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Quan Zheng

Publications and source records attributed to Quan Zheng.

34 records · Page 2Linked to original sources
arXiv2022

Uniform Complex Time Heat Kernel Estimates Without Gaussian Bounds

In this paper, first we consider the uniform complex time heat kernel estimates of $e^{-z(-Δ)^{\fracα{2}}}$ for $α>0, z\in \mathbb{C}^+$. When $\fracα{2}$ is not an integer, generally the heat kernel doest not have the Gaussian upper bounds for real time. Thus the Phragmén-Lindelöf methods fail to give the uniform complex time estimates. Instead, our first result gives the asymptotic estimates for $P(z, x)$ as $z$ tending to the imaginary axis. Then we prove the uniform complex time heat kernel estimates. Finally we also show the uniform estimates of analytic semigroup generated by $H=(-Δ)^{\fracα{2}}+V$ where $V$ belongs to higher order Kato class.

math.CA↗
arXiv2022

Dispersive estimates for the Schrödinger equation with finite rank perturbations

In this paper, we investigate dispersive estimates for the time evolution of Hamiltonians $$ H=-Δ+\sum_{j=1}^N\langle\cdot\,, φ_j\rangle φ_j\quad\,\,\,\text{in}\,\,\,\mathbb{R}^d,\,\, d\ge 1, $$ where each $φ_j$ satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant $C=C(N, d, φ_1,\ldots, φ_N)>0$ such that $$ \|e^{-itH}\|_{L^1-L^{\infty}}\leq C t^{-\frac{d}{2}}, \,\,\,\text{for}\,\,\, t>0. $$ As far as we are aware, this seems to provide the first study of $L^1-L^{\infty}$ estimates for finite rank perturbations of the Laplacian in any dimension. We first deal with rank one perturbations ($N=1$). Then we turn to the general case. The new idea in our approach is to establish the Aronszajn-Krein type formula for finite rank perturbations. This allows us to reduce the analysis to the rank one case and solve the problem in a unified manner. Moreover, we show that in some specific situations, the constant $C(N, d, φ_1,\ldots, φ_N)$ grows polynomially in $N$. Finally, as an application, we are able to extend the results to $N=\infty$ and deal with some trace class perturbations.

math.AP↗
arXiv2022

Shap-CAM: Visual Explanations for Convolutional Neural Networks based on Shapley Value

Explaining deep convolutional neural networks has been recently drawing increasing attention since it helps to understand the networks' internal operations and why they make certain decisions. Saliency maps, which emphasize salient regions largely connected to the network's decision-making, are one of the most common ways for visualizing and analyzing deep networks in the computer vision community. However, saliency maps generated by existing methods cannot represent authentic information in images due to the unproven proposals about the weights of activation maps which lack solid theoretical foundation and fail to consider the relations between each pixel. In this paper, we develop a novel post-hoc visual explanation method called Shap-CAM based on class activation mapping. Unlike previous gradient-based approaches, Shap-CAM gets rid of the dependence on gradients by obtaining the importance of each pixel through Shapley value. We demonstrate that Shap-CAM achieves better visual performance and fairness for interpreting the decision making process. Our approach outperforms previous methods on both recognition and localization tasks.

cs.CV↗
arXiv2022

Physics Informed Neural Fields for Smoke Reconstruction with Sparse Data

High-fidelity reconstruction of fluids from sparse multiview RGB videos remains a formidable challenge due to the complexity of the underlying physics as well as complex occlusion and lighting in captures. Existing solutions either assume knowledge of obstacles and lighting, or only focus on simple fluid scenes without obstacles or complex lighting, and thus are unsuitable for real-world scenes with unknown lighting or arbitrary obstacles. We present the first method to reconstruct dynamic fluid by leveraging the governing physics (ie, Navier -Stokes equations) in an end-to-end optimization from sparse videos without taking lighting conditions, geometry information, or boundary conditions as input. We provide a continuous spatio-temporal scene representation using neural networks as the ansatz of density and velocity solution functions for fluids as well as the radiance field for static objects. With a hybrid architecture that separates static and dynamic contents, fluid interactions with static obstacles are reconstructed for the first time without additional geometry input or human labeling. By augmenting time-varying neural radiance fields with physics-informed deep learning, our method benefits from the supervision of images and physical priors. To achieve robust optimization from sparse views, we introduced a layer-by-layer growing strategy to progressively increase the network capacity. Using progressively growing models with a new regularization term, we manage to disentangle density-color ambiguity in radiance fields without overfitting. A pretrained density-to-velocity fluid model is leveraged in addition as the data prior to avoid suboptimal velocity which underestimates vorticity but trivially fulfills physical equations. Our method exhibits high-quality results with relaxed constraints and strong flexibility on a representative set of synthetic and real flow captures.

