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arXiv 2023年

作者： Soner, H. Mete Yan, Qinxin Department of Operations Research and Financial Engineering Princeton University PrincetonNJ08540 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States

Dynamic programming equations for mean field control problems with a separable structure are Eikonal type equations on the Wasserstein space. Standard differentiation using linear derivatives yield a direct extension of the classical viscosity theory. We use Fourier representation of the Sobolev norms on the space of measures, together with the standard techniques from the finite dimensional theory to prove a comparison result among semicontinuous sub and super solutions, obtaining a unique characterization of the value *** Codes 35D40, 35Q89, 49L12, 49L25, 60G99 © 2023, CC BY-NC-ND.

关键词： Viscosity

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Bayesian Spectral Graph Denoising with Smoothness Prior

Bayesian Spectral Graph Denoising with Smoothness Prior

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Annual Conference on Information Sciences and Systems (CISS)

作者： Sam Leone Xingzhi Sun Michael Perlmutter Smita Krishnaswamy Program for Applied Mathematics Yale University Department of Computer Science Yale University Department of Mathematics Boise State University Department of Genetics Yale School of Medicine Wu Tsai Institute Yale University FAIR Meta AI Computational Biology and Bioinformatics Program Yale University

Here we consider the problem of denoising features associated to complex data, modeled as signals on a graph, via a smoothness prior. This is motivated in part by settings such as single-cell RNA where the data is very high-dimensional, but its structure can be captured via an affinity graph. This allows us to utilize ideas from graph signal processing. In particular, we present algorithms for the cases where the signal is perturbed by Gaussian noise, dropout, and uniformly distributed noise. The signals are assumed to follow a prior distribution defined in the frequency domain which favors signals which are smooth across the edges of the graph. By pairing this prior distribution with our three models of noise generation, we propose Maximum A Posteriori (M.A.P.) estimates of the true signal in the presence of noisy data and provide algorithms for computing the M.A.P. Finally, we demonstrate the algorithms’ ability to effectively restore signals from white noise on image data and from severe dropout in single-cell RNA sequence data.

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Plane-wave analysis of a hyperbolic system of equations with relaxation in Rd

arXiv

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arXiv 2018年

作者： de Hoop, Maarten V. Liu, Jian-Guo Markowich, Peter A. Ussembayev, Nail S. Department of Computational and Applied Mathematics Rice University HoustonTX77005 United States Department of Mathematics Department of Physics Duke University DurhamNC27708 United States Applied Mathematics and Computer Science Program CEMSE Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Faculty of Mathematics University of Vienna ViennaA-1090 Austria Applied Mathematics and Computer Science Program CEMSE Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

We consider a multi-dimensional scalar wave equation with memory corresponding to the viscoelastic material described by a generalized Zener model. We deduce that this relaxation system is an example of a non-strictly hyperbolic system satisfying Majda's block structure condition. Well-posedness of the associated Cauchy problem is established by showing that the symbol of the spatial derivatives is uniformly diagonalizable with real eigenvalues. A long-time stability result is obtained by plane-wave analysis when the memory term allows for dissipation of energy.35L40 (Primary) 74D05, 35B35 (Secondary) Copyright © 2018, The Authors. All rights reserved.

关键词： Viscoelasticity

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Single-point kinetic energy density functionals: A pointwise kinetic energy density analysis and numerical convergence investigation

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Physical Review B 2015年第4期91卷 045124-045124页

作者： Junchao Xia Emily A. Carter Department of Mechanical and Aerospace Engineering Princeton University Princeton New Jersey 08544-5263 USA Program in Applied and Computational Mathematics and Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544-5263 USA

We present a comprehensive study of single-point kinetic energy density functionals (KEDFs) to be used in orbital-free density functional theory (DFT) calculations. We first propose a form of KEDFs based on a pointwise Kohn-Sham (KS) kinetic energy density (KED) and electron localization function (ELF) analysis. We find that the ELF and modified enhancement factor have a very strong and transferable correlation with the reduced density in various bulk metals. The non-self-consistent kinetic energy errors predicted by our KEDF models are decreased greatly compared to previously reported generalized gradient approximation (GGA) KEDFs. Second, we perform self-consistent calculations with various single-point KEDFs and investigate their numerical convergence behavior. We find striking numerical instabilities for previous GGA KEDFs; most of the GGA KEDFs fail to converge and show unphysical densities during the optimization. In contrast, our KEDFs demonstrate stable convergence, and their self-consistent results of various bulk properties agree reasonably well with KSDFT. A further detailed KED analysis reveals an interesting bifurcation phenomenon in defective metals and alloys, which may shed light on directions for future KEDF development.

