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Quantitative relations among causality measures with applications to nonlinear pulse-output network reconstruction

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arXiv 2021年

作者： Tian, Zhong-Qi K. Chen, Kai Li, Songting McLaughlin, David W. Zhou, Douglas School of Mathematical Sciences Shanghai Jiao Tong University Shanghai200240 China Institute of Natural Sciences Shanghai Jiao Tong University Shanghai200240 China Ministry of Education Key Laboratory of Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai200240 China Courant Institute of Mathematical Sciences New York University New YorkNY10012 United States Center for Neural Science New York University New YorkNY10012 United States Institute of Mathematical Sciences New York University Shanghai Shanghai200122 China New York University Tandon School of Engineering New York University BrooklynNY11201 United States Neuroscience Institute of New York University Langone Health New York University New YorkNY10016 United States

The causal connectivity of a network is often inferred to understand the network function. It is arguably acknowledged that the inferred causal connectivity relies on causality measure one applies, and it may differ from the network’s underlying structural connectivity. However, the interpretation of causal connectivity remains to be fully clarified, in particular, how causal connectivity depends on causality measures and how causal connectivity relates to structural connectivity. Here, we focus on nonlinear networks with pulse signals as measured output, e.g., neural networks with spike output, and address the above issues based on four intensively utilized causality measures, i.e., time-delayed correlation, time-delayed mutual information, Granger causality, and transfer entropy. We theoretically show how these causality measures are related to one another when applied to pulse signals. Taking the simulated Hodgkin-Huxley neural network and the real mouse brain network as two illustrative examples, we further verify the quantitative relations among the four causality measures and demonstrate that the causal connectivity inferred by any of the four well coincides with the underlying network structural connectivity, therefore establishing a direct link between the causal and structural connectivity. We stress that the structural connectivity of networks can be reconstructed pairwisely without conditioning on the global information of all other nodes in a network, thus circumventing the curse of dimensionality. Our framework provides a practical and effective approach for pulse-output network reconstruction. Copyright © 2021, The Authors. All rights reserved.

关键词： Time delay

Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems

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Communications in Computational Physics 2018年第6期24卷 86-103页

作者： Dongfang Li Hong-Lin Liao Weiwei Sun Jilu Wang Jiwei Zhang School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Department of Mathematics City University of Hong KongKowloonHong Kong Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing211106China Department of Scientific Computing Florida StateUniversityTallahasseeFL 32306USA Beijing Computational Science Research Center Beijing 100094China.

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element *** analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type *** this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional *** terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear *** theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

关键词： Time-fractional nonlinear parabolic problems L1-Galerkin FEMs Error estimates discrete fractional Gronwall type inequality Linearized schemes

The effect of numerical noises on statistics of a chaotic dynamical system 8

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The effect of numerical noises on statistics of a chaotic dy...

2018 8th International Conference on Applied Physics and Mathematics, ICAPM 2018

作者： Li, Xiaoming Li, Xiaochen Liao, Shijun School of Naval Architecture Ocean and Civil Engineering Shanghai Jiao Tong University Shanghai200240 China School of Civil Engineering and Transportation South China University of Technology Guangzhou510641 China MoE Key Laboratory in Scientific and Engineering Computing Shanghai200240 China

It is well known that chaotic dynamical systems have the sensitive dependence on initial conditions (SDIC). Unfortunately, numerical noises (such as truncation error and round-off error) always exist in practice. Thus, due to the SDIC, long-term accurate prediction of chaotic dynamical systems is practically impossible. In this paper, using the so-called "clean numerical simulation" (CNS) whose numerical noises might be much smaller even than micro-level physical uncertainty and thus are negligible, we gain accurate prediction of a chaotic dynamical system in a long enough interval of time. Then, based on these reliable simulations, the influence of numerical noises on statistic computations is investigated. It is found that the influence of numerical noises is negligible when statistic results are time-independent. Unfortunately, when the statistics of a chaotic dynamical system are time-dependent, numerical noises have a great influence even on statistic computations. © Published under licence by IOP Publishing Ltd.

