116,771 research outputs found
Shape optimization with Stokes constraints over the set of axisymmetric domains
International audienceIn this paper, we are interested in the study of shape optimizations problems with Stokes constraints within the class of axisymmetric domains represented by the graph of a function. Existence results with weak assumptions on the regularity of the graph are provided. We strongly use these assumptions to get some topological properties. We formulate the (shape) optimization problem using different constraints formulations: uniform bound constraints on the function and its derivative and/or volume (global) constraint. Writing the first order optimality conditions allows to provide quasi-explicit solutions in some particular cases and to give some hints for the treatment of the generic problem. Furthermore, we extend the (negative) result of [16] dealing with the non optimality of the cylinder
The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent
The paper deals with the genericity of domain-dependent spectral properties
of the Laplacian-Dirichlet operator. In particular we prove that, generically,
the squares of the eigenfunctions form a free family. We also show that the
spectrum is generically non-resonant. The results are obtained by applying
global perturbations of the domains and exploiting analytic perturbation
properties. The work is motivated by two applications: an existence result for
the problem of maximizing the rate of exponential decay of a damped membrane
and an approximate controllability result for the bilinear Schr\"odinger
equation
Prox-DBRO-VR: A Unified Analysis on Decentralized Byzantine-Resilient Composite Stochastic Optimization with Variance Reduction and Non-Asymptotic Convergence Rates
Decentralized Byzantine-resilient stochastic gradient algorithms resolve
efficiently large-scale optimization problems in adverse conditions, such as
malfunctioning agents, software bugs, and cyber attacks. This paper targets on
handling a class of generic composite optimization problems over multi-agent
cyberphysical systems (CPSs), with the existence of an unknown number of
Byzantine agents. Based on the proximal mapping method, two variance-reduced
(VR) techniques, and a norm-penalized approximation strategy, we propose a
decentralized Byzantine-resilient and proximal-gradient algorithmic framework,
dubbed Prox-DBRO-VR, which achieves an optimization and control goal using only
local computations and communications. To reduce asymptotically the variance
generated by evaluating the noisy stochastic gradients, we incorporate two
localized variance-reduced techniques (SAGA and LSVRG) into Prox-DBRO-VR, to
design Prox-DBRO-SAGA and Prox-DBRO-LSVRG. Via analyzing the contraction
relationships among the gradient-learning error, robust consensus condition,
and optimal gap, the theoretical result demonstrates that both Prox-DBRO-SAGA
and Prox-DBRO-LSVRG, with a well-designed constant (resp., decaying) step-size,
converge linearly (resp., sub-linearly) inside an error ball around the optimal
solution to the optimization problem under standard assumptions. The trade-offs
between the convergence accuracy and the number of Byzantine agents in both
linear and sub-linear cases are characterized. In simulation, the effectiveness
and practicability of the proposed algorithms are manifested via resolving a
sparse machine-learning problem over multi-agent CPSs under various Byzantine
attacks.Comment: 14 pages, 0 figure
Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions
We consider a spectral optimal design problem involving the Neumann traces of
the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset
of . The cost functional measures the amount of energy that Dirichlet
eigenfunctions concentrate on the boundary and that can be recovered with a
bounded density function. We first prove that, assuming a constraint on
densities, the so-called {\it Rellich functions} maximize this
functional.Motivated by several issues in shape optimization or observation
theory where it is relevant to deal with bounded densities, and noticing that
the -norm of {\it Rellich functions} may be large, depending on the
shape of , we analyze the effect of adding pointwise constraints when
maximizing the same functional. We investigate the optimality of {\it
bang-bang} functions and {\it Rellich densities} for this problem. We also deal
with similar issues for a close problem, where the cost functional is replaced
by a spectral approximation.Finally, this study is completed by the
investigation of particular geometries and is illustrated by several numerical
simulations
Density-feedback control in traffic and transport far from equilibrium
A bottleneck situation in one-lane traffic-flow is typically modelled with a
constant demand of entering cars. However, in practice this demand may depend
on the density of cars in the bottleneck. The present paper studies a simple
bimodal realization of this mechanism to which we refer to as density-feedback
control (DFC): If the actual density in the bottleneck is above a certain
threshold, the reservoir density of possibly entering cars is reduced to a
different constant value. By numerical solution of the discretized viscid
Burgers equation a rich stationary phase diagram is found. In order to maximize
the flow, which is the goal of typical traffic-management strategies, we find
the optimal choice of the threshold. Analytical results are verified by
computer simulations of the microscopic TASEP with DFC.Comment: 7 pages, 5 figure
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