 Award ID(s):
 1900251
 NSFPAR ID:
 10349989
 Date Published:
 Journal Name:
 Forum of Mathematics, Pi
 Volume:
 9
 ISSN:
 20505086
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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We study stochastic approximation procedures for approximately solving a $d$dimensional linear fixed point equation based on observing a trajectory of length $n$ from an ergodic Markov chain. We first exhibit a nonasymptotic bound of the order $t_{\mathrm{mix}} \tfrac{d}{n}$ on the squared error of the last iterate of a standard scheme, where $t_{\mathrm{mix}}$ is a mixing time. We then prove a nonasymptotic instancedependent bound on a suitably averaged sequence of iterates, with a leading term that matches the local asymptotic minimax limit, including sharp dependence on the parameters $(d, t_{\mathrm{mix}})$ in the higher order terms. We complement these upper bounds with a nonasymptotic minimax lower bound that establishes the instanceoptimality of the averaged SA estimator. We derive corollaries of these results for policy evaluation with Markov noise—covering the TD($\lambda$) family of algorithms for all $\lambda \in [0, 1)$—and linear autoregressive models. Our instancedependent characterizations open the door to the design of finegrained model selection procedures for hyperparameter tuning (e.g., choosing the value of $\lambda$ when running the TD($\lambda$) algorithm).more » « less

Classical statistical mechanics has long relied on assumptions such as the equipartition theorem to understand the behavior of the complicated systems of many particles. The successes of this approach are well known, but there are also many wellknown issues with classical theories. For some of these, the introduction of quantum mechanics is necessary, e.g., the ultraviolet catastrophe. However, more recently, the validity of assumptions such as the equipartition of energy in classical systems was called into question. For instance, a detailed analysis of a simplified model for blackbody radiation was apparently able to deduce the Stefan–Boltzmann law using purely classical statistical mechanics. This novel approach involved a careful analysis of a “metastable” state which greatly delays the approach to equilibrium. In this paper, we perform a broad analysis of such a metastable state in the classical Fermi–Pasta–Ulam–Tsingou (FPUT) models. We treat both the αFPUT and βFPUT models, exploring both quantitative and qualitative behavior. After introducing the models, we validate our methodology by reproducing the wellknown FPUT recurrences in both models and confirming earlier results on how the strength of the recurrences depends on a single system parameter. We establish that the metastable state in the FPUT models can be defined by using a single degreeoffreedom measure—the spectral entropy (η)—and show that this measure has the power to quantify the distance from equipartition. For the αFPUT model, a comparison to the integrable Toda lattice allows us to define rather clearly the lifetime of the metastable state for the standard initial conditions. We next devise a method to measure the lifetime of the metastable state tm in the αFPUT model that reduces the sensitivity to the exact initial conditions. Our procedure involves averaging over random initial phases in the plane of initial conditions, the P1Q1 plane. Applying this procedure gives us a powerlaw scaling for tm, with the important result that the power laws for different system sizes collapse down to the same exponent as Eα2→0. We examine the energy spectrum E(k) over time in the αFPUT model and again compare the results to those of the Toda model. This analysis tentatively supports a method for an irreversible energy dissipation process suggested by Onorato et al.: fourwave and sixwave resonances as described by the “wave turbulence” theory. We next apply a similar approach to the βFPUT model. Here, we explore in particular the different behavior for the two different signs of β. Finally, we describe a procedure for calculating tm in the βFPUT model, a very different task than for the αFPUT model, because the βFPUT model is not a truncation of an integrable nonlinear model.more » « less

