Searcharxiv⌕ Search

arXiv subjects

Jun Geng

Publications and source records attributed to Jun Geng.

At least 19 recordsLinked to original sources
arXiv2026

Sharp error estimates in stochastic homogenization of parabolic systems with time-dependent coefficients

This article mainly proves the existence of stationary correctors under space-time spectral gap conditions, which exhibit different properties from those of elliptic operator correctors. Additionally, new flux correctors and their fluctuation estimates are introduced.Based on this, we obtain the optimal homogenization error in the sense of strong and weak norms on C1 cylinders by using the duality and distance-weighted arguments, in which the (weighted) annealed Calderon-Zygmund estimates coupled with a novel form of the minimal radius are developed. Throughout the paper, no small-scale smoothness of the coefficients is used.

math.AP↗
arXiv2025

Higher Order Elliptic Equations on Nonsmooth Domains

In 1995, D. Jerison and C. Kenig in \cite{JK-1995} considered the the inhomogeneous Dirichlet problem $\Delta u= f$ on $\Omega$, $u=0$ on $\partial\Omega$ in Lipschitz domains. One of their main results shows that the $W^{1,p}$ estimate holds for the sharp range $\frac{3}{2}-\varepsilon<p<3+\varepsilon$ for $d\geq 3$ and $\frac{4}{3}-\varepsilon<p<4+\varepsilon$ if $d=2$. Although the argument employed in \cite{JK-1995} yields optimal results, they rely on an essential fashion on the maximum principle and, as such, do not readily adapt to higher-order case. By using a new method, the aim of this paper is to establish an extension of their theorem for higher order inhomogeneous elliptic equations on bounded Lipschitz and convex domains, uniform $W^{\ell,p}$ estimates are obtained for $p$ in certain ranges. Especially, compare to the result in \cite{MM-2013} for biharmonic equation, a larger, sharp, range of $p's$ was obtained in this paper.

math.AP↗
arXiv2024

On the Schr\"odinger equations with $B_\infty$ potentials in the region above a Lipschitz graph

In this paper we investigate the $L^p$ regularity, $L^p$ Neumann and $W^{1,p}$ problems for generalized Schr\"odinger operator $-\text{div}(A\nabla )+ V $ in the region above a Lipschitz graph under the assumption that $A$ is elliptic, symmetric and $x_d-$independent. Specifically, we prove that the $L^p$ regularity problem is uniquely solvable for $$1<p<2+\varepsilon.$$ Moreover, we also establish the $W^{1,p}$ estimate for Neumann problem for $$\frac{3}{2}-\varepsilon<p<3+\varepsilon.$$ As a by-product, we also obtain that the $L^p$ Neumann problem is uniquely solvable for $1<p<2+\varepsilon.$ The only previously known estimates of this type pertain to the classical Schr\"odinger equation $-\Delta u+ Vu=0$ in $\Omega$ and $\frac{\partial u}{\partial n}=g$ on $\partial\Omega$ which was obtained by Shen [Z. Shen, On the Neumann problem for Schr\"odinger operators in Lipschitz domains, Indiana Univ. Math. J. 43 (1994)] for ranges $1<p\leq 2$. All the ranges of $p$ are sharp.

math.AP↗
arXiv2024

Neumann Problems for the Stokes Equations in Convex Domains

This paper studies the Neumann boundary value problems for the Stokes equations in a convex domain in $\mathbb{R}^d$. We obtain nontangential-maximal-function estimates in $L^p$ and $W^{1, p}$ estimates for $p$ in certain ranges depending on $d$. These ranges are larger than the known ranges for Lipschitz domains. The proof relies on a $W^{2, 2}$ estimate for the Stokes equations in convex domains.

math.AP↗
arXiv2024

Resolvent Estimates in $L^\infty$ for the Stokes Operator in Nonsmooth Domains

We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain $\Omega$ in $R^d$ under the assumptions that $\Omega$ is $C^1$ for $d\ge 3$ and Lipschitz for $d=2$. As a corollary, it follows that the Stokes operator generates a uniformly bounded analytic semigroup in the spaces of bounded solenoidal functions in $\Omega$. The smoothness conditions on $\Omega$ are sharp. The case of exterior domains with nonsmooth boundaries is also studied.The key step in the proof involves new estimates which connect the pressure to the velocity in the $L^q$ average, but only on scales above certain level.