cs.GR↗
arXiv2021

Neural Relightable Participating Media Rendering

Learning neural radiance fields of a scene has recently allowed realistic novel view synthesis of the scene, but they are limited to synthesize images under the original fixed lighting condition. Therefore, they are not flexible for the eagerly desired tasks like relighting, scene editing and scene composition. To tackle this problem, several recent methods propose to disentangle reflectance and illumination from the radiance field. These methods can cope with solid objects with opaque surfaces but participating media are neglected. Also, they take into account only direct illumination or at most one-bounce indirect illumination, thus suffer from energy loss due to ignoring the high-order indirect illumination. We propose to learn neural representations for participating media with a complete simulation of global illumination. We estimate direct illumination via ray tracing and compute indirect illumination with spherical harmonics. Our approach avoids computing the lengthy indirect bounces and does not suffer from energy loss. Our experiments on multiple scenes show that our approach achieves superior visual quality and numerical performance compared to state-of-the-art methods, and it can generalize to deal with solid objects with opaque surfaces as well.

cs.CV↗
arXiv2021

Unique continuation properties for one dimensional higher order Schrödinger equations

We study two types of unique continuation properties for the higher order Schrödinger equation with potential $$ i\partial_tu=(-Δ_x)^mu+V(t,x)u,\quad(t,x)\in\mathbb{R}^{1+n},\,2\leq m\in\mathbb{N}_+. $$ The first one says if $u$ has certain exponential decay at two times, then $u\equiv0$, and this result is sharp by constructing critical non-trivial solutions. The second one says if $u\equiv0$ in an arbitrary half-space of $\mathbb{R}^{1+n}$, then $u\equiv0$ identically. The uniqueness theorems are given when $n=1$, but we also prove partial results when $n\in\mathbb{N}_+$ for their own interests. Possibility or obstacles to proving these unique continuation properties in higher spatial dimensions are also discussed.

math.AP↗
arXiv2021

The scattering of fractional Schrödinger operators with short range potentials

For any positive real number $s$, we study the scattering theory in a unified way for the fractional Schrödinger operator $H=H_0+V$, where $H_0=(-Δ)^\frac s2$ and the real-valued potential $V$ satisfies short range condition. We prove the existence and asymptotic completeness of the wave operators $W_\pm=\mathrm{s-}\lim_{t\rightarrow\pm\infty}e^{itH}e^{-itH_0}$, the discreteness and finite multiplicity of the non-zero pure point spectrum $σ_\mathrm{pp}\setminus\{0\}$ of $H$, and the finite decay property of eigenfunctions. The short range condition is sharp with respect to the allowed decay rate of $V$, and the decay threshold for the existence and non-existence of the wave operators is faster than $|x|^{-1}$ at the infinity in some sense. Our approach is inspired by the theory of limiting absorption principle for simply characteristic operators established by S. Agmon and L. Hörmander in the 1970s.

math-ph↗
arXiv2018

$L^p$ estimates for fractional schrodinger operators with kato class potentials

Let $α>0$, $H=(-\triangle)^α+V(x)$, $V(x)$ belongs to the higher order Kato class $K_{2α}(\mathbbm{R}^n)$. For $1\leq p\leq \infty$, we prove a polynomial upper bound of $\|e^{-itH}(H+M)^{-β}\|_{L^p, L^p}$ in terms of time $t$. Both the smoothing exponent $β$ and the growth order in $t$ are almost optimal compared to the free case. The main ingredients in our proof are pointwise heat kernel estimates for the semigroup $e^{-tH}$. We obtain a Gaussian upper bound with sharp coefficient for integral $α$ and a polynomial decay for fractal $α$.

