Effective condition number and its applications

Li, Zi‐Cai; Huang, Hung‐Tsai; Chen, Jeng‐Tzong; Wei, Yimin

doi:10.1007/s00607-010-0098-8

Cited by 20 publications

(5 citation statements)

References 17 publications

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“…Our results can also be extended to cases where the interpolation matrix is ill-conditioned, arising from the possibility of having a data site that has almost zero overlap with another data site. In this case, we will define a constant δ eff such that only the eigenvalues greater than δ eff are taken into account, essentially defining an effective condition number k eff [33]. Then the filtering procedure described in [9] is employed in the phase estimation using the δ eff .…”

Section: Discussionmentioning

confidence: 99%

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Quantum radial basis function methods for scattered data fitting

Cui¹,

Xiang²

2021

Preprint

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Scattered data fitting is a frequently encountered problem for reconstructing an unknown function from given scattered data. Radial basis function (RBF) methods have proven to be highly useful to deal with this problem. We describe two quantum algorithms to efficiently fit scattered data based on globally and compactly supported RBFs respectively. For the globally supported RBF method, the core of the quantum algorithm relies on using coherent states to calculate the radial functions and a nonsparse matrix exponentiation technique for efficiently performing a matrix inversion. A quadratic speedup is achieved in the number of data over the classical algorithms. For the compactly supported RBF method, we mainly use the HHL algorithm as a subroutine to design an efficient quantum procedure that runs in time logarithmic in the number of data, achieving an exponential improvement over the classical methods.

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Section: Discussionmentioning

confidence: 99%

“…( 21), we can estimate ||Φ(x)|| from the probability of obtaining 0 on the ancilla register in Eq. (33). Furthermore, by the swap test between |c (compact) and |Φ(x) , we can evaluate on the new given data x.…”

Section: Quantum Compactly Supported Rbf Algorithm Implementationmentioning

confidence: 99%

Quantum radial basis function methods for scattered data fitting

Cui¹,

Xiang²

2021

Preprint

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“…A T also appears in the derivation of the "effective condition number" of the total least squares problem. The effective condition number is defined as [25,26,27]…”

Section: We Notice That the Term Amentioning

confidence: 99%

Randomized algorithms for total least squares problems

Xie

Xiang

Wei

2018

Numerical Linear Algebra App

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Motivated by the recently popular probabilistic methods for low-rank approximations and randomized algorithms for the least squares problems, we develop randomized algorithms for the total least squares problem with a single right-hand side. We present the Nyström method for the medium-sized problems. For the large-scale and ill-conditioned cases, we introduce the randomized truncated total least squares with the known or estimated rank as the regularization parameter. We analyze the accuracy of the algorithm randomized truncated total least squares and perform numerical experiments to demonstrate the efficiency of our randomized algorithms. The randomized algorithms can greatly reduce the computational time and still maintain good accuracy with very high probability.

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“…The goal of this article is to clarify this dilemma. To this end, we solicit the effective condition number Cond_eff in Li et al [6][7][8], and develop a new actual condition number from the maximum principle. Both of them may offer small bounds of the solution errors via perturbation, which are independent of ε.…”

Section: Introductionmentioning

confidence: 99%

“…The effective condition number is particularly significant for solving singularity problems, as well as the degenerate scales in BEM of Chen et al [13]. New stability analysis is reported in [7,14] for corner singularity problems, and in [15] for the boundary singularity of Poisson's equation. However, an important layer singularity remains missing (cf.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis for singularly perturbed differential equations by the upwind difference scheme

Wei

Huang

et al. 2013

Numerical Methods Partial

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For solving singularly perturbed differential equations (SPDE), the upwind difference scheme (UDS) and the fitted difference method are chosen. The local refinements of grids are adopted in singular layers with the minimal meshspacing h min = O(hε), where h is the maximal meshspacing, and ε( 1) a very small parameter. The traditional condition number in 2-norm is given by Cond = O(h −2 ε −1 ). For the infinitesimal ε(≤ 10 −8 ) in application, the huge Cond issues a dilemma regarding whether the numerical solutions by the UDS can be trusted. Although the UDS has been used for several decades, such a dilemma has not been clarified yet. The goal of this article is to clarify this dilemma. To this end, we solicit the effective condition number Cond_eff in Li et al. Numer Linear Algebra Appl 15 (2008), 575-690 Effective condition for Numerical Partial Different Equation, 2013, and develop a new actual condition number from the maximum principle. Both of them may offer much smaller bounds of the solution errors caused by perturbation, e.g., rounding, truncation, or discritization errors. We study the Dirichlet problems of SPDE by the UDS in a rectangle. When the Dirichlet boundary condition on the downwind side is homogeneous, we derive Cond_eff = O( 1 √ h ). When the entire Dirichlet boundary conditions are homogeneous, the extraordinary bound, Cond_eff = O(1), is achieved. Moreover, we derive the actual condition numbers as Cond_actual = O( 1 √ h ) and Cond_actual = O( 1 h ) for the homogeneous and the nonhomogeneous SPDE, respectively. Note that these bounds do not depend on ε; this is distinct from the traditional Cond. Based on the analysis of this article, the existing dilemma caused by Cond has been removed, to grant a good stability of the UDS for SPDE.

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Effective condition number and its applications

Cited by 20 publications

References 17 publications

Quantum radial basis function methods for scattered data fitting

Quantum radial basis function methods for scattered data fitting

Randomized algorithms for total least squares problems

Stability analysis for singularly perturbed differential equations by the upwind difference scheme

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