5,538 research outputs found
Well-posedness and exponential equilibration of a volume-surface reaction-diffusion system with nonlinear boundary coupling
We consider a model system consisting of two reaction-diffusion equations,
where one species diffuses in a volume while the other species diffuses on the
surface which surrounds the volume. The two equations are coupled via a
nonlinear reversible Robin-type boundary condition for the volume species and a
matching reversible source term for the boundary species. As a consequence of
the coupling, the total mass of the two species is conserved. The considered
system is motivated for instance by models for asymmetric stem cell division.
Firstly we prove the existence of a unique weak solution via an iterative
method of converging upper and lower solutions to overcome the difficulties of
the nonlinear boundary terms. Secondly, our main result shows explicit
exponential convergence to equilibrium via an entropy method after deriving a
suitable entropy entropy-dissipation estimate for the considered nonlinear
volume-surface reaction-diffusion system.Comment: 31 page
A survey of uncertainty principles and some signal processing applications
The goal of this paper is to review the main trends in the domain of
uncertainty principles and localization, emphasize their mutual connections and
investigate practical consequences. The discussion is strongly oriented
towards, and motivated by signal processing problems, from which significant
advances have been made recently. Relations with sparse approximation and
coding problems are emphasized
Efficient Approximation of Quantum Channel Capacities
We propose an iterative method for approximating the capacity of
classical-quantum channels with a discrete input alphabet and a finite
dimensional output, possibly under additional constraints on the input
distribution. Based on duality of convex programming, we derive explicit upper
and lower bounds for the capacity. To provide an -close estimate
to the capacity, the presented algorithm requires , where denotes the input alphabet size and
the output dimension. We then generalize the method for the task of
approximating the capacity of classical-quantum channels with a bounded
continuous input alphabet and a finite dimensional output. For channels with a
finite dimensional quantum mechanical input and output, the idea of a universal
encoder allows us to approximate the Holevo capacity using the same method. In
particular, we show that the problem of approximating the Holevo capacity can
be reduced to a multidimensional integration problem. For families of quantum
channels fulfilling a certain assumption we show that the complexity to derive
an -close solution to the Holevo capacity is subexponential or
even polynomial in the problem size. We provide several examples to illustrate
the performance of the approximation scheme in practice.Comment: 36 pages, 1 figur
Sample average approximation with heavier tails II: localization in stochastic convex optimization and persistence results for the Lasso
We present exponential finite-sample nonasymptotic deviation inequalities for
the SAA estimator's near-optimal solution set over the class of stochastic
optimization problems with heavy-tailed random \emph{convex} functions in the
objective and constraints. Such setting is better suited for problems where a
sub-Gaussian data generating distribution is less expected, e.g., in stochastic
portfolio optimization. One of our contributions is to exploit \emph{convexity}
of the perturbed objective and the perturbed constraints as a property which
entails \emph{localized} deviation inequalities for joint feasibility and
optimality guarantees. This means that our bounds are significantly tighter in
terms of diameter and metric entropy since they depend only on the near-optimal
solution set but not on the whole feasible set. As a result, we obtain a much
sharper sample complexity estimate when compared to a general nonconvex
problem. In our analysis, we derive some localized deterministic perturbation
error bounds for convex optimization problems which are of independent
interest. To obtain our results, we only assume a metric regular convex
feasible set, possibly not satisfying the Slater condition and not having a
metric regular solution set. In this general setting, joint near feasibility
and near optimality are guaranteed. If in addition the set satisfies the Slater
condition, we obtain finite-sample simultaneous \emph{exact} feasibility and
near optimality guarantees (for a sufficiently small tolerance). Another
contribution of our work is to present, as a proof of concept of our localized
techniques, a persistent result for a variant of the LASSO estimator under very
weak assumptions on the data generating distribution.Comment: 34 pages. Some correction
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