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Qiang Du

Publications and source records attributed to Qiang Du.

At least 73 records · Page 4Linked to original sources
arXiv2022

Feedback and control systems for future linear colliders: White Paper for Snowmass 2021 Topical Group AF07-RF

Particle accelerators for high energy physics will generate TeV-scale particle beams in large, multi-Km size machines colliding high brightness beams at the interaction point [1-4]. The high luminosity in such machines is achieved by producing very small asymmetric beam size at the interaction point, with short durations to minimize beam-beam effects. Tuning energy, timing and position of the beam for optimal performance will require high-precision controls of amplitude and phase of high-frequency electromagnetic fields and real-time processing of complex algorithms. The stability of the colliding beams has a large impact on the collider's effective luminosity. Therefore, the technology readiness level of diagnostic and control systems will be a major consideration in the collider design. The technical requirements of such systems depend on the specifics of beam parameters, such as transverse and longitudinal dimensions, charge/pulse and beam pulse format, which are driven by the accelerating technology of choice. While feedback systems with single bunch position monitor resolution below 50 nm and latency <300 ns have been demonstrated in beam test facilities, many advanced collider concepts make use of higher repetition rates, brighter beams and higher accelerating frequencies, and will require better performance, up to 1-2 order of magnitude, demanding aggressive R&D to be able to deliver and maintain the targeted luminosity.

physics.acc-ph↗
arXiv2022

A nonlocal Stokes system with volume constraints

In this paper, we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition. We use the idea of volume constraint to enforce the no-slip boundary condition and prove that the nonlocal model is well-posed. We also show that and the solution of the nonlocal system converges to the solution of the original Stokes system as the nonlocality vanishes.

math.AP↗
arXiv2022

Robustness test of the spacegroupMining model for determining space groups from atomic pair distribution function data

Machine learning models based on convolutional neural networks have been used for predicting space groups of crystal structures from their atomic pair distribution function (PDF). However, the PDFs used to train the model are calculated using a fixed set of parameters that reflect specific experimental conditions, and the accuracy of the model when given PDFs generated with different choices of these parameters is unknown. In this paper, we report that the results of the top-1 accuracy and top-6 accuracy are robust when applied to PDFs of different choices of experimental parameters $r_\text{max}$, $Q_\text{max}$, $Q_\text{damp}$ and atomic displacement parameters.

cond-mat.mtrl-sci↗
arXiv2022

The average distance problem with an Euler elastica penalization

We consider the minimization of an average distance functional defined on a two-dimensional domain $Ω$ with an Euler elastica penalization associated with $\pd Ω$, the boundary of $Ω$. The average distance is given by \begin{equation*} \int_Ω \dist^p(x,\pd Ω)\d x \end{equation*} where $p\geq 1$ is a given parameter, and $\dist(x,\pd Ω)$ is the Hausdorff distance between $\{x\}$ and $\pd Ω$. The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve ${\pd Ω}$, which is proportional to the integrated squared curvature defined on $\pd Ω$, as given by \begin{equation*} \la \int_{\pd Ω} κ_{\pd Ω}^2\d\H_{\llcorner \pd Ω}^1, \end{equation*} where $κ_{\pd Ω}$ denotes the (signed) curvature of $\pd Ω$ and $\la>0$ denotes a penalty constant. The domain $Ω$ is allowed to vary among compact, convex sets of $\mathbb{R}^2$ with Hausdorff dimension equal to $2$\tcr{.} Under no a priori assumptions on the regularity of the boundary $\pd Ω$, we prove the existence of minimizers of $E_{p,\la}$. Moreover, we establish the $C^{1,1}$-regularity of its minimizers. An original construction of a suitable family of competitors plays a decisive role in proving the regularity.

math.AP↗
arXiv2022

The average distance problem with perimeter-to-area ratio penalization

In this paper we consider the functional \begin{equation*} E_{p,\la}(Ω):=\int_Ω\dist^p(x,\pd Ω)\d x+\la \frac{\H^1(\pd Ω)}{\H^2(Ω)}. \end{equation*} Here $p\geq 1$, $\la>0$ are given parameters, the unknown $Ω$ varies among compact, convex, Hausdorff two-dimensional sets of $\R^2$, $\pd Ω$ denotes the boundary of $Ω$, and $\dist(x,\pd Ω):=\inf_{y\in\pd Ω}|x-y|$. The integral term $\int_Ω\dist^p(x,\pd Ω)\d x$ quantifies the "easiness" for points in $Ω$ to reach the boundary, while $\frac{\H^1(\pd Ω)}{\H^2(Ω)}$ is the perimeter-to-area ratio. The main aim is to prove existence and $C^{1,1}$-regularity of minimizers of $\E$.

