Algorithms of Constrained Uniform Approximation in Numerical Problems

核心概念

Efficient numerical methods for constrained uniform approximation are developed.

摘要

The content discusses algorithms for the best uniform approximation of continuous functions on convex domains using linear combinations of functions under constraints. It introduces a method based on alternance concepts and the Remez iterative procedure, proving convergence under certain assumptions. Special attention is given to complex exponents, Gaussian functions, and polynomial systems. Applications in signal processing, ODEs, dynamical systems, and inequalities are explored.

Introduction:

Discusses Chebyshev systems for function approximation.
Introduces non-Chebyshev systems and challenges in uniform polynomial approximation.

The Roadmap of Main Results:

Defines spaces and polynomials for approximations.

The Generalized Alternance:

Formulates criteria for best uniform approximation by non-Chebyshev systems.

Algorithm of Best Approximation - Regular Case:

Describes an algorithm with linear convergence rate under regularity assumptions.

Algorithm in the General Case - Regularization:

Presents a regularization method when simplices become degenerate.

Data Extraction:

"A system of N vectors b1, . . . , bN in RN"
"Let vectors u(t1), . . . , u(tn−1) span a hyperplane H ⊂Rn."

Quotations:

"The efficiency of the new approach is approved by numerical experiments."

Inquiry and Critical Thinking:
How do non-Chebyshev systems impact numerical problems?
What are the implications of degenerate configurations in approximation algorithms?
How can these methods be extended to higher-dimensional spaces?

統計資料

"A system of N vectors b1, . . . , bN in RN"
"Let vectors u(t1), . . . , u(tn−1) span a hyperplane H ⊂Rn."

引述

"The efficiency of the new approach is approved by numerical experiments."

從以下內容提煉的關鍵洞見

Algorithms of constrained uniform approximation

by Vladimir Yu.... 於 arxiv.org 03-26-2024

https://arxiv.org/pdf/2403.16330.pdf

深入探究

How do non-Chebyshev systems impact numerical problems

Non-Chebyshev systems impact numerical problems by introducing challenges in the approximation of functions on compact domains. Unlike Chebyshev systems, non-Chebyshev systems do not always have unique solutions for the best approximation of a function. This lack of uniqueness can complicate numerical computations and make it harder to find optimal approximations efficiently.

What are the implications of degenerate configurations in approximation algorithms

Degenerate configurations in approximation algorithms can lead to slow convergence or even cause the algorithm to get stuck. When a configuration becomes degenerate, it means that certain properties or conditions necessary for efficient computation are no longer valid. In such cases, the algorithm may struggle to progress towards finding an optimal solution due to ill-conditioned linear systems or other technical issues.

How can these methods be extended to higher-dimensional spaces

These methods can be extended to higher-dimensional spaces by adapting them to work with multivariate systems. In higher-dimensional spaces, the concept of generalized alternance and regularization techniques can still be applied to improve convergence rates and overcome degeneracy issues. By considering sets of points in multidimensional domains and adjusting the algorithms accordingly, these methods can effectively handle complex numerical problems in higher dimensions as well.