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Gautam Iyer

Publications and source records attributed to Gautam Iyer.

At least 19 recordsLinked to original sources
arXiv2026

Convergence of Langevin AIS for multimodal distributions

We study convergence rates of the annealed importance sampling algorithm (Neal '01) combined with Langevin Monte Carlo when the target is a multimodal Gibbs measure. The main result shows that for a fixed error threshold, the time complexity is quadratic in the inverse temperature. We identify a simple and useful quantity that controls the sampling error for AIS in a general setting, and then bound this quantity in our setting using spectral estimates. We also study an autonormalized version and obtain bounds for the time complexity in terms of the inverse temperature.

math.PR↗
arXiv2026

Time-complexity of sampling from a multimodal distribution using sequential Monte Carlo

We study a sequential Monte Carlo algorithm to sample from the Gibbs measure with a non-convex energy function at a low temperature. We use the practical and popular geometric annealing schedule, and use a Langevin diffusion at each temperature level. The Langevin diffusion only needs to run for a time that is long enough to ensure local mixing within energy valleys, which is much shorter than the time required for global mixing. Our main result shows convergence of Monte Carlo estimators with time complexity that, approximately, scales like the fourth power of the inverse temperature, and the square of the inverse allowed error. We also study this algorithm in an illustrative model scenario where more explicit estimates can be given.

math.ST↗
arXiv2025

Convergence of Empirical Measures for i.i.d. samples in $W^{-α, p}$

Given $N$ i.i.d. samples from a probability measure $μ$ on $\mathbf{R}^d$, we study the rate of convergence of the empirical measure $μ_N \to μ$ in the negative Sobolev space $W^{-α, p}$. When $W^{-α, p}$ contains point measures (i.e. when $αp > (p-1)d$), we show $\mathbf{E} \| μ_N - μ\|_{W^{-α, p}}^p \leq C_d / N^{p/2}$ for an explicit dimensional constant $C_d$, and obtain a Gaussian tail bound. When $0 < αp \leq d(p-1)$, we prove a similar result for Gaussian regularizations.

math.PR↗
arXiv2025

Residual Diffusivity for Expanding Bernoulli Maps

Consider a discrete time Markov process $X^ε$ on $\mathbf R^d$ that makes a deterministic jump based on its current location, and then takes a small Gaussian step of variance $ε^2$. We study the behavior of the asymptotic variance as $ε\to 0$. In some situations (for instance if there were no jumps), then the asymptotic variance vanishes as $ε\to 0$. When the jumps are "chaotic", however, the asymptotic variance may be bounded from above and bounded away from $0$, as $ε\to 0$. This phenomenon is known as residual diffusivity, and we prove this occurs when the jumps are determined by certain expanding Bernoulli maps.

math.PR↗
arXiv2025

Incompressible 2D Euler equations with non-decaying random initial vorticity

Consider a random initial vorticity $ω_0(x) = \sum_{n\in \mathbb{Z}^2} a_n ϕ(x-n)$, where $ϕ$ is bounded and compactly supported and $\{a_n\}$ are independent, uniformly bounded, mean $0$, variance $1$ random variables (i.e. $ω_0$ is an array of randomly weighted vortex blobs). We prove global well-posedness of weak solutions to the Euler equations in $\mathbf{R}^2$ for almost every such initial vorticity. The main contribution of our work is the construction of a corresponding initial velocity field that grows slowly at infinity, which enables us to apply a recent well-posedness result of Cobb and Koch.

math.AP↗
arXiv2025

Exponentially mixing flows with slow enhanced dissipation

Consider a passive scalar which is advected by an incompressible flow $u$ and has small molecular diffusivity $κ$. Previous results show that if $u$ is exponentially mixing and $C^1$, then the dissipation time is $O(|\log κ|^2)$. We produce a family of incompressible flows which are $C^0$ and exponentially mixing, uniformly in $κ$; however have a dissipation time of order $1/κ$ (i.e. exhibits no enhanced dissipation). We also estimate the dissipation time of mixing flows, and obtain improved bounds in terms of the mixing rate with explicit constants, and allow for a time inhomogeneous mixing rate which is typical for random constructions of mixing flows.

