The AZ Algorithm for Least Squares Systems with a Known Incomplete Generalized Inverse

Coppé, Vincent; Huybrechs, Daan; Matthysen, Roel; Webb, Marcus

doi:10.1137/19m1306385

Cited by 15 publications

(20 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We anticipate, however, that a fast implementation may be possible, as it is with Fourier extensions [26][27][28]. One potential idea in this direction is the AZ algorithm [17].…”

Section: Discussionmentioning

confidence: 99%

Fast and stable approximation of analytic functions from equispaced samples via polynomial frames

Adcock¹,

Shadrin²

2021

Preprint

View full text Add to dashboard Cite

We consider approximating analytic functions on the interval [−1, 1] from their values at a set of m+1 equispaced nodes. A result of Platte, Trefethen & Kuijlaars states that fast and stable approximation from equispaced samples is generally impossible. In particular, any method that converges exponentially fast must also be exponentially ill-conditioned. We prove a positive counterpart to this 'impossibility' theorem. Our 'possibility' theorem shows that there is a well-conditioned method that provides exponential decay of the error down to a finite, but user-controlled tolerance ǫ > 0, which in practice can be chosen close to machine epsilon. The method is known as polynomial frame approximation or polynomial extensions. It uses algebraic polynomials of degree n on an extended interval [−γ, γ], γ > 1, to construct an approximation on [−1, 1] via a SVD-regularized leastsquares fit. A key step in the proof of our possibility theorem is a new result on the maximal behaviour of a polynomial of degree n on [−1, 1] that is simultaneously bounded by one at a set of m + 1 equispaced nodes in [−1, 1] and 1/ǫ on the extended interval [−γ, γ]. We show that linear oversampling, i.e., m = cn log(1/ǫ)/ γ 2 − 1, is sufficient for uniform boundedness of any such polynomial on [−1, 1]. This result aside, we also prove an extended impossibility theorem, which shows that the possibility theorem (and consequently the method of polynomial frame approximation) is essentially optimal.

show abstract

“…We anticipate, however, that a fast implementation may be possible, as it is with Fourier extensions [26][27][28]. One potential idea in this direction is the AZ algorithm [17].…”

Section: Discussionmentioning

confidence: 99%

Fast and stable approximation of analytic functions from equispaced samples via polynomial frames

Adcock¹,

Shadrin²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…However, as mentioned in the introduction, more efficient algorithms exist. The AZ algorithm is the most general form of these efficient algorithms and it was introduced in [18]. The AZ algorithm is compatible with the techniques developed below, for several types of frames.…”

Section: Computational Costmentioning

confidence: 99%

“…For example, it reduces the computational complexity to O(N log 2 N ) for univariate Fourier extensions. The complexity for higher-dimensional problems remains higher than (log-)linear, but better than cubic -we refer to [18] for details.…”

Section: Computational Costmentioning

confidence: 99%

See 1 more Smart Citation

On the adaptive spectral approximation of functions using redundant sets and frames

Coppé

Huybrechs

2021

IMA Journal of Numerical Analysis

Self Cite

View full text Add to dashboard Cite

The approximation of smooth functions with a spectral basis typically leads to rapidly decaying coefficients, where the rate of decay depends on the smoothness of the function and vice versa. The optimal number of degrees of freedom in the approximation can be determined with relative ease by truncating the coefficients once a threshold is reached. Recent approximation schemes based on redundant sets and frames extend the applicability of spectral approximations to functions defined on irregular geometries and to certain nonsmooth functions. However, due to their inherent redundancy, the expansion coefficients in frame approximations do not necessarily decay even for very smooth functions. In this paper, we highlight this lack of equivalence between smoothness and coefficient decay, and we explore approaches to determine an optimal number of degrees of freedom for such redundant approximations.

show abstract

“…However, a regularizing singular value decomposition (SVD), truncated with a small threshold , can be employed to find accurate and stable approximations [3,2,1]. While the computation of the SVD has cubic complexity O N 3 , more recently algorithms for Fourier extensions have been proposed with lower computational complexity [20,21,22], culminating in the more general formulation of the so-called AZ algorithm for ill-conditioned least squares problems [12].…”

Section: Introductionmentioning

confidence: 99%

Efficient function approximation on general bounded domains using splines on a Cartesian grid

Coppé¹,

Huybrechs²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a simple, regular grid that is defined on a bounding box. This approach allows the use of high order and highly structured splines as a basis for piecewise polynomials. The methodology is analogous to that of Fourier extensions, using Fourier series on a bounding box, which leads to spectral accuracy for smooth functions. However, Fourier extension approximations involve solving a highly illconditioned linear system, and this is an expensive step. The computational complexity of recent algorithms is O N log 2 (N ) in 1 dimension and O N 2 log 2 (N ) in two dimensions. We show that the compact support of B-splines enables improved complexity for multivariate approximations, namely O(N ) in 1-D, O N 3 2 in 2-D and more generally O N 3(d−1) d in d-D with d > 1. This comes at the cost of achieving only algebraic rates of convergence. Our statements are corroborated with numerical experiments and Julia code is available.

show abstract

The AZ Algorithm for Least Squares Systems with a Known Incomplete Generalized Inverse

Cited by 15 publications

References 14 publications

Fast and stable approximation of analytic functions from equispaced samples via polynomial frames

Fast and stable approximation of analytic functions from equispaced samples via polynomial frames

On the adaptive spectral approximation of functions using redundant sets and frames

Efficient function approximation on general bounded domains using splines on a Cartesian grid

Contact Info

Product

Resources

About