arXiv subjects

Guodong Wang

Publications and source records attributed to Guodong Wang.

At least 55 records · Page 3Linked to original sources

arXiv2021

Student-Teacher Feature Pyramid Matching for Anomaly Detection

Anomaly detection is a challenging task and usually formulated as an one-class learning problem for the unexpectedness of anomalies. This paper proposes a simple yet powerful approach to this issue, which is implemented in the student-teacher framework for its advantages but substantially extends it in terms of both accuracy and efficiency. Given a strong model pre-trained on image classification as the teacher, we distill the knowledge into a single student network with the identical architecture to learn the distribution of anomaly-free images and this one-step transfer preserves the crucial clues as much as possible. Moreover, we integrate the multi-scale feature matching strategy into the framework, and this hierarchical feature matching enables the student network to receive a mixture of multi-level knowledge from the feature pyramid under better supervision, thus allowing to detect anomalies of various sizes. The difference between feature pyramids generated by the two networks serves as a scoring function indicating the probability of anomaly occurring. Due to such operations, our approach achieves accurate and fast pixel-level anomaly detection. Very competitive results are delivered on the MVTec anomaly detection dataset, superior to the state of the art ones.

cs.CV↗

arXiv2021

Nonlinear stability of planar steady Euler flows associated with semistable solutions of elliptic problems

This paper is devoted to the study of nonlinear stability of steady incompressible Euler flows in two dimensions. We prove that a steady Euler flow is nonlinearly stable in $L^p$ norm of the vorticity if its stream function is a semistable solution of some semilinear elliptic problem with strictly increasing nonlinearity. The idea of the proof is to show that such a flow has strict local maximum energy among flows whose vorticities are rearrangements of a given function, with the help of an improvement version of Wolansky and Ghil's stability theorem. The result can be regarded as an extension of Arnol'd's second stability theorem.

math.AP↗

arXiv2021

Global existence of weak solutions to the compressible Navier-Stokes equations with temperature-depending viscosity coefficients

This paper is devoted to the global existence of weak solutions to the three-dimensional compressible Navier-Stokes equations with heat-conducting effects in a bounded domain. The viscosity and the heat conductivity coefficients are assumed to be functions of the temperature, and the shear viscosity coefficient may vanish as the temperature goes to zero. The proof is to apply Galerkin method to a suitable approximate system with several parameters and obtain uniform estimates for the approximate solutions. The key ingredient in obtaining the required estimates is to apply De Giorgi's iteration to the modified temperature equation, from which we can get a lower bound for the temperature not depending on the artificial viscosity coefficient introduced in the modified momentum equation, which makes the compactness argument available as the artificial viscous term vanishes.

math.AP↗

arXiv2021

Location of concentrated vortices in planar steady Euler flows

In this paper, we study two-dimensional steady incompressible Euler flows in which the vorticity is sharply concentrated in a finite number of regions of small diameter in a bounded domain. Mathematical analysis of such flows is an interesting and physically important research topic in fluid mechanics. The main purpose of this paper is to prove that in such flows the locations of these concentrated blobs of vorticity must be in the vicinity of some critical point of the Kirchhoff-Routh function, which is determined by the geometry of the domain. The vorticity is assumed to be only in $L^{4/3},$ which is the optimal regularity for weak solutions to make sense. As a by-product, we prove a nonexistence result for concentrated multiple vortex flows in convex domains.

math.AP↗

arXiv2020

A Novel Video Salient Object Detection Method via Semi-supervised Motion Quality Perception

Previous video salient object detection (VSOD) approaches have mainly focused on designing fancy networks to achieve their performance improvements. However, with the slow-down in development of deep learning techniques recently, it may become more and more difficult to anticipate another breakthrough via fancy networks solely. To this end, this paper proposes a universal learning scheme to get a further 3\% performance improvement for all state-of-the-art (SOTA) methods. The major highlight of our method is that we resort the "motion quality"---a brand new concept, to select a sub-group of video frames from the original testing set to construct a new training set. The selected frames in this new training set should all contain high-quality motions, in which the salient objects will have large probability to be successfully detected by the "target SOTA method"---the one we want to improve. Consequently, we can achieve a significant performance improvement by using this new training set to start a new round of network training. During this new round training, the VSOD results of the target SOTA method will be applied as the pseudo training objectives. Our novel learning scheme is simple yet effective, and its semi-supervised methodology may have large potential to inspire the VSOD community in the future.