math.PR↗
arXiv2025

A Harris theorem for enhanced dissipation, and an example of Pierrehumbert

In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium -- a phenomenon known as enhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certain Harris conditions. If $κ$ denotes the strength of the diffusion, then we show that with probability at least $1 - o(κ^N)$ enhanced dissipation occurs on time scales of order $|\ln κ|$, a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surely independent of diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert '94.

math.DS↗
arXiv2024

Residual Diffusivity for Noisy Bernoulli Maps

Consider a discrete time Markov process $X^\varepsilon$ on $\mathbb R^d$ that makes a deterministic jump prescribed by a map $φ\colon \mathbb R^d \to \mathbb R^d$, and then takes a small Gaussian step of variance $\varepsilon^2$. For certain chaotic maps $φ$, the effective diffusivity of $X^\varepsilon$ may be bounded away from $0$ as $\varepsilon \to 0$. This is known as residual diffusivity, and in this paper we prove residual diffusivity occurs for a class of maps $φ$ obtained from piecewise affine expanding Bernoulli maps.

math.DS↗
arXiv2023

Speeding up Langevin Dynamics by Mixing

We study an overdamped Langevin equation on the $d$-dimensional torus with stationary distribution proportional to~$p = e^{-U / κ}$. When~$U$ has multiple wells the mixing time of the associated process is exponentially large (of size~$e^{O(1/κ)}$). We add a drift to the Langevin dynamics (without changing the stationary distribution) and obtain quantitative estimates on the mixing time. We show that an exponentially mixing drift can be rescaled to make the mixing time of the Langevin system arbitrarily small. For numerical purposes, it is useful to keep the size of the imposed drift small, and we show that the smallest allowable rescaling ensures that the mixing time is $O( d/κ^2)$, which is an order of magnitude smaller than~$e^{O(1/κ)}$. We provide one construction of an exponentially mixing drift, although with rate constants whose~$κ$-dependence is unknown.Heuristics (from discrete time) suggest that $κ$-dependence of the mixing rate is such that the imposed drift is of size~$O(d / κ^3)$. The large amplitude of the imposed drift increases the numerical complexity, and thus we expect this method will be most useful in the initial phase of Monte Carlo methods to rapidly explore the state space.

math.PR↗
arXiv2023

Quantifying the dissipation enhancement of cellular flows

We study the dissipation enhancement by cellular flows. Previous work by Iyer, Xu, and Zlatoš produces a family of cellular flows that can enhance dissipation by an arbitrarily large amount. We improve this result by providing quantitative bounds on the dissipation enhancement in terms of the flow amplitude, cell size and diffusivity. Explicitly we show that the mixing time is bounded by the exit time from one cell when the flow amplitude is large enough, and by the reciprocal of the effective diffusivity when the flow amplitude is small. This agrees with the optimal heuristics. We also prove a general result relating the dissipation time of incompressible flows to the mixing time. The main idea behind the proof is to study the dynamics probabilistically and construct a successful coupling.

math.AP↗
arXiv2023

An oscillator driven by algebraically decorrelating noise

We consider a stochastically forced nonlinear oscillator driven by a stationary Gaussian noise that has an algebraically decaying covariance function. It is well known that such noise processes can be renormalized to converge to fractional Brownian motion, a process that has memory. In contrast, we show that the renormalized limit of the nonlinear oscillator driven by this noise converges to diffusion driven by standard (not fractional) Brownian motion, and thus retains no memory in the scaling limit. The proof is based on the study of a fast-slow system using the perturbed test function method.

math.PR↗
arXiv2023

Using Bernoulli maps to accelerate mixing of a random walk on the torus

We study the mixing time of a random walk on the torus, alternated with a Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the mixing time of the random walk alone is $O(1/ε^2)$, where $ε$ is the step size. Our main results show that for a class of Bernoulli maps, when the random walk is alternated with the Bernoulli map $φ$ the mixing time becomes $O(|\ln ε|)$. We also study the \emph{dissipation time} of this process, and obtain $O(|\ln ε|)$ upper and lower bounds with explicit constants.

