On partial derivatives of multivariate Bernstein polynomials

Veretennikov, A. Yu.; Veretennikova, E. V.

doi:10.3103/s1055134416040039

Cited by 10 publications

(3 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There exists a sequence of multivariate polynomials [81]), where we will choose l > 0 large enough later. Fix n ∈ AE for the moment and write P n in the form…”

Section: Diagonal Continuitymentioning

confidence: 99%

Probabilistic Descriptions of Fluid Flow: A Survey

Gallenmüller¹,

Wagner²,

Wiedemann³

2023

Preprint

View full text Add to dashboard Cite

Fluids can behave in a highly irregular, turbulent way. It has long been realised that, therefore, some weak notion of solution is required when studying the fundamental partial differential equations of fluid dynamics, such as the compressible or incompressible Navier-Stokes or Euler equations. The standard concept of weak solution (in the sense of distributions) is still a deterministic one, as it gives exact values for the state variables (like velocity or density) for almost every point in time and space. However, observations and mathematical theory alike suggest that this deterministic viewpoint has certain limitations. Thus, there has been an increased recent interest in the mathematical fluids community in probabilistic concepts of solution. Due to the considerable number of such concepts, it has become challenging to navigate the corresponding literature, both classical and recent. We aim here to give a reasonably concise yet fairly detailed overview of probabilistic formulations of fluid equations, which can roughly be split into measure-valued and statistical frameworks. We discuss both approaches and their relationship, as well as the interrelations between various statistical formulations, focusing on the compressible and incompressible Euler equations.

show abstract

“…There exists a sequence of multivariate polynomials [81]), where we will choose l > 0 large enough later. Fix n ∈ AE for the moment and write P n in the form…”

Section: Diagonal Continuitymentioning

confidence: 99%

Probabilistic Descriptions of Fluid Flow: A Survey

Gallenmüller¹,

Wagner²,

Wiedemann³

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…We now provide a brief description of how Bernstein polynomials can approximate smooth functions. For a more in-depth overview of the field we refer to [13]. Now, recalling from Section II that we P d (R n , R) as the set of d-degree polynomials we next define the Bernstein operator that maps onto P d (R n , R).…”

Section: A Bernstein Approximation Of Smooth Functionsmentioning

confidence: 99%

Combining Trajectory Data with Analytical Lyapunov Functions for Improved Region of Attraction Estimation

Lugnani¹,

Jones²,

Alberto³

et al. 2021

Preprint

View full text Add to dashboard Cite

The increasing penetration of inverter based sources (IBS) has been raising a lot of concerns for the power system system operators around the world. The high level of complexity of this kind of generators make model-based stability analysis a difficult task for operators. This work presents a method to estimate the Region of Attraction (ROA) based on sychrophasor measurements, through trajectory polynomial fitting using Bernstein polynomials, of a converse Lyapunov function of the system and its derivative. The method is derived from the expansion of a previous ROA, which guarantees an inner approximation and is not dependent on the chosen approximation method, in this case the Bernstein polynomials. The mathematical derivation is described and applied to the Single Machine Infinite Bus (SMIB) system.

show abstract

“…See [35,Theorem 4] for a proof. Let m be fixed now and large enough for the above inequalities to hold.…”

Section: Approximation Of Homogeneous Functions By Rational Functionsmentioning

confidence: 99%

On Algebraic Proofs of Stability for Homogeneous Vector Fields

Ahmadi

Khadir

2020

IEEE Trans. Automat. Contr.

View full text Add to dashboard Cite

We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sum of squares certificates and hence such a Lyapunov function can always be found by semidefinite programming. This generalizes the classical fact that an asymptotically stable linear system admits a quadratic Lyapunov function which satisfies a certain linear matrix inequality. In addition to homogeneous vector fields, the result can be useful for showing local asymptotic stability of non-homogeneous systems by proving asymptotic stability of their lowest order homogeneous component.This paper also includes some negative results: We show that (i) in absence of homogeneity, globally asymptotically stable polynomial vector fields may fail to admit a global rational Lyapunov function, and (ii) in presence of homogeneity, the degree of the numerator of a rational Lyapunov function may need to be arbitrarily high (even for vector fields of fixed degree and dimension). On the other hand, we also give a family of homogeneous polynomial vector fields that admit a low-degree rational Lyapunov function but necessitate polynomial Lyapunov functions of arbitrarily high degree. This shows the potential benefits of working with rational Lyapunov functions, particularly as the ones whose existence we guarantee have structured denominators and are not more expensive to search for than polynomial ones.Index Terms. Converse Lyapunov theorems, nonlinear dynamics, algebraic methods in control, semidefinite programming, rational Lyapunov functions.

show abstract

On partial derivatives of multivariate Bernstein polynomials

Abstract: It is shown that Bernstein polynomials for a multivariate function converge to this function along with partial derivatives provided that the latter derivatives exist and are continuous. This result may be useful in some issues of stochastic calculus.

Cited by 10 publications

References 7 publications

Probabilistic Descriptions of Fluid Flow: A Survey

Probabilistic Descriptions of Fluid Flow: A Survey

Combining Trajectory Data with Analytical Lyapunov Functions for Improved Region of Attraction Estimation

On Algebraic Proofs of Stability for Homogeneous Vector Fields

Contact Info

Product

Resources

About