math.AP↗
arXiv2024

Quantitative estimates in almost periodic homogenization of parabolic systems

We consider a family of second-order parabolic operators $\partial_t+\mathcal{L}_\varepsilon$ in divergence form with rapidly oscillating, time-dependent and almost-periodic coefficients. We establish uniform interior and boundary H\"older and Lipschitz estimates as well as convergence rate. The estimates of fundamental solution and Green's function are also established. In contrast to periodic case, the main difficulty is that the corrector equation $(\partial_s+\mathcal{L}_1)(\chi^\beta_{j})=-\mathcal{L}_1(P^\beta_j) $ in $\mathbb{R}^{d+1}$ may not be solvable in the almost periodic setting for linear functions $P(y)$ and $\partial_t \chi_S$ may not in $B^2(\mathbb{R}^{d+1})$. Our results are new even in the case of time-independent coefficients.

math.AP↗
arXiv2024

Resolvent Estimates for the Stokes Operator in Bounded and Exterior $C^1$ Domains

We establish resolvent estimates in $L^q$ spaces for the Stokes operator in a bounded $C^1$ domain $\Omega$ in $\mathbb{R}^d$. As a corollary, it follows that the Stokes operator generates a bounded analytic semigroup in $L^q(\Omega; \mathbb{C}^d)$ for any $1< q< \infty$ and $d\ge 2$. The case of an exterior $C^1$ domain is also studied.

math.AP↗
arXiv2021

Homogenization of locally periodic parabolic operators with non-self-similar scales

We investigate quantitative estimates in homogenization of the locally periodic parabolic operator with multiscales $$ \partial_t- \text{div} (A(x,t,x/\varepsilon,t/\kappa^2) \nabla ),\qquad \varepsilon>0,\, \kappa>0. $$ Under proper assumptions, we establish the full-scale interior and boundary Lipschitz estimates. These results are new even for the case $\kappa=\varepsilon$, and for the periodic operators $ \partial_t-\text{div}(A(x/\varepsilon, t/\varepsilon^{\ell}) \nabla ),$ $0<\varepsilon,\ell<\infty, $ of which the large-scale Lipschitz estimate down to $\varepsilon+\varepsilon^{\ell/2}$ was recently established by the first author and Shen in Arch. Ration. Mech. Anal. 236(1): 145--188 (2020). Due to the non-self-similar structure, the full-scale estimates do not follow directly from the large-scale estimates and the blow-up argument. As a byproduct, we also derive the convergence rates for the corresponding initial-Dirichlet problems, which extend the results in the aforementioned literature to more general settings.

math.AP↗
arXiv2019

Performance of regression models as a function of experiment noise

A challenge in developing machine learning regression models is that it is difficult to know whether maximal performance has been reached on a particular dataset, or whether further model improvement is possible. In biology this problem is particularly pronounced as sample labels (response variables) are typically obtained through experiments and therefore have experiment noise associated with them. Such label noise puts a fundamental limit to the performance attainable by regression models. We address this challenge by deriving a theoretical upper bound for the coefficient of determination (R2) for regression models. This theoretical upper bound depends only on the noise associated with the response variable in a dataset as well as its variance. The upper bound estimate was validated via Monte Carlo simulations and then used as a tool to bootstrap performance of regression models trained on biological datasets, including protein sequence data, transcriptomic data, and genomic data. Although we study biological datasets in this work, the new upper bound estimates will hold true for regression models from any research field or application area where response variables have associated noise.

q-bio.BM↗
arXiv2019

Homogenization of Parabolic Equations with Non-self-similar Scales

This paper is concerned with quantitative homogenization of second-order parabolic systems with periodic coefficients varying rapidly in space and time, in different scales. We obtain large-scale interior and boundary Lipschitz estimates as well as interior $C^{1, \alpha}$ and $C^{2, \alpha}$ estimates by utilizing higher-order correctors. We also investigate the problem of convergence rates for initial-boundary value problems.

math.AP↗
arXiv2019

Oscillatory integrals and periodic homogenization of Robin boundary value problems

In this paper, we consider a family of second-order elliptic systems subject to a periodically oscillating Robin boundary condition. We establish the qualitative homogenization theorem on any Lipschitz domains satisfying a non-resonance condition. We also use the quantitative estimates of oscillatory integrals to obtain the dimension-dependent convergence rates in $L^2$, assuming that the domain is smooth and strictly convex.

math.AP↗
arXiv2018

Quick Best Action Identification in Linear Bandit Problems

In this paper, we consider a best action identification problem in the stochastic linear bandit setup with a fixed confident constraint. In the considered best action identification problem, instead of minimizing the accumulative regret as done in existing works, the learner aims to obtain an accurate estimate of the underlying parameter based on his action and reward sequences. To improve the estimation efficiency, the learner is allowed to select his action based his historical information; hence the whole procedure is designed in a sequential adaptive manner. We first show that the existing algorithms designed to minimize the accumulative regret is not a consistent estimator and hence is not a good policy for our problem. We then characterize a lower bound on the estimation error for any policy. We further design a simple policy and show that the estimation error of the designed policy achieves the same scaling order as that of the derived lower bound.

