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IEEE Transactions on Privacy 2024年第1期1卷 45-56页

作者： Zhao, Xingyuan Zhou, Ruyu Liu, Fang University of Notre Dame Department of Applied and Computational Mathematics and Statistics Notre DameIN46656 United States

Applying a randomized algorithm to a subset rather than the entire dataset amplifies privacy guarantees. We propose a class of subsampling methods "MUltistage Sampling Technique (MUST)"for privacy amplification (PA) in the context of differential privacy (DP). We conduct comprehensive analyses of the PA effects and utility for several 2-stage MUST procedures through newly introduced concepts including strong vs weak PA effects and aligned privacy profile. We provide the privacy loss composition analysis over repeated applications of MUST via the Fourier accountant algorithm. Our theoretical and empirical results suggest that MUST offers stronger PA in ϵ than the common one-stage sampling procedures including Poisson sampling, sampling without replacement, and sampling with replacement, while the results on δ vary case by case. Our experiments show that MUST is non-inferior in the utility and stability of privacy-preserving (PP) outputs to one-stage subsampling methods at similar privacy loss while enhancing the computational efficiency of algorithms that require complex function calculations on distinct data points. MUST can be seamlessly integrated into stochastic optimization algorithms or procedures that involve parallel or simultaneous subsampling when DP guarantees are necessary. © 2024 IEEE.

关键词： Differential privacy

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A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems

A hybrid symbolic-numeric approach to exceptional sets of ge...

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2015 International Workshop on Parallel Symbolic Computation, PASCO 2015

作者： Hauenstein, Jonathan D. Liddell, Alan C. Department of Applied and Computational Mathematics and Statistics University of Notre Dame United States

ISBN: (纸本)9781450335997

Exceptional sets are the sets where the dimension of the fiber of a map is larger than the generic fiber dimension, which we assume is zero. Such situations naturally arise in kinematics, for example, when designing a mechanism that moves when the generic case is rigid. In 2008, Sommese and Wampler showed that one can use fiber products to promote such sets to become irreducible components. We propose an alternative approach using rank constraints on Macaulay matrices. Symbolic computations are used to construct the proper Macaulay matrices, while numerical computations are used to solve the rank-constraint problem. Various exceptional sets are computed, including exceptional RR dyads, lines on surfaces in C3, and exceptional planar pentads. © 2015 ACM.

关键词： Matrix algebra

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AN EMBEDDED BOUNDARY METHOD FOR ELLIPTIC AND PARABOLIC PROBLEMS WITH INTERFACES AND APPLICATION TO MULTI-MATERIAL SYSTEMS WITH PHASE TRANSITIONS

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Acta Mathematica Scientia 2010年第2期30卷 499-521页

作者： Shuqiang Wang Roman Samulyak Tongfei Guo Department of Applied Mathematics and Statistics Stony Brook University Stony Brook NY 11794 USA Computational Science Center Brookhaven National Laboratory Upton NY 11973 USA

The embedded boundary method for solving elliptic and parabolic problems in geometrically complex domains using Cartesian meshes by Johansen and Colella （1998, J. Comput. Phys. 147, 60） has been extended for elliptic and parabolic problems with interior boundaries or interfaces of discontinuities of material properties or solutions. Second order accuracy is achieved in space and time for both stationary and moving interface problems. The method is conservative for elliptic and parabolic problems with fixed interfaces. Based on this method, a front tracking algorithm for the Stefan problem has been developed. The accuracy of the method is measured through comparison with exact solution to a two-dimensional Stefan problem. The algorithm has been used for the study of melting and solidification problems.

关键词： embedded boundary method elliptic interface problem front tracking Ste-fan problem

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PARAMETER AND QUANTILE ESTIMATION FOR THE GENERALIZED EXTREME-VALUE DISTRIBUTION

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ENVIRONMETRICS 1994年第4期5卷 417-432页

作者： CASTILLO, E HADI, AS 1Department of Applied Mathematics and Computational Sciences University of Cantabria 39005 Santander Spain2Department of Statistics Cornell University U.S.A.