cs.CV↗

arXiv2019

Singularity for a multidimensional variational wave equation arising from nematic liquid crystals

This article is focused on a multidimensional nonlinear variational wave equation which is the Euler-Lagrange equation of a variational principle arising form the theory of nematic liquid crystals. By using the method of characteristics, we show that the smooth solutions for the spherically-symmetric variational wave equation breakdown in finite time, even for the arbitrarily small initial energy.

math.AP↗

arXiv2019

Steady vortex flows of perturbation type in a planar bounded domain

In this paper, we investigate steady Euler flows in a two-dimensional bounded domain. By an adaption of the vorticity method, we prove that for any nonconstant harmonic function $q$, which corresponds to a nontrivial irrotational flow, there exists a family of steady Euler flows with small circulation in which the vorticity is continuous and supported in a small neighborhood of the set of maximum points of $q$ near the boundary, and the corresponding stream function satisfies a semilinear elliptic equation with a given profile function. Moreover, if $q$ has $k$ isolated maximum points $\{\bar{x}_1,\cdot\cdot\cdot,\bar{x}_k\}$ on the boundary, we show that there exists a family of steady Euler flows whose vorticity is continuous and supported in $k$ disjoint regions of small diameter, and each of them is contained in a small neighborhood of $\bar{x}_i$, and in each of these small regions the stream function satisfies a semilinear elliptic equation with a given profile function.

math.AP↗

arXiv2019

Asymptotic behavior of global vortex rings

In this paper, we are concerned with nonlinear desingularization of steady vortex rings in $\mathbb{R}^3$ with given general nonlinearity $f$. Using the improved vorticity method, we construct a family of steady vortex rings which constitute a desingularization of the classical circular vortex filament in the whole space. The requirements on $f$ are very general, and $f$ may have a simple discontinuity at the origin. Some qualitative and asymptotic properties are also established.

math.AP↗

arXiv2019

Desingularization of vortices for 2D steady Euler flows via the vorticity method

In this paper, we consider steady Euler flows in a planar bounded domain in which the vorticity is sharply concentrated in a finite number of disjoint regions of small diameter. Such flows are closely related to the point vortex model and can be regarded as desingularization of point vortices. By an adaption of the vorticity method, we construct a family of steady Euler flows in which the vorticity is concentrated near a global minimum point of the Robin function of the domain, and the corresponding stream function satisfies a semilinear elliptic equation with a given profile function. Furthermore, for any given isolated minimum point $(\bar{x}_1,\cdot\cdot\cdot,\bar{x}_k)$ of the Kirchhoff-Routh function of the domain, we prove that there exists a family of steady Euler flows whose vorticity is supported in $k$ small regions near $\bar{x}_i$, and near each $\bar{x}_i$ the corresponding stream function satisfies a semilinear elliptic equation with a given profile function.

math.AP↗

arXiv2019

On radial symmetry of rotating vortex patches in the disc

In this note, we consider the radial symmetry property of rotating vortex patches for the 2D incompressible Euler equations in the unit disc. By choosing a suitable vector field to deform the patch, we show that each simply-connected rotating vortex patch $D$ with angular velocity $Ω$, $Ω\geq \max\{{1}/{2},({2 l^2})/{(1-l^2)^2}\}$ or $Ω\leq -({2 l^2})/{(1-l^2)^2}$, where $l=\sup_{x\in D}|x|$, must be a disc. The main idea of the proof, which has a variational flavor, comes from a very recent paper of Gómez-Serrano--Park--Shi--Yao, arXiv:1908.01722, where radial symmetry of rotating vortex patches in the whole plane was studied.

math.AP↗

arXiv2019

Existence of steady symmetric vortex patch in a disk

In this paper we construct a family of steady symmetric vortex patches for the incompressible Euler equations in an open disk. The result is obtained by studying a variational problem in which the kinetic energy of the fluid is maximized subject to some appropriate constraints for the vorticity. Moreover, we show that these vortex patches shrink to a given minimum point of the corresponding Kirchhoff-Routh function as the vorticity strength parameter goes to infinity.

