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Register a new userMinimum energy with infinite horizon: from stationary to non-stationary states
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We study a non standard infinite horizon, infinite dimensional linear-quadratic control problem arising in the physics of non-stationary states (see e.g. cite{BDGJL4,BertiniGabrielliLebowitz05}): finding the minimum energy to drive a given stationary state $bar x=0$ (at time $t=-infty$) into an arbitrary non-stationary state $x$ (at time $t=0$). This is the opposite to what is commonly studied in the literature on null controllability (where one drives a generic state $x$ into the equilibrium state $bar x=0$). Consequently, the Algebraic Riccati Equation (ARE) associated to this problem is non-standard since the sign of the linear part is opposite to the usual one and since it is intrinsically unbounded. Hence the standard theory of AREs does not apply. The analogous finite horizon problem has been studied in the companion paper cite{AcquistapaceGozzi17}. Here, similarly to such paper, we prove that the linear selfadjoint operator associated to the value function is a solution of the above mentioned ARE. Moreover, differently to cite{AcquistapaceGozzi17}, we prove that such solution is the maximal one. The first main result (Theorem ref{th:maximalARE}) is proved by approximating the problem with suitable auxiliary finite horizon problems (which are different from the one studied in cite{AcquistapaceGozzi17}). Finally in the special case where the involved operators commute we characterize all solutions of the ARE (Theorem ref{th:sol=proj}) and we apply this to the Landau-Ginzburg model.
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In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between $L_{rho}^2({mathbb{R}^{d}};{mathbb{R}^{1}}) otimes L_{rho}^2({mathbb{R}^{d}};{mathbb{R}^{d}})$ valued solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. Moreover, we prove the existence and uniqueness of the solutions of BDSDEs on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary solutions (independent of any initial value) of SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.
We investigate stochastic optimization problems under relaxed assumptions on the distribution of noise that are motivated by empirical observations in neural network training. Standard results on optimal convergence rates for stochastic optimization assume either there exists a uniform bound on the moments of the gradient noise, or that the noise decays as the algorithm progresses. These assumptions do not match the empirical behavior of optimization algorithms used in neural network training where the noise level in stochastic gradients could even increase with time. We address this behavior by studying convergence rates of stochastic gradient methods subject to changing second moment (or variance) of the stochastic oracle as the iterations progress. When the variation in the noise is known, we show that it is always beneficial to adapt the step-size and exploit the noise variability. When the noise statistics are unknown, we obtain similar improvements by developing an online estimator of the noise level, thereby recovering close variants of RMSProp. Consequently, our results reveal an important scenario where adaptive stepsize methods outperform SGD.
Potential buyers of a product or service tend to read reviews from previous consumers before making their decisions. This behavior is modeled by a market of Bayesian consumers with heterogeneous preferences, who sequentially decide whether to buy an item of unknown quality, based on previous buyers reviews. The quality is multi-dimensional and the reviews can assume one of different forms and can also be multi-dimensional. The belief about the items quality in simple uni-dimensional settings is known to converge to its true value. Our paper extends this result to the more general case of a multidimensional quality, possibly in a continuous space, and provides anytime convergence rates. In practice, the quality of an item may vary over time, due to some change in the production process or the need to keep up with the competition. This paper also studies the learning dynamic when the unknown quality changes at random times and shows that the cost of learning is rather small
We study the structure of stationary non equilibrium states for interacting particle systems from a microscopic viewpoint. In particular we discuss two different discrete geometric constructions. We apply both of them to determine non reversible transition rates corresponding to a fixed invariant measure. The first one uses the equivalence of this problem with the construction of divergence free flows on the transition graph. Since divergence free flows are characterized by cyclic decompositions we can generate families of models from elementary cycles on the configuration space. The second construction is a functional discrete Hodge decomposition for translational covariant discrete vector fields. According to this, for example, the instantaneous current of any interacting particle system on a finite torus can be canonically decomposed in a gradient part, a circulation term and an harmonic component. All the three components are associated to functions on the configuration space. This decomposition is unique and constructive. The stationary condition can be interpreted as an orthogonality condition with respect to an harmonic discrete vector field and we use this decomposition to construct models having a fixed invariant measure.
Systems kept out of equilibrium in stationary states by an external source of energy store an energy $Delta U=U-U_0$. $U_0$ is the internal energy at equilibrium state, obtained after the shutdown of energy input. We determine $Delta U$ for two model systems: ideal gas and Lennard-Jones fluid. $Delta U$ depends not only on the total energy flux, $J_U$, but also on the mode of energy transfer into the system. We use three different modes of energy transfer where: the energy flux per unit volume is (i) constant; (ii) proportional to the local temperature (iii) proportional to the local density. We show that $Delta U /J_U=tau$ is minimized in the stationary states formed in these systems, irrespective of the mode of energy transfer. $tau$ is the characteristic time scale of energy outflow from the system immediately after the shutdown of energy flux. We prove that $tau$ is minimized in stable states of the Rayleigh-Benard cell.
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