关键词： Statistics

Analysis and approximations of dirichlet boundary control of stokes flows in the energy space

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arXiv 2020年

作者： Gong, Wei Mateos, Mariano Singler, John R. Zhang, Yangwen State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics National Center for Mathematics and Interdisciplinary Sciences Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Dpto. de Matemáticas Universidad de Oviedo Campus de Gijón Spain Department of Mathematics and Statistics Missouri University of Science and Technology RollaMO United States Department of Mathematical Science University of Delaware NewarkDE United States

We study Dirichlet boundary control of Stokes flows in 2D polygonal domains. We consider cost functionals with two different boundary control regularization terms: the L2 norm and an energy space seminorm. We prove well-posedness and regularity results for both problems, develop finite element discretizations for both problems, and prove finite element error estimates for the latter problem. The motivation to study the energy space problem follows from our analysis: we prove that the choice of the control space L2(Γ) can lead to an optimal control with discontinuities at the corners, even when the domain is convex. We observe this phenomenon in numerical experiments. This behavior does not occur in Dirichlet boundary control problems for the Poisson equation on convex polygonal domains, and may not be desirable in real applications. For the energy space problem, we derive the first order optimality conditions, and show that the solution of the control problem is more regular than the solution of the problem with the L2(Γ) regularization. We also prove a priori error estimates for the control in the energy norm, and present several numerical experiments for both control problems on convex and nonconvex domains. © 2020, CC0.

关键词： Poisson equation

A DYNAMICAL-TOPOLOGICAL OBSTRUCTION for SMOOTH ISOMETRIC EMBEDDINGS of RIEMANNIAN MANIFOLDS VIA INCOMPRESSIBLE EULER EQUATIONS

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arXiv 2020年

作者： Li, Siran School of Mathematical Sciences Key Laboratory of Scientific and Engineering Computing Ministry of Education Shanghai Jiao Tong University No. 6 Science Buildings 800 Dongchuan Road Minhang District Shanghai200240 China New York University - Shanghai Office 1146 1555 Century Avenue Pudong District Shanghai200122 China NYU-ECNU Institute of Mathematical Sciences Room 340 Geography Building 3663 North Zhongshan Road Shanghai200062 China

We obtain a dynamical-topological obstruction for the existence of isometric embedding of a Riemannian manifold-with-boundary (M, g): if the first real homology of M is nontrivial, if the centre of the fundamental group is trivial, and if M is isometrically embedded into a Euclidean space of dimension at least 3, then the isometric embedding must violate a certain dynamical, kinetic energy-related condition (the "rigid isotopy extension property" in Definition 1.1). The arguments are motivated by the incompressible Euler equations with prescribed initial and terminal configurations in *** Codes 76B03, 35Q35, 58B05, 57R57 Copyright © 2020, The Authors. All rights reserved.

关键词： Euler equations

VARIATIONAL STRUCTURE AND TWO-DIMENSIONAL SUBSONIC JET FLOWS FOR COMPRESSIBLE EULER SYSTEM WITH GENERAL INCOMING FLOWS

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arXiv 2020年

作者： Li, Yan Shi, Wenhui Tang, Lan Xie, Chunjing School of Mathematical Sciences Shanghai Jiao Tong University 800 Dongchuan Road Shanghai200240 China School of Mathematics Monash University 9 Rainforest Walk ClaytonVIC3800 Australia School of Mathematics and Statistics Central China Normal University Hubei Wuhan430079 China School of mathematical Sciences Institute of Natural Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing CMA-Shanghai Shanghai Jiao Tong University 800 Dongchuan Road Shanghai China

In this paper, we proved the well-posedness theory of compressible subsonic jet flows for two-dimensional steady Euler system with general incoming horizontal velocity as long as the flux is larger than a critical value. One of the key observations is that the stream function formulation for two-dimensional compressible steady Euler system enjoys a variational structure even when the flows have nontrivial vorticity, so that the jet problem can be reformulated as a domain variation problem. This variational structure helps to adapt the framework developed by Alt, Caffarelli, and Friedman to study the jet problem, which is a Bernoulli type free boundary problem. A major technical point to analyze the jet flows is that the inhomogeneous terms in the rescaled equation near the free boundary are always small, even when the vorticity of the flows is big. Copyright © 2020, The Authors. All rights reserved.

