Concentration inequalities for random tensors Vershynin, Roman
Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability,
11/2020, Letnik:
26, Številka:
4
Journal Article
We propose and analyze a new stabilized cut finite element method for the Laplace–Beltrami operator on a closed surface. The new stabilization term provides control of the full R3 gradient on the ...active mesh consisting of the elements that intersect the surface. Compared to face stabilization, based on controlling the jumps in the normal gradient across faces between elements in the active mesh, the full gradient stabilization is easier to implement and does not significantly increase the number of nonzero elements in the mass and stiffness matrices. The full gradient stabilization term may be combined with a variational formulation of the Laplace–Beltrami operator based on tangential or full gradients and we present a simple and unified analysis that covers both cases. The full gradient stabilization term gives rise to a consistency error which, however, is of optimal order for piecewise linear elements, and we obtain optimal order a priori error estimates in the energy and L2 norms as well as an optimal bound of the condition number. Finally, we present detailed numerical examples where we in particular study the sensitivity of the condition number and error on the stabilization parameter.
The conditioning theory of the ML -weighted least squares and ML -weighted pseudoinverse problems is explored in this article. We begin by introducing three types of condition numbers for the ML ...-weighted pseudoinverse problem: normwise, mixed, and componentwise, along with their explicit expressions. Utilizing the derivative of the ML -weighted pseudoinverse problem, we then provide explicit condition number expressions for the solution of the ML -weighted least squares problem. To ensure reliable estimation of these condition numbers, we employ the small-sample statistical condition estimation method for all three algorithms. The article concludes with numerical examples that highlight the results obtained.
In this paper, we considered the condition number theory of a new generalized ridge regression model. The explicit expressions of different types of condition numbers were derived to measure the ...ill-conditionness of the generalized ridge regression problem with respect to different circumstances. To overcome the computational difficulty of computing the exact value of the condition number, we employed the statistical condition estimation theory to design efficient condition number estimators, and the numerical examples were also given to illustrate its efficiency.
In this work we first review the (phased) inverse scattering problem and then pursue the phaseless reconstruction from far-field data with the help of the concept of scattering coefficients. We ...perform sensitivity, resolution, and stability analysis of both phased and phaseless problems and compare the degree of ill-posedness of the phased and phaseless reconstructions. The phaseless reconstruction is highly nonlinear and much more severely ill-posed. Algorithms are provided to solve both the phased and the phaseless reconstructions in the linearized case. Stability is studied by estimating the condition number of the inversion process for both the phased and the phaseless cases. An optimal strategy is suggested to attain the infimum of the condition numbers of the phaseless reconstruction, which may provide an important guidance for efficient phaseless measurements in practical applications. To the best of our knowledge, the stability analysis in terms of condition numbers is new for the phased and phaseless inverse scattering problems and is very important to help us understand the degree of ill-posedness of these inverse problems. Numerical experiments are provided to illustrate the theoretical asymptotic behavior, as well as the effectiveness and robustness of the phaseless reconstruction algorithm.
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.
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.
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.
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.
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