math.AP↗
arXiv2022

Numerical simulation of singularity propagation modeled by linear convection equations with spatially heterogeneous nonlocal interactions

We study the propagation of singularities in solutions of linear convection equations with spatially heterogeneous nonlocal interactions. A spatially varying nonlocal horizon parameter is adopted in the model, which measures the range of nonlocal interactions. Via heterogeneous localization, this can lead to the seamless coupling of the local and nonlocal models. We are interested in understanding the impact on singularity propagation due to the heterogeneities of nonlocal horizon and the local and nonlocal transition. We first analytically derive equations to characterize the propagation of different types of singularities for various forms of nonlocal horizon parameters in the nonlocal regime. We then use asymptotically compatible schemes to discretize the equations and carry out numerical simulations to illustrate the propagation patterns in different scenarios.

math.NA↗
arXiv2021

Discovering State Variables Hidden in Experimental Data

All physical laws are described as relationships between state variables that give a complete and non-redundant description of the relevant system dynamics. However, despite the prevalence of computing power and AI, the process of identifying the hidden state variables themselves has resisted automation. Most data-driven methods for modeling physical phenomena still assume that observed data streams already correspond to relevant state variables. A key challenge is to identify the possible sets of state variables from scratch, given only high-dimensional observational data. Here we propose a new principle for determining how many state variables an observed system is likely to have, and what these variables might be, directly from video streams. We demonstrate the effectiveness of this approach using video recordings of a variety of physical dynamical systems, ranging from elastic double pendulums to fire flames. Without any prior knowledge of the underlying physics, our algorithm discovers the intrinsic dimension of the observed dynamics and identifies candidate sets of state variables. We suggest that this approach could help catalyze the understanding, prediction and control of increasingly complex systems. Project website is at: https://www.cs.columbia.edu/~bchen/neural-state-variables

math.DS↗
arXiv2021

Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localization

In this work, we prove some trace theorems for function spaces with a nonlocal character that contain the classical $W^{s,p}$ space as a subspace. The result we obtain generalizes well known trace theorems for $W^{s,p}(Ω)$ functions which has a well-defined $W^{1-s/p, p}$ trace on the boundary of a domain with sufficient regularity. The new generalized spaces are associated with norms that are characterized by nonlocal interaction kernels defined heterogeneously with a special localization feature on the boundary. Intuitively speaking, the class contains functions that are as rough as an $L^p$ function inside the domain of definition but as smooth as a $W^{s,p}$ function near the boundary. Our result is that the $W^{1-s/p, p}$ norm of the trace on the boundary such functions is controlled by the nonlocal norms that are weaker than the classical $W^{s, p}$ norm. These results are improvement and refinement of the classical results since the boundary trace can be attained without imposing regularity of the function in the interior of the domain. They also extend earlier results in the case of $p=2$. In the meantime, we prove Hardy-type inequalities for functions in the new generalized spaces that vanish on the boundary, showing them having the same decay rate to the boundary as functions in the smaller space $W^{s,p}(Ω)$. A Poincaré-type inequality is also derived. An application of the new theory leads to the well-posedness of a nonlinear variational problem that allows possible singular behavior in the interior with imposed smoother data on the boundary.

math.AP↗
arXiv2021

A Physics-Informed Deep Learning Paradigm for Traffic State and Fundamental Diagram Estimation

Traffic state estimation (TSE) bifurcates into two categories, model-driven and data-driven (e.g., machine learning, ML), while each suffers from either deficient physics or small data. To mitigate these limitations, recent studies introduced a hybrid paradigm, physics-informed deep learning (PIDL), which contains both model-driven and data-driven components. This paper contributes an improved version, called physics-informed deep learning with a fundamental diagram learner (PIDL+FDL), which integrates ML terms into the model-driven component to learn a functional form of a fundamental diagram (FD), i.e., a mapping from traffic density to flow or velocity. The proposed PIDL+FDL has the advantages of performing the TSE learning, model parameter identification, and FD estimation simultaneously. We demonstrate the use of PIDL+FDL to solve popular first-order and second-order traffic flow models and reconstruct the FD relation as well as model parameters that are outside the FD terms. We then evaluate the PIDL+FDL-based TSE using the Next Generation SIMulation (NGSIM) dataset. The experimental results show the superiority of the PIDL+FDL in terms of improved estimation accuracy and data efficiency over advanced baseline TSE methods, and additionally, the capacity to properly learn the unknown underlying FD relation.