关键词： Vorticity

Optimal provable robustness of quantum classification via quantum hypothesis testing

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arXiv 2020年

作者： Weber, Maurice Liu, Nana Li, Bo Zhang, Ce Zhao, Zhikuan Department of Computer Science ETH Zürich Universitätstrasse 6 Zürich8092 Switzerland Institute of Natural Sciences Shanghai Jiao Tong University Shanghai200240 China Ministry of Education Key Laboratory in Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai200240 China University of Michigan-Shanghai Jiao Tong University Joint Institute Shanghai200240 China Department of Computer Science University of Illinois UrbanaIL61801 United States

Quantum machine learning models have the potential to offer speedups and better predictive accuracy compared to their classical counterparts. However, these quantum algorithms, like their classical counterparts, have been shown to also be vulnerable to input perturbations, in particular for classification problems. These can arise either from noisy implementations or, as a worst-case type of noise, adversarial attacks. In order to develop defence mechanisms and to better understand the reliability of these algorithms, it is crucial to understand their robustness properties in presence of natural noise sources or adversarial manipulation. From the observation that measurements involved in quantum classification algorithms are naturally probabilistic, we uncover and formalize a fundamental link between binary quantum hypothesis testing and provably robust quantum classification. This link leads to a tight robustness condition which puts constraints on the amount of noise a classifier can tolerate, independent of whether the noise source is natural or adversarial. Based on this result, we develop practical protocols to optimally certify robustness. Finally, since this is a robustness condition against worst-case types of noise, our result naturally extends to scenarios where the noise source is known. Thus, we also provide a framework to study the reliability of quantum classification protocols beyond the adversarial, worst-case noise scenarios. Copyright © 2020, The Authors. All rights reserved.

关键词： Quantum theory

High order explicit local time-stepping methods for hyperbolic conservation laws

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arXiv 2019年

作者： Hoang, Thi-Thao-Phuong Ju, Lili Leng, Wei Wang, Zhu Department of Mathematics and Statistics Auburn University AL36849 United States Department of Mathematics University of South Carolina ColumbiaSC29208 United States State Key Laboratory of Scientific and Engineering Computing Chinese Academy of Sciences Beijing100190 China

In this paper we present and analyze a general framework for constructing high order explicit local time stepping (LTS) methods for hyperbolic conservation laws. In particular, we consider the model problem discretized by Runge-Kutta discontinuous Galerkin (RKDG) methods and design LTS algorithms based on strong stability preserving Runge-Kutta (SSP-RK) schemes, that allow spatially variable time step sizes to be used for time integrations in different regions. The proposed algorithms are of predictor-corrector type, in which the interface information along the time direction is first predicted based on the SSP-RK approximations and Taylor expansions, and then the fluxes over the region of interface are corrected to conserve mass exactly at each time step. Following the proposed framework, we detail the corresponding LTS schemes with accuracy up to the fourth order, and prove their conservation property and nonlinear stability for the scalar conservation laws. Numerical experiments are also presented to demonstrate excellent performance of the proposed LTS algorithms. Copyright © 2019, The Authors. All rights reserved.

关键词： Galerkin methods

A NEW PHYSICS-PRESERVING IMPES SCHEME FOR INCOMPRESSIBLE AND IMMISCIBLE TWO-PHASE FLOW IN HETEROGENEOUS POROUS MEDIA

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arXiv 2019年

作者： Chen, Huangxin Shuyu, S.U.N. School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling High Performance Scientific Computing Xiamen University Fujian361005 China Computational Transport Phenomena Laboratory Division of Physical Science Engineering King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this work we consider a new efficient IMplicit Pressure Explicit Saturation (IMPES) scheme for the simulation of incompressible and immiscible two-phase flow in heterogeneous porous media with capillary pressure. Compared with the conventional IMPES schemes, the new IMPES scheme is inherently physics-preserving, namely, the new algorithm is locally mass conservative for both phases and it also enjoys another appealing feature that the total velocity is continuous in the normal direction. Moreover, the new scheme is unbiased with regard to the two phases and the saturations of both phases are bounds-preserving if the time step size is smaller than a certain value. The key ideas in the new scheme include that the Darcy flows for both phases are rewritten in the formulation based on the total velocity and an auxiliary velocity referring to as the capillary potential gradient, and the total discretized conservation equation is obtained by the summation of the discretized conservation equation for each phase. The upwind strategy is applied to update the saturations explicitly, and the upwind mixed finite element methods are used to solve the pressure-velocity systems which can be decoupled. We also present some interesting examples to demonstrate the efficiency and robustness of the new algorithm. Copyright © 2019, The Authors. All rights reserved.

关键词： Capillarity