null (Ed.)ABSTRACT Understanding the evolution of selfgravitating, isothermal, magnetized gas is crucial for star formation, as these physical processes have been postulated to set the initial mass function (IMF). We present a suite of isothermal magnetohydrodynamic (MHD) simulations using the gizmo code that follow the formation of individual stars in giant molecular clouds (GMCs), spanning a range of Mach numbers found in observed GMCs ($\mathcal {M} \sim 10\!\!50$). As in past works, the mean and median stellar masses are sensitive to numerical resolution, because they are sensitive to lowmass stars that contribute a vanishing fraction of the overall stellar mass. The massweighted median stellar mass M50 becomes insensitive to resolution once turbulent fragmentation is well resolved. Without imposing Larsonlike scaling laws, our simulations find $M_\mathrm{50} \,\, \buildrel\propto \over \sim \,\,M_\mathrm{0} \mathcal {M}^{3} \alpha _\mathrm{turb}\, \mathrm{SFE}^{1/3}$ for GMC mass M0, sonic Mach number $\mathcal {M}$, virial parameter αturb, and star formation efficiency SFE = M⋆/M0. This fit agrees well with previous IMF results from the ramses, orion2, and sphng codes. Although M50 has no significant dependence on the magnetic field strength at the cloud scale, MHD is necessary to prevent a fragmentation cascade that results in nonconvergent stellar masses. For initial conditions and SFE similar to starforming GMCs in our Galaxy, we predict M50 to be $\gt 20 \, \mathrm{M}_{\odot }$, an order of magnitude larger than observed ($\sim 2 \, \mathrm{M}_\odot$), together with an excess of brown dwarfs. Moreover, M50 is sensitive to initial cloud properties and evolves strongly in time within a given cloud, predicting much larger IMF variations than are observationally allowed. We conclude that physics beyond MHD turbulence and gravity are necessary ingredients for the IMF.more » « less

Consider the linear transport equation in 1D under an external confining potential
:\begin{document}$ \Phi $\end{document} For
(with\begin{document}$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $\end{document} small), we prove phase mixing and quantitative decay estimates for\begin{document}$ \varepsilon >0 $\end{document} , with an inverse polynomial decay rate\begin{document}$ {\partial}_t \varphi : =  \Delta^{1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $\end{document} . In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in\begin{document}$ O({\langle} t{\rangle}^{2}) $\end{document} D under the external potential\begin{document}$ 1 $\end{document} .\begin{document}$ \Phi $\end{document} 
null (Ed.)Recent theoretical and computational progress has led to unprecedented understanding of symmetrybreaking instabilities in 2D dynamic fracture. At the heart of this progress resides the identification of two intrinsic, near crack tip length scales — a nonlinear elastic length scale ℓ and a dissipation length scale ξ — that do not exist in Linear Elastic Fracture Mechanics (LEFM), the classical theory of cracks. In particular, it has been shown that at a propagation velocity v of about 90% of the shear wavespeed, cracks in 2D brittle materials undergo an oscillatory instability whose wavelength varies linearly with ℓ, and at larger loading levels (corresponding to yet higher propagation velocities), a tipsplitting instability emerges, both in agreements with experiments. In this paper, using phasefield models of brittle fracture, we demonstrate the following properties of the oscillatory instability: (i) It exists also in the absence of neartip elastic nonlinearity, i.e. in the limit ℓ→0, with a wavelength determined by the dissipation length scale ξ. This result shows that the instability crucially depends on the existence of an intrinsic length scale associated with the breakdown of linear elasticity near crack tips, independently of whether the latter is related to nonlinear elasticity or to dissipation. (ii) It is a supercritical Hopf bifurcation, featuring a vanishing oscillations amplitude at onset. (iii) It is largely independent of the phenomenological forms of the degradation functions assumed in the phasefield framework to describe the cohesive zone, and of the velocitydependence of the fracture energy Γ(v) that is controlled by the dissipation time scale in the GinzburgLandautype evolution equation for the phasefield. These results substantiate the universal nature of the oscillatory instability in 2D. In addition, we provide evidence indicating that the tipsplitting instability is controlled by the limiting rate of elastic energy transport inside the crack tip region. The latter is sensitive to the wavespeed inside the dissipation zone, which can be systematically varied within the phasefield approach. Finally, we describe in detail the numerical implementation scheme of the employed phasefield fracture approach, allowing its application in a broad range of materials failure problems.more » « less