cs.LG↗
arXiv2021

Censored Stable Subordinators and Fractional Derivatives

Based on the popular Caputo fractional derivative of order $β$ in $(0,1)$, we define the censored fractional derivative on the positive half-line $\mathbb R_+$. This derivative proves to be the Feller generator of the censored (or resurrected) decreasing $β$-stable process in $\mathbb R_+$. We provide a series representation for the inverse of this censored fractional derivative, which we use to study general censored initial value problems. We are then able to prove that this censored process hits the boundary in a finite time $τ_\infty$, whose expectation is proportional to that of the first passage time of the $β$-stable subordinator. We also show that the censored relaxation equation is solved by the Laplace transform of $τ_\infty$. This relaxation solution proves to be a completely monotone series, with algebraic decay one order faster than its Caputo counterpart, leading, surprisingly, to a new regime of fractional relaxation models. Lastly, we discuss how this work identifies a new sub-diffusion model.

math.CA↗
arXiv2021

Second order accurate Dirichlet boundary conditions for linear nonlocal diffusion problems

We present an approach to handle Dirichlet type nonlocal boundary conditions for nonlocal diffusion models with a finite range of nonlocal interactions. Our approach utilizes a linear extrapolation of prescribed boundary data. A novelty is, instead of using local gradients of the boundary data that are not available a priori, we incorporate nonlocal gradient operators into the formulation to generalize the finite differences-based methods which are pervasive in literature; our particular choice of the nonlocal gradient operators is based on the interplay between a constant kernel function and the geometry of nonlocal interaction neighborhoods. Such an approach can be potentially useful to address similar issues in peridynamics, smoothed particle hydrodynamics and other nonlocal models. We first show the well-posedness of the newly formulated nonlocal problems and then analyze their asymptotic convergence to the local limit as the nonlocality parameter shrinks to zero. We justify the second order localization rate, which is the optimal order attainable in the absence of physical boundaries.

math.AP↗
arXiv2021

Nonlocal Trace Spaces and Extension Results for Nonlocal Calculus

For a given Lipschitz domain $Ω$, it is a classical result that the trace space of $W^{1,p}(Ω)$ is $W^{1-1/p,p}(\partialΩ)$, namely any $W^{1,p}(Ω)$ function has a well-defined $W^{1-1/p,p}(\partialΩ)$ trace on its codimension-1 boundary $\partialΩ$ and any $W^{1-1/p,p}(\partialΩ)$ function on $\partialΩ$ can be extended to a $W^{1,p}(Ω)$ function. Recently, Dyda and Kassmann (2019) characterize the trace space for nonlocal Dirichlet problems involving integrodifferential operators with infinite interaction ranges, where the boundary datum is provided on the whole complement of the given domain $\mathbb{R}^d\backslashΩ$. In this work, we study function spaces for nonlocal Dirichlet problems with a finite range of nonlocal interactions, which naturally serves a bridging role between the classical local PDE problem and the nonlocal problem with infinite interaction ranges. For these nonlocal Dirichlet problems, the boundary conditions are normally imposed on a region with finite thickness volume which lies outside of the domain. We introduce a function space on the volumetric boundary region that serves as a trace space for these nonlocal problems and study the related extension results. Moreover, we discuss the consistency of the new nonlocal trace space with the classical $W^{1-1/p,p}(\partialΩ)$ space as the size of nonlocal interaction tends to zero. In making this connection, we conduct an investigation on the relations between nonlocal interactions on a larger domain and the induced interactions on its subdomain. The various forms of trace, embedding and extension theorems may then be viewed as consequences in different scaling limits.

math.AP↗
arXiv2021

Dynamic driving and routing games for autonomous vehicles on networks: A mean field game approach

This paper aims to answer the research question as to optimal design of decision-making processes for autonomous vehicles (AVs), including dynamical selection of driving velocity and route choices on a transportation network. Dynamic traffic assignment (DTA) has been widely used to model travelers's route choice or/and departure-time choice and predict dynamic traffic flow evolution in the short term. However, the existing DTA models do not explicitly describe one's selection of driving velocity on a road link. Driving velocity choice may not be crucial for modeling the movement of human drivers but it is a must-have control to maneuver AVs. In this paper, we aim to develop a game-theoretic model to solve for AVs's optimal driving strategies of velocity control in the interior of a road link and route choice at a junction node. To this end, we will first reinterpret the DTA problem as an N-car differential game and show that this game can be tackled with a general mean field game-theoretic framework. The developed mean field game is challenging to solve because of the forward and backward structure for velocity control and the complementarity conditions for route choice. An efficient algorithm is developed to address these challenges. The model and the algorithm are illustrated on the Braess network and the OW network with a single destination. On the Braess network, we first compare the LWR based DTA model with the proposed game and find that the driving and routing control navigates AVs with overall lower costs. We then compare the total travel cost without and with the middle link and find that the Braess paradox may still arise under certain conditions. We also test our proposed model and solution algorithm on the OW network.

