The Experts below are selected from a list of 273 Experts worldwide ranked by ideXlab platform
Miroslav Krstic - One of the best experts on this subject based on the ideXlab platform.
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Maximizing Map Sensitivity and Higher Derivatives Via Extremum Seeking
IEEE Transactions on Automatic Control, 2018Co-Authors: Greg Mills, Miroslav KrsticAbstract:We present a generalization to the scalar Newton-based extremum seeking algorithm which, through perturbation-induced measurements of an unknown steady-state input-to-output map, maximizes the map's higher Derivatives. As with other extremum seeking (ES) problems, the scheme relies on Derivative estimators and learning dynamics operating in different time scales. We provide analysis for the typical deterministic (sinusoidal) perturbation of the optimal parameter estimate. Also, we alternatively include a stochastic method where the map is perturbed via the sinusoid of Brownian motion about the boundary of a circle. By properly demodulating the map output corresponding to the manner in which it is perturbed, the ES algorithm maximizes the Nth Derivative only through measurements of the map. The Newton-based ES approach removes the dependence of the convergence rate on the unknown Hessian of the higher Derivative, an effort to improve performance over standard gradient-based ES. Our design stems from the existing multivariable Newton-based ES algorithm where a differential Riccati equation estimates the inverse Hessian of the function to be maximized. Algebraically computing a direct estimate of the inverse Hessian is susceptible to singularity, whereas employing the Riccati filter removes that potential. We prove local stability of the algorithm for general nonlinear equilibrium profiles of dynamic maps and compare the Newton-based method against the gradient-based method. We also extend this abstraction to multiplayer noncooperative games but limit our attention to a two-player game for notational simplicity and length considerations.
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Maximizing higher Derivatives of unknown maps with stochastic extremum seeking
2016 American Control Conference (ACC), 2016Co-Authors: Greg Mills, Miroslav KrsticAbstract:We present a stochastic generalization to the scalar Newton-based extremum seeking algorithm which through measurements of an unknown map, maximizes the map's higher Derivatives. The proposed method perturbs the estimate of the optimal parameter with the sinusoid of Brownian motion about the boundary of a circle. Then by properly demodulating the map output of the randomly perturbed estimate, the extremum seeking algorithm maximizes the Nth Derivative only through measurements of the map. The Newton-based extremum seeking approach removes the dependence of the convergence rate on the unknown Hessian of the higher Derivative, an effort to improve performance over standard gradient-based extremum seeking. Our design stems from the existing multivariable Newton-based extremum seeking algorithm where a differential Riccati equation estimates the inverse Hessian of the function to be maximized. Algebraically computing a direct estimate of the inverse Hessian is susceptible to singularity, where-as employing the Riccati filter removes that potential. We prove local stability of the algorithm for general nonlinear static maps via stochastic averaging theory developed for extremum seeking.
J.m. Gryn - One of the best experts on this subject based on the ideXlab platform.
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ICIP (3) - Three-dimensional Nth Derivative of Gaussian separable steerable filters
IEEE International Conference on Image Processing 2005, 2020Co-Authors: K.g. Derpanis, J.m. GrynAbstract:This paper details the construction of three-dimensional separable steerable filters. The approach presented is an extension of the construction of two-dimensional separable steerable filters outlined in W.T. Freeman and E.H. Adelson (1991). Additionally, three-dimensional separable steerable filters, both continuous and discrete versions, for the second Derivative of the Gaussian and its Hilbert transform are reported. Experimental evaluation demonstrates that the errors in the constructed separable filters are negligible.
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Three-dimensional Nth Derivative of Gaussian separable steerable filters
IEEE International Conference on Image Processing 2005, 2005Co-Authors: K.g. Derpanis, J.m. GrynAbstract:This paper details the construction of three-dimensional separable steerable filters. The approach presented is an extension of the construction of two-dimensional separable steerable filters outlined in W.T. Freeman and E.H. Adelson (1991). Additionally, three-dimensional separable steerable filters, both continuous and discrete versions, for the second Derivative of the Gaussian and its Hilbert transform are reported. Experimental evaluation demonstrates that the errors in the constructed separable filters are negligible.
Mhenni M Benghorbal - One of the best experts on this subject based on the ideXlab platform.