cs.LG↗
arXiv2017

Multi-Chart Detection Procedure for Bayesian Quickest Change-Point Detection with Unknown Post-Change Parameters

In this paper, the problem of quickly detecting an abrupt change on a stochastic process under Bayesian framework is considered. Different from the classic Bayesian quickest change-point detection problem, this paper considers the case where there is uncertainty about the post-change distribution. Specifically, the observer only knows that the post-change distribution belongs to a parametric distribution family but he does not know the true value of the post-change parameter. In this scenario, we propose two multi-chart detection procedures, termed as M-SR procedure and modified M-SR procedure respectively, and show that these two procedures are asymptotically optimal when the post-change parameter belongs to a finite set and are asymptotically $\epsilon-$optimal when the post-change parameter belongs to a compact set with finite measure. Both algorithms can be calculated efficiently as their detection statistics can be updated recursively. We then extend the study to consider the multi-source monitoring problem with unknown post-change parameters. When those monitored sources are mutually independent, we propose a window-based modified M-SR detection procedure and show that the proposed detection method is first-order asymptotically optimal when post-change parameters belong to finite sets. We show that both computation and space complexities of the proposed algorithm increase only linearly with respect to the number of sources.

cs.IT↗
arXiv2016

Boundary Korn Inequality and Neumann Problems in Homogenization of Systems of Elasticity

This paper concerns with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization. We establish uniform optimal regularity estimates for solutions of Neumann problems in a bounded Lipschitz domain with $L^2$ boundary data. The proof relies on a boundary Korn inequality for solutions of systems of linear elasticity and uses a large-scale Rellich estimate obtained in \cite{Shen-2016}.

math.AP↗
arXiv2014

Bayesian Quickest Change Point Detection with Sampling Right Constraints

In this paper, Bayesian quickest change detection problems with sampling right constraints are considered. Specifically, there is a sequence of random variables whose probability density function will change at an unknown time. The goal is to detect this change in a way such that a linear combination of the average detection delay and the false alarm probability is minimized. Two types of sampling right constrains are discussed. The first one is a limited sampling right constraint, in which the observer can take at most $N$ observations from this random sequence. Under this setup, we show that the cost function can be written as a set of iterative functions, which can be solved by Markov optimal stopping theory. The optimal stopping rule is shown to be a threshold rule. An asymptotic upper bound of the average detection delay is developed as the false alarm probability goes to zero. This upper bound indicates that the performance of the limited sampling right problem is close to that of the classic Bayesian quickest detection for several scenarios of practical interest. The second constraint discussed in this paper is a stochastic sampling right constraint, in which sampling rights are consumed by taking observations and are replenished randomly. The observer cannot take observations if there are no sampling rights left. We characterize the optimal solution, which has a very complex structure. For practical applications, we propose a low complexity algorithm, in which the sampling rule is to take observations as long as the observer has sampling rights left and the detection scheme is a threshold rule. We show that this low complexity scheme is first order asymptotically optimal as the false alarm probability goes to zero.

cs.IT↗
arXiv2013

Quickest Search over Multiple Sequences with Mixed Observation

The problem of sequentially finding an independent and identically distributed (i.i.d.) sequence that is drawn from a probability distribution $f_1$ by searching over multiple sequences, some of which are drawn from $f_1$ and the others of which are drawn from a different distribution $f_0$, is considered. The observer is allowed to take one observation at a time. It has been shown in a recent work that if each observation comes from one sequence, the cumulative sum test is optimal. In this paper, we propose a new approach in which each observation can be a linear combination of samples from multiple sequences. The test has two stages. In the first stage, namely scanning stage, one takes a linear combination of a pair of sequences with the hope of scanning through sequences that are unlikely to be generated from $f_1$ and quickly identifying a pair of sequences such that at least one of them is highly likely to be generated by $f_1$. In the second stage, namely refinement stage, one examines the pair identified from the first stage more closely and picks one sequence to be the final sequence. The problem under this setup belongs to a class of multiple stopping time problems. In particular, it is an ordered two concatenated Markov stopping time problem. We obtain the optimal solution using the tools from the multiple stopping time theory. The optimal solution has a rather complex structure. For implementation purpose, a low complexity algorithm is proposed, in which the observer adopts the cumulative sum test in the scanning stage and adopts the sequential probability ratio test in the refinement stage. The performance of this low complexity algorithm is analyzed when the prior probability of $f_{1}$ occurring is small. Both analytical and numerical simulation results show that this search strategy can significantly reduce the searching time when $f_{1}$ is rare.

cs.IT↗