math.OC↗
arXiv2021

A fast two-stage algorithm for non-negative matrix factorization in streaming data

In this article, we study algorithms for nonnegative matrix factorization (NMF) in various applications involving streaming data. Utilizing the continual nature of the data, we develop a fast two-stage algorithm for highly efficient and accurate NMF. In the first stage, an alternating non-negative least squares (ANLS) framework is used, in combination with active set method with warm-start strategy for the solution of subproblems. In the second stage, an interior point method is adopted to accelerate the local convergence. The convergence of the proposed algorithm is proved. The new algorithm is compared with some existing algorithms in benchmark tests using both real-world data and synthetic data. The results demonstrate the advantage of our algorithm in finding high-precision solutions.

math.OC↗
arXiv2021

Physics-Informed Deep Learning for Traffic State Estimation

Traffic state estimation (TSE), which reconstructs the traffic variables (e.g., density) on road segments using partially observed data, plays an important role on efficient traffic control and operation that intelligent transportation systems (ITS) need to provide to people. Over decades, TSE approaches bifurcate into two main categories, model-driven approaches and data-driven approaches. However, each of them has limitations: the former highly relies on existing physical traffic flow models, such as Lighthill-Whitham-Richards (LWR) models, which may only capture limited dynamics of real-world traffic, resulting in low-quality estimation, while the latter requires massive data in order to perform accurate and generalizable estimation. To mitigate the limitations, this paper introduces a physics-informed deep learning (PIDL) framework to efficiently conduct high-quality TSE with small amounts of observed data. PIDL contains both model-driven and data-driven components, making possible the integration of the strong points of both approaches while overcoming the shortcomings of either. This paper focuses on highway TSE with observed data from loop detectors, using traffic density as the traffic variables. We demonstrate the use of PIDL to solve (with data from loop detectors) two popular physical traffic flow models, i.e., Greenshields-based LWR and three-parameter-based LWR, and discover the model parameters. We then evaluate the PIDL-based highway TSE using the Next Generation SIMulation (NGSIM) dataset. The experimental results show the advantages of the PIDL-based approach in terms of estimation accuracy and data efficiency over advanced baseline TSE methods.

cs.LG↗
arXiv2020

Numerical Discretization of Variational Phase Field Model for Phase Transitions in Ferroelectric Thin Films

Phase field methods have been widely used to study phase transitions and polarization switching in ferroelectric thin films. In this paper, we develop an efficient numerical scheme for the variational phase field model based on variational forms of the electrostatic energy and the relaxation dynamics of the polarization vector. The spatial discretization combines the Fourier spectral method with the finite difference method to handle three-dimensional mixed boundary conditions. It allows for an efficient semi-implicit discretization for the time integration of the relaxation dynamics. This method avoids explicitly solving the electrostatic equilibrium equation (a Poisson equation) and eliminates the use of associated Lagrange multipliers. We present several numerical examples including phase transitions and polarization switching processes to demonstrate the effectiveness of the proposed method.

physics.comp-ph↗
arXiv2020

A Game-Theoretic Framework for Autonomous Vehicles Velocity Control: Bridging Microscopic Differential Games and Macroscopic Mean Field Games

This paper proposes an efficient computational framework for longitudinal velocity control of a large number of autonomous vehicles (AVs) and develops a traffic flow theory for AVs. Instead of hypothesizing explicitly how AVs drive, our goal is to design future AVs as rational, utility-optimizing agents that continuously select optimal velocity over a period of planning horizon. With a large number of interacting AVs, this design problem can become computationally intractable. This paper aims to tackle such a challenge by employing mean field approximation and deriving a mean field game (MFG) as the limiting differential game with an infinite number of agents. The proposed micro-macro model allows one to define individuals on a microscopic level as utility-optimizing agents while translating rich microscopic behaviors to macroscopic models. Different from existing studies on the application of MFG to traffic flow models, the present study offers a systematic framework to apply MFG to autonomous vehicle velocity control. The MFG-based AV controller is shown to mitigate traffic jam faster than the LWR-based controller. MFG also embodies classical traffic flow models with behavioral interpretation, thereby providing a new traffic flow theory for AVs.

math.OC↗