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a unified formula for the Nth Derivative and the Nth anti Derivative of the bessel function of real orders
American Journal of Applied Mathematics and Statistics, 2015Co-Authors: Mhenni M BenghorbalAbstract:A complete solution to the problem of finding the Nth Derivative and the Nth anti-Derivative of elementary and special functions has been given. It deals with the problem of finding formulas for the Nth Derivative and the Nth anti-Derivative of elementary and special functions. We do not limit n to be an integer, it can be a real number. In general, the solution is given through unified formulas in terms of the Fox H-function and the Miejer G-function which, in many cases, can be simplified to less general functions. This, in turn, makes the first real use of these two special functions in the literature and shows the need of such functions. In this talk, we would like to present the idea on the Bessel function which is a well known special function. One of the key points in this work is that the approach does not depend on integration techniques. We adopt the classical definitions for generalization of differentiation and integration. Namely, the Nth order of differentiation is found according to the Riemann-Liouville definition where (k − 1 . The generalized Cauchy n-fold integral is adopted for the Nth order of integration
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A Unified Formula for the Nth Derivative and the Nth Anti-Derivative of the Power-Logarithmic Class
2009 International Conference on Computing Engineering and Information, 2009Co-Authors: Mhenni M BenghorbalAbstract:We give a complete solution to the problem of finding the Nth Derivative and the Nth anti-Derivative, where n is a real number or a symbol, of elementary and special classes of functions. In general, the solutions are given through unified formulas in terms of the Fox H-function which in many cases can be simplified for less general functions. In this work, we consider the class of the power-logarithmic class { f(x):f(x)=Sigmaj=1 lscrpj(xalpha j)ln(betajxgamma j+1)} (1) where alphajisinCopf, betajisinCopf\{0}, gammajisinRopf\{0}, and pj's are polynomials of certain degrees.One of the key points in this work is that the approach does not depend on integration techniques. The arbitrary order of differentiation is found according to the Riemann-Liouville definition, whereas the generalized Cauchy n-fold integral is adopted for arbitrary order of integration. A software exhibition will be within the talk using the computer algebra system Maple.
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a unified formula for the Nth Derivative and the Nth anti Derivative of the power logarithmic class
2009 International Conference on Computing Engineering and Information, 2009Co-Authors: Mhenni M BenghorbalAbstract:We give a complete solution to the problem of finding the $n$thDerivative and the $n$th anti-Derivative, where $n$ is a real number or a symbol,of elementary and special classes of functions. In general, the solutions are given through unified formulas in terms of the Fox H-function~\cite{Fox:1961} which in many cases can be simplified for less general functions. In this work, we consider the class of the \textsl{power-logarithmic class}\begin{equation}\label{power-logarithmic-class}\left\{ f(x):f(x)=\sum_{j=1}^{\ell}p_j(x^{\alpha_j})\ln({\beta_j x^{\gamma_j}}+1)\right\}\end{equation}where $\alpha_j \in \mathbb{C}$,$\beta_j \in \mathbb{C} \backslash \{0\}$,$\gamma_j \in\mathbb{R}\backslash \{0\}$, and $p_j$'s are polynomials of certain degrees.One of the key points in this work is that the approach does not depend on integration techniques. The arbitrary order of differentiation is found according to the Riemann-Liouville definition, whereas the generalized Cauchy $n$-fold integral is adopted for arbitrary order of integration. A software exhibition will be within the talk using the computer algebra system Maple.
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Unified formulas for arbitrary order symbolic Derivatives and anti-Derivatives of the power-inverse hyperbolic class 1
ACM Communications in Computer Algebra, 2009Co-Authors: Mhenni M BenghorbalAbstract:We continue on tackling and giving a complete solution to the problem of finding the Nth Derivative and the Nth anti-Derivative, where n can be an integer, a fraction, a real, or a symbol, of elementary and special classes of functions. In general, the solutions are given through unified formulas in terms of the Fox H-function which in many cases can be simplified to less general functions. In this work, we consider two subclasses of the power-inverse hyperbolic class. Namely, the power-inverse hyperbolic sine class {f(x) : f(x) = Σlj=1Pj(xαj)arcsinh(βjxγj), αj ∈ C, βj ∈ C\{0},γj ∈ R\{0}, (1) and the power-inverse hyperbolic cosine class {f(x) : f(x) = Σlj=1Pj(xαj)arccosh(βjxγj), αj ∈ C, βj ∈ C\{0},γj ∈ R\{0}, (2) where pj's are polynomials of certain degrees. One of the key points in this work is that the approach does not depend on integration techniques The arbitrary order of differentiation is found according to the Riemann-Liouville definition, whereas the generalized Cauchy n-fold integral is adopted for arbitrary order of integration. The motivation of this work comes from the area of symbolic computation. The idea is that: Given a function f in a variable x, can CAS find a formula for the Nth Derivative, the Nth anti-Derivative, or both of f? This enhances the power of integration and differentiation of CAS. In Maple, the formulas correspond to invoking the commands diff(f(x) for the Nth Derivative and int(f(x), x$n) for the Nth anti-Derivative. A software exhibition will be given using Maple. Example: A unified formula for arcsinh(√x) in terms of the Meijer G-function (arcsinh(√x)) (n) = x(1/2--nover2√π G1,2over1,2 (1/2,1/2over0,n--1/2│x) , │x│ The above G-function reduces to the original function if n = 0. It gives Derivatives of any order if n > 0 and anti-Derivatives of any order if n
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The n th Derivative
ACM Sigsam Bulletin, 2002Co-Authors: Mhenni M Benghorbal, Robert M. CorlessAbstract:The following problem is one that many first year calculus students find quite difficult:Given a formula for a function f in a variable x, find a formula for its Nth Derivative.Example 1.1: [1, p. 229] Iff(x) = xm, (1)then its Nth Derivative isf(n) (x) = m-nxm-n, (2)wherem-n = m(m - 1) (m - 2) ṡṡṡ (m - n + 1).The difficulties for students include, first, the discovery of a formula valid for all integers n and, second, the proof (for example, by induction) that the formula is correct. Can computer algebra systems do better?It is certain that Macsyma could (we remember commands for infinite Taylor series expansion of elementary functions, and that necessarily involves discovering a correct formula for the Nth Derivative of the input). Currently, Maple cannot---at least, not without help, just with one command, now that the Formal Power Series package in the share library is no longer supported.