math.PR↗
arXiv2022

Global Dynamics and Photon Loss in the Kompaneets Equation

The Kompaneets equation governs dynamics of the photon energy spectrum in certain high temperature (or low density) plasmas. We prove several results concerning the long-time convergence of solutions to Bose--Einstein equilibria and the failure of photon conservation. In particular, we show the total photon number can decrease with time via an outflux of photons at the zero-energy boundary. The ensuing accumulation of photons at zero energy is analogous to Bose--Einstein condensation. We provide two conditions that guarantee that photon loss occurs, and show that once loss is initiated then it persists forever. We prove that as $t\to \infty$, solutions necessarily converge to equilibrium and we characterize the limit in terms of the total photon loss. Additionally, we provide a few results concerning the behavior of the solution near the zero-energy boundary, an Oleinik inequality, a comparison principle, and show that the solution operator is a contraction in $L^1$. None of these results impose a boundary condition at the zero-energy boundary.

math.AP↗
arXiv2022

Growth of Sobolev norms and loss of regularity in transport equations

We consider transport of a passive scalar advected by an irregular divergence free vector field. Given any non-constant initial data $\bar ρ\in H^1_\text{loc}({\mathbb R}^d)$, $d\geq 2$, we construct a divergence free advecting velocity field $v$ (depending on $\bar ρ$) for which the unique weak solution to the transport equation does not belong to $H^1_\text{loc}({\mathbb R}^d)$ for any positive positive time. The velocity field $v$ is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space $W^{s,p}$ that does not embed into the Lipschitz class. The velocity field $v$ is constructed by pulling back and rescaling an initial data dependent sequence of sine/cosine shear flows on the torus. This loss of regularity result complements that in [Ann. PDE, 5(1):Paper No. 9, 19, 2019].

math.AP↗
arXiv2021

Bounds on the heat transfer rate via passive advection

In heat exchangers, an incompressible fluid is heated initially and cooled at the boundary. The goal is to transfer the heat to the boundary as efficiently as possible. In this paper we study a related steady version of this problem where a steadily stirred fluid is uniformly heated in the interior and cooled on the boundary. For a given large Péclet number, how should one stir to minimize some norm of the temperature? This version of the problem was previously studied by Marcotte, Doering et\ al.\ (SIAM Appl.\ Math '18) in a disk, where the authors used matched asymptotics to show that when the Péclet number, $\pe$, is sufficiently large one can stir the fluid in a manner that ensures the total heat is $O(1/\pe)$. In this paper we confirm their results with rigorous proofs, and also provide an almost matching lower bound. For simplicity, we work on the infinite strip instead of the unit disk and the proof uses probabilistic techniques.

math.AP↗
arXiv2021

Scheduling fixed length quarantines to minimize the total number of fatalities during an epidemic

We consider a susceptible, infected, removed (SIR) system where the transmission rate may be temporarily reduced for a fixed amount of time. We show that in order to minimize the total number of fatalities, the transmission rate should be reduced on a single contiguous time interval, and we characterize this interval via an integral condition. We conclude with a few numerical simulations showing the actual reduction obtained.

math.DS↗
arXiv2020

Convection-Induced Singularity Suppression in the Keller-Segel and Other Non-linear PDEs

In this paper we study the effect of the addition of a convective term, and of the resulting increased dissipation rate, on the growth of solutions to a general class of non-linear parabolic PDEs. In particular, we show that blow-up in these models can always be prevented if the added drift has a small enough dissipation time. We also prove a general result relating the dissipation time and the effective diffusivity of stationary cellular flows, which allows us to obtain examples of simple incompressible flows with arbitrarily small dissipation times. As an application, we show that blow-up in the Keller-Segel model of chemotaxis can always be prevented if the velocity field of the ambient fluid has a sufficiently small dissipation time. We also study reaction-diffusion equations with ignition-type nonlinearities, and show that the reaction can always be quenched by the addition of a convective term with a small enough dissipation time, provided the average initial temperature is initially below the ignition threshold.

math.AP↗