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Difference Methods for Differential Inclusions: A Survey

Asen Dontchev, Frank Lempio
1992 SIAM Review  
Farkhi, Sofia, and A. Kastner-Maresch, Bayreuth.  ...  Suppose that the map F : I x Rn ===? Rn satisfies the following conditions. THEOREM 6.3 ([22]). On the conditions of Theorem 6.2 haus(X, Xh) = O(h) in C(I)n.  ...  Sometimes it may be convenient to use an approximation of the set-valued map F in (10), e.g., by polygons. Let "'N (·)satisfy J ·-o 1 ...  ... 
doi:10.1137/1034050 fatcat:lx3gvoxv5bf4piucw2smw5snqy

Lipschitzian Stability in Nonlinear Control and Optimization

Asen L. Dontchev, William W. Hager
1993 SIAM Journal of Control and Optimization  
Recently, a different approach to sensitivity based on nonsmooth analysis and the differentiability properties of set-valued maps was developed by Aubin [4], Aubin and Frankowska [6], Rockafellar [39]  ...  A motivation for the nonsmooth approach to Lipschitz stability is given by Rockafellar in [40] . An outline of our paper follows, while detailed comments connecting specific  ...  The work was performed while the first author was a visitor at the University of Florida. Then z* is a strict local minimizer for (85).  ... 
doi:10.1137/0331026 fatcat:lw2hkazoujcj5ixl7m2lph4q6q

First-order conditions for the optimal control of learning-informed nonsmooth PDEs [article]

Guozhi Dong and Michael Hintermüller and Kostas Papafitsoros and Kathrin Völkner
2022 arXiv   pre-print
Based on approximation capabilities of the pertinent networks, we address fundamental questions regarding approximating properties of the learning-informed control-to-state map and the solution of the  ...  In this paper we study the optimal control of a class of semilinear elliptic partial differential equations which have nonlinear constituents that are only accessible by data and are approximated by nonsmooth  ...  ) satisfying (3.4), with the corresponding control-to-state-map S n satisfying (i)-(iii) of Proposition 3.1 (with global Lipschitz constants).  ... 
arXiv:2206.00297v1 fatcat:xq5tamh4ifaphismnyoc3pbxgu

Representing smooth functions as compositions of near-identity functions with implications for deep network optimization [article]

Peter L. Bartlett, Steven N. Evans, Philip M. Long
2018 arXiv   pre-print
,h_m, any critical point of a quadratic criterion in this near-identity region must be a global minimizer.  ...  We show that any smooth bi-Lipschitz h can be represented exactly as a composition h_m ∘ ... ∘ h_1 of functions h_1,...  ...  Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS).  ... 
arXiv:1804.05012v2 fatcat:adbunxjlkvexvg7umy7b2by6su

GAN Estimation of Lipschitz Optimal Transport Maps [article]

Alberto González-Sanz
2022 arXiv   pre-print
Furthermore, we show through a number of numerical experiments that the learnt mapping has promising performances.  ...  Building on theoretical and practical advances in the field of Lipschitz neural networks, we define a Lipschitz-constrained generative adversarial network penalized by the quadratic transportation cost  ...  -1 distance, and G n is a class of Lipschitz generators parametrizing the space of feasible mappings.  ... 
arXiv:2202.07965v1 fatcat:xbssal4lfjevfndtdt6owruaka

Set-valued solutions to the Cauchy problem for hyperbolic systems of partial differential inclusions

Jean-Pierre Aubin, Halina Frankowska
1997 NoDEA. Nonlinear differential equations and applications (Printed ed.)  
We prove the existence of global set-valued solutions to the Cauchy problem for partial differential equations and inclusions, with either single-valued or set-valued initial conditions.  ...  The method is based on the equivalence between this problem and problem of finding viability tubes of the associated characteristic system of ordinary differential equations or differential inclusions.  ...  We also adapt the proof of Theorem 4.2 with Ql(t, x) := f (t, x, u(t, x)), zl(s) := z(s), Qz := Q and Q2 := Q, to show that the estimates stated in the theorem hold true.  ... 
doi:10.1007/pl00001413 fatcat:azu2i5gv7revhaw5vqkjrswvpa

Inexact alternating projections on nonconvex sets [article]

Dmitriy Drusvyatskiy, Adrian S. Lewis
2018 arXiv   pre-print
Given two arbitrary closed sets in Euclidean space, a simple transversality condition guarantees that the method of alternating projections converges locally, at linear rate, to a point in the intersection  ...  On the other hand, if one set is a smooth manifold represented through local coordinates, then the approximate projection resulting from linearizing the coordinate system around the preceding iterate on  ...  (u, v), as a function of the point z, it is globally 1-Lipschitz (being a projection).  ... 
arXiv:1811.01298v1 fatcat:wpphraydqnfbzcosb5g6ouye5q

Strong convergence and stability of Picard iteration sequences for a general class of contractive-type mappings