Greg Mills - One of the best experts on this subject based on the ideXlab platform.
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Maximizing Map Sensitivity and Higher Derivatives Via Extremum Seeking
IEEE Transactions on Automatic Control, 2018Co-Authors: Greg Mills, Miroslav KrsticAbstract:We present a generalization to the scalar Newton-based extremum seeking algorithm which, through perturbation-induced measurements of an unknown steady-state input-to-output map, maximizes the map's higher Derivatives. As with other extremum seeking (ES) problems, the scheme relies on Derivative estimators and learning dynamics operating in different time scales. We provide analysis for the typical deterministic (sinusoidal) perturbation of the optimal parameter estimate. Also, we alternatively include a stochastic method where the map is perturbed via the sinusoid of Brownian motion about the boundary of a circle. By properly demodulating the map output corresponding to the manner in which it is perturbed, the ES algorithm maximizes the Nth Derivative only through measurements of the map. The Newton-based ES approach removes the dependence of the convergence rate on the unknown Hessian of the higher Derivative, an effort to improve performance over standard gradient-based ES. Our design stems from the existing multivariable Newton-based ES algorithm where a differential Riccati equation estimates the inverse Hessian of the function to be maximized. Algebraically computing a direct estimate of the inverse Hessian is susceptible to singularity, whereas employing the Riccati filter removes that potential. We prove local stability of the algorithm for general nonlinear equilibrium profiles of dynamic maps and compare the Newton-based method against the gradient-based method. We also extend this abstraction to multiplayer noncooperative games but limit our attention to a two-player game for notational simplicity and length considerations.
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Maximizing higher Derivatives of unknown maps with stochastic extremum seeking
2016 American Control Conference (ACC), 2016Co-Authors: Greg Mills, Miroslav KrsticAbstract:We present a stochastic generalization to the scalar Newton-based extremum seeking algorithm which through measurements of an unknown map, maximizes the map's higher Derivatives. The proposed method perturbs the estimate of the optimal parameter with the sinusoid of Brownian motion about the boundary of a circle. Then by properly demodulating the map output of the randomly perturbed estimate, the extremum seeking algorithm maximizes the Nth Derivative only through measurements of the map. The Newton-based extremum seeking approach removes the dependence of the convergence rate on the unknown Hessian of the higher Derivative, an effort to improve performance over standard gradient-based extremum seeking. Our design stems from the existing multivariable Newton-based extremum seeking algorithm where a differential Riccati equation estimates the inverse Hessian of the function to be maximized. Algebraically computing a direct estimate of the inverse Hessian is susceptible to singularity, where-as employing the Riccati filter removes that potential. We prove local stability of the algorithm for general nonlinear static maps via stochastic averaging theory developed for extremum seeking.
K.g. Derpanis - One of the best experts on this subject based on the ideXlab platform.
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ICIP (3) - Three-dimensional Nth Derivative of Gaussian separable steerable filters
IEEE International Conference on Image Processing 2005, 2020Co-Authors: K.g. Derpanis, J.m. GrynAbstract:This paper details the construction of three-dimensional separable steerable filters. The approach presented is an extension of the construction of two-dimensional separable steerable filters outlined in W.T. Freeman and E.H. Adelson (1991). Additionally, three-dimensional separable steerable filters, both continuous and discrete versions, for the second Derivative of the Gaussian and its Hilbert transform are reported. Experimental evaluation demonstrates that the errors in the constructed separable filters are negligible.
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Three-dimensional Nth Derivative of Gaussian separable steerable filters
IEEE International Conference on Image Processing 2005, 2005Co-Authors: K.g. Derpanis, J.m. GrynAbstract:This paper details the construction of three-dimensional separable steerable filters. The approach presented is an extension of the construction of two-dimensional separable steerable filters outlined in W.T. Freeman and E.H. Adelson (1991). Additionally, three-dimensional separable steerable filters, both continuous and discrete versions, for the second Derivative of the Gaussian and its Hilbert transform are reported. Experimental evaluation demonstrates that the errors in the constructed separable filters are negligible.
