894 research outputs found
Stochastic homogenization of nonconvex unbounded integral functionals with convex growth
We consider the well-travelled problem of homogenization of random integral
functionals. When the integrand has standard growth conditions, the qualitative
theory is well-understood. When it comes to unbounded functionals, that is,
when the domain of the integrand is not the whole space and may depend on the
space-variable, there is no satisfactory theory. In this contribution we
develop a complete qualitative stochastic homogenization theory for nonconvex
unbounded functionals with convex growth. We first prove that if the integrand
is convex and has -growth from below (with , the dimension), then it
admits homogenization regardless of growth conditions from above. This result,
that crucially relies on the existence and sublinearity at infinity of
correctors, is also new in the periodic case. In the case of nonconvex
integrands, we prove that a similar homogenization result holds provided the
nonconvex integrand admits a two-sided estimate by a convex integrand (the
domain of which may depend on the space-variable) that itself admits
homogenization. This result is of interest to the rigorous derivation of rubber
elasticity from polymer physics, which involves the stochastic homogenization
of such unbounded functionals.Comment: 64 pages, 2 figure
Fixed-Time Stable Proximal Dynamical System for Solving MVIPs
In this paper, a novel modified proximal dynamical system is proposed to
compute the solution of a mixed variational inequality problem (MVIP) within a
fixed time, where the time of convergence is finite, and is uniformly bounded
for all initial conditions. Under the assumptions of strong monotonicity and
Lipschitz continuity, it is shown that a solution of the modified proximal
dynamical system exists, is uniquely determined and converges to the unique
solution of the associated MVIP within a fixed time. As a special case for
solving variational inequality problems, the modified proximal dynamical system
reduces to a fixed-time stable projected dynamical system. Furthermore, the
fixed-time stability of the modified projected dynamical system continues to
hold, even if the assumption of strong monotonicity is relaxed to that of
strong pseudomonotonicity. Connections to convex optimization problems are
discussed, and commonly studied dynamical systems in the continuous-time
optimization literature follow as special limiting cases of the modified
proximal dynamical system proposed in this paper. Finally, it is shown that the
solution obtained using the forward-Euler discretization of the proposed
modified proximal dynamical system converges to an arbitrarily small
neighborhood of the solution of the associated MVIP within a fixed number of
time steps, independent of the initial conditions. Two numerical examples are
presented to substantiate the theoretical convergence guarantees.Comment: 12 pages, 5 figure
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