Charles E Chidume
2014 Fixed Point Theory and Applications  
Let (E, · ) be a normed linear space, T : E → E be a mapping of E into itself satisfying the following contractive condition: It is shown that the Picard iteration process converges strongly to the unique  ...  In particular, it improves a recent result of Akewe et al. (Fixed Point Theory Appl 2014:45, 2014, and a host of other results.  ...  All attempts to use the Mann formula, which has been successfully employed for nonexpansive mappings, to approximate a fixed point of a Lipschitz pseudocontractive map even on a compact convex domain in  ... 
doi:10.1186/1687-1812-2014-233 fatcat:bw2yaewfnbavjosbbg3mgzpwwm

Tropical Dynamic Programming for Lipschitz Multistage Stochastic Programming [article]

Marianne Akian and Jean-Philippe Chancelier and Benoît Tran
2020 arXiv   pre-print
We prove, for every Lipschitz MSP, the asymptotic convergence of the generated approximating functions of TDP to the Bellman value functions on sets of interest.  ...  Here, Tropical Dynamic Programming builds upper (resp. lower) approximations of a given value function as min-plus linear (resp. max-plus linear) combinations of "basic functions".  ...  A.1 Recalls on POMDP P u t (x i , x j ) = P{X t+1 = x j | X t = x i , U t = u} , (13) and conditional law of the observations Q t+1 (o | x, u) = P{O t+1 = o | X t+1 = x, U t = u} , (14) a real-valued cost  ... 
arXiv:2010.10619v1 fatcat:ocufnjmygvdmldb3mx3gnf232m

Stability Theory for Parametric Generalized Equations and Variational Inequalities Via Nonsmooth Analysis

Boris Mordukhovich
1994 Transactions of the American Mathematical Society  
This approach allows us to obtain effective sufficient conditions as well as necessary and sufficient conditions for a natural Lipschitzian behavior of solution maps.  ...  Our main concern is Lipschitzian stability of multivalued solution maps depending on parameters.  ...  Acknowledgments The author gratefully acknowledges many fruitful discussions with Professor Terry Rockafellar on the material of this paper.  ... 
doi:10.2307/2154734 fatcat:c5l6h7mfuzaujfsyfihuop6yhq

Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis

Boris Mordukhovich
1994 Transactions of the American Mathematical Society  
This approach allows us to obtain effective sufficient conditions as well as necessary and sufficient conditions for a natural Lipschitzian behavior of solution maps.  ...  Our main concern is Lipschitzian stability of multivalued solution maps depending on parameters.  ...  Acknowledgments The author gratefully acknowledges many fruitful discussions with Professor Terry Rockafellar on the material of this paper.  ... 
doi:10.1090/s0002-9947-1994-1242786-4 fatcat:6yger2hizfdixpt7dxup3eqj5a

An asymptotic viscosity selection result for the regularized Newton dynamic [article]

Boushra Abbas
2015 arXiv   pre-print
When Φ is a convex differentiable function whose gradient is Lipschitz continuous, we show that the strong convergence property is satisfied.  ...  Precisely, when C=argminΦ≠∅, and ε (·) is a "slow" control, i.e., ∫_0^+∞ε(t)dt=+∞, then each trajectory of the system converges weakly, as t goes to +∞, to the element of minimal norm of the closed convex  ...  Hence, in this situation (H 1 ) ǫ is satisfied if the following condition on ǫ(·) is satisfied: +∞ 0 ǫ 2 (t) < +∞.  ... 
arXiv:1504.07793v1 fatcat:kkv4nuooxnc6rohrftdllgvgxa

Infinitesimal Bishop-Gromov condition for Alexandrov spaces [article]

Kazuhiro Kuwae, Takashi Shioya
2010 arXiv   pre-print
We prove the infinitesimal version of Bishop-Gromov volume comparison condition for Alexandrov spaces.  ...  The triangle comparison condition implies the local Lipschitz continuity of the map Φ p,t : W p,t → M . We call Φ p,t the radial expansion map.  ...  By the triangle comparison condition, exp p : U p ∩B(o p , r) → M is Lipschitz continuous with Lipschitz constant 1 + Θ(r|p).  ... 
arXiv:1003.4334v1 fatcat:6qxxl3qfjnbenmiwyp6bzbq6b4

A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions

Adam M. Oberman
2004 Mathematics of Computation  
This article considers the problem of building absolutely minimizing Lipschitz extensions to a given function.  ...  A convergent difference scheme for the infinity Laplacian equation is introduced, which arises by minimizing the discrete Lipschitz constant of the solution at every grid point.  ...  Definition (Explicit Euler map and nonlinear CFL condition). Let F be a Lipschitz continuous, discretely elliptic scheme, with Lipschitz constant K.  ... 
doi:10.1090/s0025-5718-04-01688-6 fatcat:3vywq377wvh3zfg22jqeaqqgda

On the first integral conjecture of René Thom

Jacky Cresson, Aris Daniilidis, Masahiro Shiota
2008 Bulletin des Sciences Mathématiques  
In this work, using a recent nonsmooth extension of Sard theorem we establish the validity of Thom's conjecture for locally Lipschitz first integrals, interpreting genericity in the C 1 sense. (J.  ...  More than half a century ago R.  ...  Acknowledgement Part of this work has been made during a research visit of the second author to the University of Pau (June 2007).  ... 
doi:10.1016/j.bulsci.2007.10.002 fatcat:pqiwtamskndpbc3hssv6ktd3du
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