math.AP↗

arXiv2019

Rotating vortex patches for the planar Euler equations in a disk

We construct a family of rotating vortex patches with fixed angular velocity for the two-dimensional Euler equations in a disk. As the vorticity strength goes to infinity, the limit of these rotating vortex patches is a rotating point vortex whose motion is described by the Kirchhoff-Routh equation. The construction is performed by solving a variational problem for the vorticity which is based on an adaption of Arnold's variational principle. We also prove nonlinear orbital stability of the set of maximizers in the variational problem under $L^p$ perturbation when $p\in[{3}/{2},+\infty)$.

math.AP↗

arXiv2019

Steady vortex patches near a rotating flow with constant vorticity in a planar bounded domain

In this paper, we study steady vortex patch solutions to the incompressible Euler equations in a planar bounded domain $D$. Let $ψ_0$ be the solution of the elliptic problem $-Δψ_{0} =1$ in $D$; $ψ_0=0$ on $\partial D$. We prove that for any finite collection of isolated maximum points of $ψ_0$, say $\{x_1,\cdot\cdot\cdot,x_k\},$ and any $k$-tuple $\vecκ=(κ_1,\cdot,\cdot,\cdot,κ_k)$ with $κ_i>0$ and $|\vecκ|:=\sum_{i=1}^kκ_i<<1,$ there exists a steady solution of the Euler equations such that the vorticity has the form $ω^{\vecκ}=1-I_{\cup_{i=1}^k A^{\vecκ}_i}$, where $I$ denotes the characteristic function, $|A^{\vecκ}_i|=κ_i$ and $A^{\vecκ}_i$ "shrinks" to $x_i$ as $|\vecκ|\to 0$.

math.AP↗

arXiv2019

Nonlinear Stability of Planar Vortex Patches in an ideal fluid

In this paper, we prove nonlinear stability of planar vortex patches concentrated near an isolated minimum point of the Robin function in a general bounded domain. These vortex patches are stationary solutions of the two-dimensinal incompressible Euler equations. The result is obtained by showing that these concentrated vortex patches are in fact isolated maximizers of the kinetic energy among isovortical patches.

math.AP↗

arXiv2019

Steady vortex patches near a nontrivial irrotational flow

In this paper, we study the vortex patch problem in an ideal fluid in a planar bounded domain. By solving a certain minimization problem and studying the limiting behavior of the minimizer, we prove that for any harmonic function $q$ corresponding to a nontrivial irrotational flow, there exists a family of steady vortex patches approaching the set of extremum points of $q$ on the boundary of the domain. Furthermore, we show that each finite collection of strict extreme points of $q$ corresponds to a family of steady multiple vortex patches approaching it.

math.AP↗

arXiv2018

Existence of steady multiple vortex patches to the vortex-wave system

In this paper we prove the existence of steady multiple vortex patch solutions to the vortex-wave system in a planar bounded domain. The construction is performed by solving a certain variational problem for the vorticity and studying its asymptotic behavior as the vorticity strength goes to infinity.

math.AP↗

arXiv2018

Steady vortex patch solutions to the vortex-wave system

The vortex-wave system describes the motion of a two-dimensional ideal fluid in which the vorticity includes continuously distributed vorticity, which is called the background vorticity, and a finite number of concentrated vortices. In this paper we restrict ourselves to the case of a single point vortex in bounded domains. We prove the existence of steady vortex patch solutions to this system with prescribed distribution for the background vorticity. Moreover, we show that the supports of these solutions "shrink" to a minimum point of the Kirchhoff-Routh function as the strength parameter of the background vorticity goes to infinity.

math.AP↗

arXiv2018

Nonlinear Orbital Stability for Planar Vortex Patches

In this paper, we prove nonlinear orbital stability for steady vortex patches that maximize the kinetic energy among isovortical rearrangements in a planar bounded domain. As a result, nonlinear stability for an isolated vortex patch is proved. The proof is based on conservation of energy and vorticity, which is an analogue of the classical Liapunov function method.

math.AP↗