math.AP↗
arXiv2017

Remarks on $L^p$-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems

This paper comprises two parts. We first investigate a $L^p$ type of limiting absorption principle for Schrödinger operators $H=-Δ+V$, i.e., In $\mathbb{R}^n$ ($n\ge 3$) we prove the $ε-$uniform $L^{\frac{2(n+1)}{n+3}}$-$L^{\frac{2(n+1)}{n-1}}$ estimates of the resolvent $(H-λ\pm iε)^{-1}$ for all $λ>0$ when the potential $V$ belongs to some integrable spaces and a spectral condition of $H$ at zero is assumed. As an application, we establish a sharp spectral multiplier theorem and $L^p$ bound of Bochner-Riesz means associated with Schrödinger operators $H$. Next, we consider the fractional Schrödinger operator $H=(-Δ)^α+V$ ($0<2α<n$) and prove a uniform Hardy-Littlewood-Sobolev inequality for $(-Δ)^α$, which generalizes the corresponding result of Kenig-Ruiz-Sogge \cite{KRS}.

math.AP↗
arXiv2017

Inhomogeneous Oscillatory Integrals and Global Smoothing Effects for Dispersive Equations

We study oscillatory integrals of the type ${\mathcal F}^{-1}(e^{ita(\cdot)}ψ(\cdot))$ where $a$ is a general function satisfying some elliptic type and non-degenerate conditions at both the origin and infinity, and $ψ$ belongs to some symbol class. Point-wise estimates in space-time are gained with partial sharpness. As applications, global smoothing effects of $L^p-L^q$ as well as Strichartz type for dispersive equations are studied. An application to fractional Schrödinger equations is also given.

math.AP↗
arXiv2016

Distance, Borders, and Time: The Diffusion and Permeability of Political Violence in North and West Africa

This paper explores the spatial and temporal diffusion of political violence in North and West Africa. It does so by endeavoring to represent the mental landscape that lives in the back of a group leader's mind as he contemplates strategic targeting. We assume that this representation is a combination of the physical geography of the target environment, and the mental and physical cost of following a seemingly random pattern of attacks. Focusing on the distance and time between attacks and taking into consideration the transaction costs that state boundaries impose, we wish to understand what constrains a group leader to attack at a location other than the one that would seem to yield the greatest overt payoff. By its very nature, the research problem defies the collection of a full set of structural data. Instead, we leverage functional data from the Armed Conflict Location and Event Data project (ACLED) dataset that, inter alia, meticulously catalogues violent extremist incidents in North and West Africa since 1997, to generate a network whose nodes are administrative regions. These nodes are connected by edges of qualitatively different types: undirected edges representing geographic distance, undirected edges representing borders, and directed edges representing consecutive attacks by the same group at the two endpoints. We analyze the resulting network using novel spectral embedding techniques that are able to account fully for the different types of edges. The result is a map of North and West Africa that depicts the permeability to violence. A better understanding of how location, time, and borders condition attacks enables planning, prepositioning, and response.

cs.SI↗
arXiv2015

Quantitative uniqueness of some higher order elliptic equations

We study the quantitative unique continuation property of some higher order elliptic operators. In the case of $P=(-Δ)^m$, where $m$ is a positive integer, we derive lower bounds of decay at infinity for any nontrivial solutions under some general assumptions. Furthermore, in dimension 2, we can obtain essentially sharp lower bounds for some forth order elliptic operators, the sharpness is shown by constructing a Meshkov type example.

math.AP↗
arXiv2004

Convex Hypersurfaces and $L^p$ Estimates for Schrödinger Equations

This paper is concerned with Schrödinger equations whose principal operators are homogeneous elliptic. When the corresponding level hypersurface is convex, we show the $L^p$-$L^q$ estimate of solution operator in free case. This estimate, combining with the results of fractionally integrated groups, allows us to further obtain the $L^p$ estimate of solutions for the initial data belonging to a dense subset of $L^p$ in the case of integrable potentials.

math.AP↗
arXiv1999

The number of ramified covering of a Riemann surface by Riemann surface

Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann surface by Riemann surface with elementary branch points and prescribed ramification type over a special point.

math.AG↗