arXiv subjects

Jiajie Chen

Publications and source records attributed to Jiajie Chen.

41 records · Page 3Linked to original sources

arXiv2020

On the Slightly Perturbed De Gregorio Model on $S^1$

It is conjectured that the generalization of the Constantin-Lax-Majda model (gCLM) $ω_t + a uω_x = u_x ω$ due to Okamoto, Sakajo and Wunsch can develop a finite time singularity from smooth initial data for $a < 1$. For the endpoint case where $a$ is close to and less than $1$, we prove finite time asymptotically self-similar blowup of gCLM on a circle from a class of smooth initial data. For the gCLM on a circle with the same initial data, if the strength of advection $a$ is slightly larger than $1$, we prove that the solution exists globally with $|| ω(t)||_{H^1}$ decaying in a rate of $O(t^{-1})$ for large time. The transition threshold between two different behaviors is $a=1$, which corresponds to the De Gregorio model.

math.AP↗

arXiv2019

Singularity Formation and Global Well-Posedness for the Generalized Constantin-Lax-Majda Equation with Dissipation

We study a generalization due to De Gregorio and Wunsch et.al. of the Constantin-Lax-Majda equation (gCLM) on the real line \[ ω_t + a u ω_x = u_x ω- νΛ^γ ω, \quad u_x = H ω, \] where $H$ is the Hilbert transform and $Λ= (-\partial_{xx})^{1/2}$. We use the method in \cite{chen2019finite} to prove finite time self-similar blowup for $a$ close to $\frac{1}{2}$ and $γ=2$ by establishing nonlinear stability of an approximate self-similar profile. For $a>-1$, we discuss several classes of initial data and establish global well-posedness and an one-point blowup criterion for different initial data. For $a\leq-1$, we prove global well-posedness for gCLM with critical and supercritical dissipation.

math.AP↗

arXiv2019

A Pseudo Knockoff Filter for Correlated Features

In 2015, Barber and Candes introduced a new variable selection procedure called the knockoff filter to control the false discovery rate (FDR) and prove that this method achieves exact FDR control. Inspired by the work of Barber and Candes (2015), we propose and analyze a pseudo-knockoff filter that inherits some advantages of the original knockoff filter and has more flexibility in constructing its knockoff matrix. Moreover, we perform a number of numerical experiments that seem to suggest that the pseudo knockoff filter with the half Lasso statistic has FDR control and offers more power than the original knockoff filter with the Lasso Path or the half Lasso Statistic for the numerical examples that we consider in this paper. Although we cannot establish rigorous FDR control for the pseudo knockoff filter, we provide some partial analysis of the pseudo knockoff filter with the half Lasso statistic and establish a uniform FDP bound and an expectation inequality.

stat.ME↗

arXiv2019

A Prototype Knockoff Filter for Group Selection with FDR Control

In many applications, we need to study a linear regression model that consists of a response variable and a large number of potential explanatory variables and determine which variables are truly associated with the response. In 2015, Barber and Candes introduced a new variable selection procedure called the knockoff filter to control the false discovery rate (FDR) and proved that this method achieves exact FDR control. In this paper, we propose a prototype knockoff filter for group selection by extending the Reid-Tibshirani prototype method. Our prototype knockoff filter improves the computational efficiency and statistical power of the Reid-Tibshirani prototype method when it is applied for group selection. In some cases when the group features are spanned by one or a few hidden factors, we demonstrate that the PCA prototype knockoff filter outperforms the Dai-Barber group knockoff filter. We present several numerical experiments to compare our prototype knockoff filter with the Reid-Tibshirani prototype method and the group knockoff filter. We have also conducted some analysis of the knockoff filter. Our analysis reveals that some knockoff path method statistics, including the Lasso path statistic, may lead to loss of power for certain design matrices and a specially designed response even if their signal strengths are still relatively strong.

stat.ME↗

arXiv2014

Validating Sample Average Approximation Solutions with Negatively Dependent Batches

Sample-average approximations (SAA) are a practical means of finding approximate solutions of stochastic programming problems involving an extremely large (or infinite) number of scenarios. SAA can also be used to find estimates of a lower bound on the optimal objective value of the true problem which, when coupled with an upper bound, provides confidence intervals for the true optimal objective value and valuable information about the quality of the approximate solutions. Specifically, the lower bound can be estimated by solving multiple SAA problems (each obtained using a particular sampling method) and averaging the obtained objective values. State-of-the-art methods for lower-bound estimation generate batches of scenarios for the SAA problems independently. In this paper, we describe sampling methods that produce negatively dependent batches, thus reducing the variance of the sample-averaged lower bound estimator and increasing its usefulness in defining a confidence interval for the optimal objective value. We provide conditions under which the new sampling methods can reduce the variance of the lower bound estimator, and present computational results to verify that our scheme can reduce the variance significantly, by comparison with the traditional Latin hypercube approach.

stat.OT↗