Directional derivatives of the maximum function

Борисенко, О. Ф.; Minchenko, L. I.

doi:10.1007/bf01126219

Cited by 4 publications

(6 citation statements)

References 2 publications

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“…In fact, by definition, the function is directionally differentiable with respect to all its variables. Hence, using the results in 29 , 30 , we can deduce that is directionally differentiable with respect to ω 1 and ω 2 . This property can be exploited if one tries to solve ( 4.18 ) using a gradient based optimization method 31 .…”

Section: Analysis Of the If Algorithmmentioning

confidence: 85%

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Intrinsic Frequency Analysis and Fast Algorithms

Tavallali

Koorehdavoudi

Krupa³

2018

Sci Rep

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Intrinsic Frequency (IF) has recently been introduced as an ample signal processing method for analyzing carotid and aortic pulse pressure tracings. The IF method has also been introduced as an effective approach for the analysis of cardiovascular system dynamics. The physiological significance, convergence and accuracy of the IF algorithm has been established in prior works. In this paper, we show that the IF method could be derived by appropriate mathematical approximations from the Navier-Stokes and elasticity equations. We further introduce a fast algorithm for the IF method based on the mathematical analysis of this method. In particular, we demonstrate that the IF algorithm can be made faster, by a factor or more than 100 times, using a proper set of initial guesses based on the topology of the problem, fast analytical solution at each point iteration, and substituting the brute force algorithm with a pattern search method. Statistically, we observe that the algorithm presented in this article complies well with its brute-force counterpart. Furthermore, we will show that on a real dataset, the fast IF method can draw correlations between the extracted intrinsic frequency features and the infusion of certain drugs.

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Section: Analysis Of the If Algorithmmentioning

confidence: 85%

“…In fact, by definition, the function Q + p1 − f 2 2 is directionally differentiable with respect to all its variables. Hence, using the results in [5,28], we can deduce that (4. 19)…”

Section: Analysis Of the If Algorithmmentioning

confidence: 90%

Intrinsic Frequency Analysis and Fast Algorithms

Tavallali

Koorehdavoudi

Krupa³

2018

Sci Rep

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“…For an arbitrarily chosen b ∈ B we use the abbreviation For the first result we need a technical lemma. Early investigations on directional derivatives of sup inf functions can be found in [4] (compare also [2]).…”

Section: Necessary Conditions For Set Inequalitiesmentioning

confidence: 99%

Characterizations of the Set Less Order Relation in Nonconvex Set Optimization

Jahn

2022

J Optim Theory Appl

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“…and we search the expression of its derivative. For this, we use the main Theorem obtained in Pshenichny (1971) or Theorem 1 in Borisenko and Minchenko (1992) after verifying that the hypotheses of the Theorem are satisfied.…”

Section: Geometric Properties For Non Blow-up Of the Solutionsmentioning

confidence: 99%

“…Since Borisenko and Minchenko (1992)), by using also (41) we obtain the expression of the derivative of ϕ given for any t ∈ [0, T ] by,…”

Section: Geometric Properties For Non Blow-up Of the Solutionsmentioning

confidence: 99%

A new path to the non blow-up of incompressible flows

Agélas

2019

Ann. Inst. H. Poincaré Anal. Non Linéaire

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One of the most challenging questions in fluid dynamics is whether the threedimensional (3D) incompressible Navier-Stokes, 3D Euler and two-dimensional Quasi-Geostrophic (2D QG) equations can develop a finite-time singularity from smooth initial data. Recently, from a numerical point of view, Luo & Hou presented a class of potentially singular solutions to the Euler equations in a fluid with solid boundary Luo and Hou (2014a,b). Furthermore, in two recent papers Tao (2016a,b), Tao indicates a significant barrier to establishing global regularity for the 3D Euler and Navier-Stokes equations, in that any method for achieving this, must use the finer geometric structure of these equations. In this paper, we show that the singularity discovered by Luo & Hou which lies right on the boundary is not relevant in the case of the whole domain R 3. We reveal also that the translation and rotation invariance present in the Euler, Navier-Stokes and 2D QG equations are the key for the non blow-up in finite time of the solutions. The translation and rotation invariance of these equations combined with the anisotropic structure of regions of high vorticity allowed to establish a new geometric non blow-up criterion which yield us to the non blow-up of the solutions in all the Kerr's numerical experiments and to show that the potential mechanism of blow-up introduced in Brenner et al. (2016) cannot lead to the blow-up in finite time of solutions of Euler equations. Contents 1 Introduction 2 2 Some notations and definitions 8 3 Local regularity of the solutions 9 3.1 Local regularity for 3D Navier-Stokes or 3D Euler equations. . 10 3.

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Directional derivatives of the maximum function

Cited by 4 publications

References 2 publications

Intrinsic Frequency Analysis and Fast Algorithms

Intrinsic Frequency Analysis and Fast Algorithms

Characterizations of the Set Less Order Relation in Nonconvex Set Optimization

A new path to the non blow-up of incompressible flows

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