The generalized extreme-value distribution (GEVD) was introduced by Jenkinson (1955). It is now widely used to model extremes of natural and environmental data. The GEVD has three parameters: a location parameter (- infinity < lambda < infinity), a scale parameter (alpha > 0) and a shape parameter (- infinity < k < infinity). The traditional methods of estimation (e.g., the maximum likelihood and the moments-based methods) have problems either because the range of the distribution depends on the parameters, or because the mean and higher moments do not exist when k less-than-or-equal-to -1. The currently favoured estimators are those obtained by the method of probability-weighted moments (PWM). The PWM estimators are good for cases where - 1/2 < k < 1/2. Outside this range of k, the PWM estimates may not exist and if they do exist they cannot be recommended because their performance worsens as k increases. In this paper, we propose a method for estimating the parameters and quantiles of the GEVD. The estimators are well-defined for all possible combinations of parameter and sample values. They are also easy to compute as they are based on equations which involve only one variable (rather than three). A simulation study is implemented to evaluate the performance of the proposed method and to compare it with the PWM. The simulation results seem to indicate that the proposed method is comparable to the PWM for - 1/2 < k < 1/2 but outside this range it gives a better performance. Two real-life environmental data sets are used to illustrate the methodology.

关键词： LEAST MEDIAN OF SQUARES MAXIMUM LIKELIHOOD MEDIAN METHOD OF MOMENTS ORDER statistics PROBABILITY-WEIGHTED MOMENTS

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A localized Fourier collocation method for 2D and 3D elliptic partial differential equations:Theory and MATLAB code

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International Journal of Mechanical System Dynamics 2022年第4期2卷 339-351页

作者： Yan Gu Zhuojia Fu Mikhail V.Golub Department of Computational Mathematics School of Mathematics and StatisticsQingdao UniversityQingdaoChina Department of Engineering Mechanics College of Mechanics and MaterialsHohai UniversityNanjingChina Department of Applied Mathematics Institute for MathematicsMechanics and InformaticsKuban State UniversityKrasnodarRussian Federation

A localized Fourier collocation method is proposed for solving certain types of elliptic boundary value *** method first discretizes the entire domain into a set of overlapping small subdomains,and then in each of the subdomains,the unknown functions and their derivatives are approximated using the pseudo-spectral Fourier collocation *** key idea of the present method is to combine the merits of the quick convergence of the pseudo-spectral method and the high sparsity of the localized discretization technique to yield a new framework that may be suitable for large-scale *** present method can be viewed as a competitive alternative for solving numerically large-scale boundary value problems with complex-shape *** numerical experiments involving Poisson,Helmholtz,and modified-Helmholtz equations in both two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.

关键词： meshless method collocation method Fourier basis functions large-scale problem

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THE CONTAGION NUMBER: HOW FAST CAN A DISEASE SPREAD?

Decision Making: Applications in Management and Engineering

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Decision Making: Applications in Management and Engineering 2023年第1期6卷 219-239页

作者： Blessley, Misty Davila, Randy Hale, Trevor Pepper, Ryan Statistics Operations and Data Science Department Temple University United States Computational and Applied Mathematics Department Rice University United States Information and Operations Management Department Texas A&M University United States Mathematics and Statistics Department University of Houston - Downtown United States

The overall purpose of this paper is to define a new metric on the spreadability of a disease. Herein, we define a variant of the well-known graph-theoretic burning number (BN) metric that we coin the contagion number (CN). We aver that the CN is a better metric to model disease spread than the BN as the CN concentrates on first time infections. This is important because the Centers for Disease Control and Prevention report that COVID-19 reinfections are rare. This paper delineates a novel methodology to solve for the CN of any tree, in polynomial time, which addresses how fast a disease could spread (i.e., a worst-cast analysis). We then employ Monte Carlo simulation to determine the average contagion number (ACN) (i.e., a most-likely analysis) of how fast a disease would spread. The latter is analyzed on scale-free graphs, which are specifically designed to model human social networks (sociograms). We test our method on some randomly generated scale-free graphs and our findings indicate the CN to be a robust, tractable (the BN is NP-hard even for a tree), and effective disease spread metric for decision makers. The contributions herein advance disease spread understanding and reveal the importance of the underlying network structure. Understanding disease spreadability informs public policy and the associated managerial allocation decisions. © 2023 by the authors.

