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Bounds-constrained polynomial approximation using the Bernstein basis.
Larry Allen,Robert C. Kirby +1 more
TL;DR: In this paper, the problem of approximating functions by polynomials whose Bernstein coefficients with respect to a given degree satisfy such bounds is considered, which implies such bounds on the approximant.
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Abstract: A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In this paper, we consider the problem of approximating functions by polynomials whose Bernstein coefficients with respect to a given degree satisfy such bounds, which implies such bounds on the approximant. We frame the problem as an inequality-constrained optimization problem and give an algorithm for finding the Bernstein coefficients of the exact solution. Additionally, our method can be modified slightly to include equality constraints such as mass preservation. It also extends naturally to multivariate polynomials over a simplex.
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Citations
Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints
Brendan Keith,Thomas M. Surowiec +1 more
TL;DR: This paper introduces the proximal Galerkin method, a high-order, low-iteration numerical method preserving structure for pointwise bound constraints, and applies it to free boundary problems, optimal design, and topology optimization, combining ideas from nonlinear programming, functional analysis, and differential geometry.
1
Approximating the Level Curves on Pascal's Surface
TL;DR: In this article , the authors present a selection of approximations for the level curves problem on Pascal's surface, and present fresh simulations supporting the claim that some of these could be quite useful, as being both reasonably easy to calculate as well as fairly accurate.
1
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Density theorems with applications in quantum signal processing
TL;DR: In this article, the approximation capabilities of two families of univariate polynomials that arise in applications of quantum signal processing were studied and it was shown that the additional constraints do not hinder the ability of these polynomial families to approximate arbitrarily well any continuous function in the supremum norm.
Non-dissipative and structure-preserving emulators via spherical optimization
TL;DR: In this paper , the authors present a new framework that enforces via optimization such structure on approximations and is simultaneously norm-preserving, which results in a conceptually simple convex optimization problem on the sphere.
High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities
Robert C. Kirby,Daniel Shapero +1 more
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