning theory of the generalized inverse CA‡ is considered in this article. First, we introduce three kinds of condition numbers for the generalized inverse CA‡, i.e., normwise, mixed and componentwise ones, and present their explicit expressions. Then, using the intermediate result, which is the derivative of CA‡, we can recover the explicit condition number expressions for the solution of the equality constrained indefinite least squares problem. Furthermore, using the augment system, we investigate the componentwise perturbation analysis of the solution and residual of the equality constrained indefinite least squares problem. To estimate these condition numbers with high reliability, we choose the probabilistic spectral norm estimator to devise the first algorithm and the small-sample statistical condition estimation method for the other two algorithms. In the end, the numerical examples illuminate the obtained results.

Structured conditioning theory for the total least squares problem with linear equality constraint and their estimation

<abstract><p>This article is devoted to the structured and unstructured condition numbers for the total least squares with linear equality constraint (TLSE) problem. By making use of the dual techniques, we investigate three distinct kinds of unstructured condition numbers for a linear function of the TLSE solution and three structured condition numbers for this problem, i.e., normwise, mixed, and componentwise ones, and present their explicit expressions under both unstructured and structured componentwise perturbations. In addition, the relations between structured and unstructured normwise, componentwise, and mixed condition numbers for the TLSE problem are investigated. Furthermore, using the small-sample statistical condition estimation method, we also consider the statistical estimation of both unstructured and structured condition numbers and propose three algorithms. Theoretical and experimental results show that structured condition numbers are always smaller than the corresponding unstructured condition numbers.</p></abstract>

Perturbation analysis and condition numbers for the Tikhonov regularization of total least squares problem and their statistical estimation

Condition number plays an important role in perturbation analysis, the latter is a tool to judge whether a numerical solution makes sense, especially for ill-posed problems. In this paper, perturbation analysis of the Tikhonov regularization of total least squares problem (TRTLS) is considered. The explicit expressions of normwise, mixed and componentwise condition numbers for the TRTLS problem are first presented. With the intermediate result, i.e. normwise condition number, we can recover the upper bound of TRTLS problem. To improve the computational efficiency in calculating the normwise condition number, a new compact and tight upper bound of the TRTLS problem is introduced. In addition, we also derive the normwise, mixed and componentwise condition numbers for TRTLS problem when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed. We choose the probabilistic spectral norm estimator and the small-sample statistical condition estimation method to estimate these condition numbers with high reliability. Numerical experiments are provided to verify the obtained results.

An analysis of the mixed least squares-total least squares problems

In this paper, we first get further consideration of the first order perturbation with normwise condition number of the MTLS problem. For easy estimation, we show a lower bound for the normwise condition number which is proved to be optimal. In order to overcome the problems encountered in calculating the normwise condition number, we give an upper bound for computing more effectively and nonstandard and unusual perturbation bounds for the MTLS problem. Both of the two types of the perturbation bounds can enjoy storage and computational advantages. For getting more insight into the sensitivity of the MTLS technique with respect to perturbations in all data, we analyze the corrections applied by MTLS to the data in Ax ? b to make the set compatible and indicate how closely the data A, b fit the so-called general errors-in-variables model. On how to estimate the conditioning of the MTLS problem more effectively, we propose statistical algorithms by taking advantage of the superiority of small sample statistical condition estimation (SCE) techniques.

Condition Numbers for a Linear Function of the Solution to the Constrained and Weighted Least Squares Problem and Their Statistical Estimation

In this paper, we consider the condition number theory for a linear function of the solution to the constrained and weighted least squares problem. We first present two explicit expressions without Kronecker product of normwise condition number using the classical method for condition numbers. Then, we derive the explicit expression of mixed and componentwise condition numbers by the dual techniques. To estimate these condition numbers with high reliability, we choose the probabilistic spectral norm estimator and the small-sample statistical condition estimation method and devise three algorithms. Numerical experiments are provided to illustrate the obtained results.

Structured condition numbers and statistical condition estimation for the LDU factorization

In this article, we consider the structured condition numbers for LDU, factorization by using the modified matrix-vector approach and the differential calculus, which can be represented by sets of parameters. By setting the specific norms and weight parameters, we present the expressions of the structured normwise, mixed, componentwise condition numbers and the corresponding results for unstructured ones. In addition, we investigate the statistical estimation of condition numbers of LDU factorization using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, we compare the structured condition numbers with the corresponding unstructured ones in numerical experiments.

Sensitivity analysis for the generalized Cholesky block downdating problem

In this article, first we will present the new rigorous perturbation bounds with normwise perturbation for the generalized Cholesky block downdating problem by combining the unified matrix–vector equation approach with the method of Lyapunov majorant functions and the Banach fixed point theorem. Then, we will derive the explicit expressions for the six distinct kinds of condition numbers, i.e. four normwise ones, mixed and componentwise ones. Furthermore, with the help of probabilistic spectral norm estimator and the small-sample statistical condition estimation method, these condition numbers can be estimated with high accuracy. At the end, the obtained results are illuminated by the numerical examples.

Small-sample statistical condition estimation of rational Riccati equations

We study the small-sample statistical condition estimation of the rational Riccati equation, which may be incorporated into a direct solver applying the homotopy method. Our proposed condition estimation algorithms are efficient for small and medium size rational Riccati equations. Illustrative numerical examples are presented.