关键词： COVID-19

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Parametric structure of probabilities in Bayesian networks 3rd

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3rd European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty, ECSQARU 1995

作者： Castillo, Enrique Gutiérrez, José Manuel Hadi, Ali S. Department of Applied Mathematics and Computational Science University of Cantabria Spain Department of Statistics Cornell University United States

ISBN: (纸本)3540601120

The paper presents a method for uncertainty propagation in Bayesian networks in symbolic, as opposed to numeric, form. The algebraic structure of probabilities is characterized. The prior probabilities of instantiations and the marginal probabilities are shown to be rational functions of the parameters, where the polynomials appearing in the numerator and the denominator are at the most first degree in each of the parameters. It is shown that numeric propagation algorithms can be adapted for symbolic computations by means of canonical components. Furthermore, the same algorithms can be used to build automatic code generators for symbolic propagation of evidence. An example of uncertainty propagation in a clique tree is used to illustrate all the steps and the corresponding code in Mathematica is given. Finally, it is shown that upper and lower bounds for the marginal probabilities of nodes are attained at one of the canonical components. © Springer-Verlag Berlin Heidelberg 1995.

关键词： Bayesian networks

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Vanishing capillarity limit of the compressible fluid models of korteweg type to the navier-stokes equations

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SIAM Journal on Mathematical Analysis 2014年第2期46卷 1633-1650页

作者： Bian, Dongfen Yao, Lei Zhu, Changjiang Institute of Applied Physics and Computational Mathematics Beijing 100088 China Center for Nonlinear Studies Department of Mathematics Northwest University Xi'an 710127 China School of Mathematics and Statistics Central China Normal University Wuhan 430079 China

In this paper, we consider the three-dimensional isentropic compressible fluid models of Korteweg type, called the compressible Navier-Stokes-Korteweg system. We mainly present the vanishing capillarity limit of the smooth solution to the initial value problem. Precisely, we first establish the uniform estimates of the global smooth solution with respect to the capillary coefficient κ. Then by the Lions-Aubin lemma, we show that the unique smooth solution of the three-dimensional Navier-Stokes-Korteweg system converges globally in time to the smooth solution of the three-dimensional Navier-Stokes system as κ tends to zero. Also, we give the convergence rate estimates for any given positive time. © 2014 Society for Industrial and applied mathematics.

关键词： Capillarity

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Certification using Newton-invariant subspaces 7th

Certification using Newton-invariant subspaces

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7th International Conference on Mathematical Aspects of Computer and Information Sciences, MACIS 2017

作者： Hauenstein, Jonathan D. Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States

ISBN: (纸本)9783319724522

For a square system of analytic equations, a Newton-invariant subspace is a set which contains the resulting point of a Newton iteration applied to each point in the subspace. For example, if the equations have real coefficients, then the set of real points form a Newton-invariant subspace. Starting with any point for which Newton’s method quadratically converges to a solution, this article uses Smale’s α -theory to certifiably determine if the corresponding solution lies in a given Newton-invariant subspace or its complement. This approach generalizes the method developed in collaboration with F. Sottile for deciding the reality of the solution in the special case that the Newton iteration defines a real map. A description of the implementation in alphaCertified is presented along with examples. © Springer International Publishing AG 2017.

关键词： Iterative methods

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Privacy-Preserving Data Synthesis via Differentially Private Normalizing Flows with Application to Electronic Health Records Data

Privacy-Preserving Data Synthesis via Differentially Private...

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2023 AAAI Summer Symposium Series, SuSS 2023

作者： Su, Bingyue Wang, Yu Schiavazzi, Daniele Liu, Fang Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46530 United States

Medical data often contain sensitive personal information about individuals, posing significant limitations to it being shared or released for downstream learning and inferential tasks. We use normalizing flows (NF), a family of deep generative models, to estimate the probability density of a dataset with differential privacy (DP) guarantees, from which privacy-preserving synthetic data are generated and released. We apply the technique to an electronic health records dataset containing patients with pulmonary hypertension. We assess the learning and inferential utility of synthetic data by comparing the accuracy of hypertension predictions and the variational posterior distribution of the parameters in a physicsbased model. The results suggest that synthetic data generated via NF with DP can yield good utility at a reasonable privacy cost. Our study provides evidence and adds to the growing literature on the feasibility of generating synthetic medical data for sharing or obtaining inferences from medical data using deep generative models with formal privacy guarantees. © 2023, Association for the Advancement of Artificial Intelligence (***). All rights reserved.

关键词： Records management

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