Condition numbers for the K-weighted pseudoinverse and their statistical estimation

The explicit expressions of normwise, mixed and componentwise condition numbers for the K-weighted pseudoinverse are first presented. With the intermediate result, i.e. the derivative of , we can recover the explicit expressions of condition numbers for solution of least squares problem with equality constraint. Then, we investigate the statistical estimation of condition numbers of using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, numerical examples are provided to illustrate these results.

Sensitivity analysis for the block Cholesky downdating problem

ABSTRACTSome improved rigorous perturbation bounds with normwise perturbation for the block Cholesky downdating problem are first derived by combining the modified matrix-vector equation approach with the strategy for Lyapunov majorant function and the Banach fixed point theorem. Then, we investigate four distinct kinds of condition numbers, i.e. two normwise ones, and mixed and componentwise ones, for this problem, and present their explicit expressions. Furthermore, using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, we also consider the statistical estimation of these condition numbers and design two algorithms. The obtained results are illustrated by numerical examples.

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, we first present an expression of the partial unified condition number when the data space is measured by a 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. 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.

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.

Small sample statistical condition estimation for the total least squares problem

In this paper, under the genericity condition, we study the condition estimation of the total least squares (TLS) problem based on small sample condition estimation (SCE), which can be incorporated into the direct solver for the TLS problem via the singular value decomposition (SVD) of the augmented matrix [A, b]. Our proposed condition estimation algorithms are efficient for the small and medium size TLS problem because they utilize the computed SVD of [A, b] during the numerical solution to the TLS problem. Numerical examples illustrate the reliability of the algorithms. Both normwise and componentwise perturbations are considered. Moreover, structured condition estimations are investigated for the structured TLS problem.

Structured condition numbers of structured Tikhonov regularization problem and their estimations

Both structured componentwise and structured normwise perturbation analysis of the Tikhonov regularization are presented. The structured matrices under consideration include: Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Structured normwise, mixed and componentwise condition numbers for the Tikhonov regularization are introduced and their explicit expressions are derived. For the general linear structure, based on the derived expressions, we prove structured condition numbers are smaller than their corresponding unstructured counterparts. By means of the power method and small sample statistical condition estimation (SCE), fast condition estimation algorithms are proposed. Our estimation methods can be integrated into Tikhonov regularization algorithms that use the generalized singular value decomposition (GSVD). For large scale linear structured Tikhonov regularization problems, we show how to incorporate the SCE into the preconditioned conjugate gradient (PCG) method to get the posterior error estimations. The structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts. Our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones, and the proposed condition estimation algorithms are reliable. Also, an image restoration example is tested to show the effectiveness of the SCE for large scale linear structured Tikhonov regularization problems.

Effective condition numbers and small sample statistical condition estimation for the generalized Sylvester equation

In this paper, we investigate the effective condition numbers for the generalized Sylvester equation (AX − YB, DX − YE) = (C, F), where A, D ∈ ℝm×m, B, E ∈ ℝn×n and C, F ∈ ℝm×n. We apply the small sample statistical method for the fast condition estimation of the generalized Sylvester equation, which requires O(m2n + mn2) flops, comparing with O(m3 + n3) flops for the generalized Schur and generalized Hessenberg-Schur methods for solving the generalized Sylvester equation. Numerical examples illustrate the sharpness of our perturbation bounds.

Mixed, componentwise condition numbers and small sample statistical condition estimation of Sylvester equations

SUMMARYWe present a componentwise perturbation analysis for the continuous‐time Sylvester equations. Componentwise, mixed condition numbers and new perturbation bounds are derived for the matrix equations. The small sample statistical method can also be applied for the condition estimation. These condition numbers and perturbation bounds are tested on numerical examples and compared with the normwise condition number. The numerical examples illustrate that the mixed condition number gives sharper bounds than the normwise one. Copyright © 2011 John Wiley & Sons, Ltd.

Error Estimation for Reduced‐Order Models of Dynamical Systems

The use of reduced‐order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most important, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error, while the derived bounds are within an order of magnitude.

Error Estimation for Reduced-Order Models of Dynamical Systems

The use of reduced-order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most importantly, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error, while the derived bounds are within an order of magnitude.

Statistical Condition Estimation for Linear Systems

The standard approach to measuring the condition of a linear system compresses all sensitivity information into one number. Thus a loss of information can occur in situations in which the standard condition number with respect to inversion does not accurately reflect the actual sensitivity of a solution or particular entries of a solution. It is shown that a new method for estimating the sensitivity of linear systems addresses these difficulties. The new procedure measures the effects on the solution of small random changes in the input data and, by properly scaling the results, obtains reliable condition estimates for each entry of the computed solution. Moreover, this approach, which is referred to as small-sample statistical condition estimation, is no more costly than the standard 1-norm or power method 2-norm condition estimates, and it has the advantage of considerable flexibility. For example, it easily accommodates restrictions on, or structure associated with, allowable perturbations. The method also has a rigorous statistical theory available for the probability of accuracy of the condition estimates. However, it gives no estimate of an approximate null vector for nearly singular systems. The theory of this approach is discussed along with several illustrative examples.

Condition Estimates

Condition Estimates