关键词： KINETIC energy DENSITY functionals DENSITY functional theory ELECTRON research BIFURCATION theory

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Riccati conditioning and sensitivity for a MinMax controlled cable-mass system

Riccati conditioning and sensitivity for a MinMax controlled...

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IEEE Conference on Decision and Control

作者： Lizette Zietsman Katie A. Evans J. Teye Brown Richard A. Idowu Faculty of the Interdisciplinary Center for Applied Mathematics Virginia Technology Blacksburg VA USA Faculty of Mathematics and Statistics Louisiana Tech University Ruston LA USA Computational Analysis and Modeling Program Louisiana Tech University Ruston LA USA

ISBN: (纸本)9781424431236

In this paper we present a numerical study that investigates the relationship between the parameter q, used in the design of the MinMax controller, and the conditioning of the approximate algebraic Riccati equations, the sensitivity of the eigenvalues of I-¿ 2 P¿ to ¿ as well as the effect of q on the stability radia and the stability margin of the system. In order to guarantee accurate numerical solutions to the approximate Riccati equations, the Riccati equations must remain well-conditioned for the values of ¿ that are considered. This condition number reflects the combined sensitivity of the Riccati equations to the system inputs A, B, R, C and ¿. In addition, we also consider the sensitivity of the eigenvalues of I-¿ 2 P¿ to ¿. We study the possibility of these sensitivities serving as an indication of the largest value of ¿ for which I-¿ 2 P¿ remains positive definite. This sensitivity could also serve as an indication of the accuracy of the computation of I-¿ 2 P¿. Lastly, in order to design efficient low order controllers, it is important to ensure the robustness of the design. Stability radius and stability margin serve as measures of the robustness of the controller. A one-dimensional nonlinear cable mass system is considered to illustrate these ideas and numerical results are presented.

关键词： Riccati equations Minimax techniques Control systems Eigenvalues and eigenfunctions Robust control Robust stability Optimal control Contracts mathematics Military computing

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Reply to “Comment on ‘Single-point kinetic energy density functionals: A pointwise kinetic energy density analysis and numerical convergence investigation’ ”

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Physical Review B 2015年第11期92卷 117102-117102页

We find that the multivalued character of the G factor as a function of the reduced gradient (s) still exists after accounting for pseudopotential artifacts and the kinetic energy global upper bound. We also find that the VT84F functional indeed exhibits stable convergence and more reasonable results for self-consistent bulk properties compared to other generalized gradient approximation (GGA) kinetic energy density functionals (KEDFs) that we tested earlier. However, VT84F generally yields overestimated equilibrium volumes, which may result from its inability (as with all GGAs) to reproduce the G−s multivalued character. The analogous failure to predict the multivalued character of G as a function of the reduced density (d) is also likely to be responsible for the inaccuracy of our vWGTF functionals reported earlier. Our multivaluedness analysis therefore does not impugn any particular GGA KEDF. Instead, it merely confirms the importance of pointwise analysis for improving KEDFs by emphasizing the need to resolve the multivaluedness of G with respect to various density variables.

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Density-decomposed orbital-free density functional theory for covalently bonded molecules and materials

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Physical Review B 2012年第23期86卷 235109-235109页

We propose a density decomposition scheme using a Wang-Govind-Carter- (WGC-) based kinetic energy density functional (KEDF) to accurately and efficiently simulate various covalently bonded molecules and materials within orbital-free (OF) density functional theory (DFT). By using a local, density-dependent scale function, the total density is decomposed into a highly localized density within covalent bond regions and a flattened delocalized density, with the former described by semilocal KEDFs and the latter treated by the WGC KEDF. The new model predicts reasonable equilibrium volumes, bulk moduli, and phase-ordering energies for various semiconductors compared to Kohn-Sham (KS) DFT benchmarks. The decomposition formalism greatly improves numerical stability and accuracy, while retaining computational speed compared to simply applying the original WGC KEDF to covalent materials. The surface energy of Si(100) and various diatomic molecule properties can be stably calculated and also agree well with KSDFT benchmarks. This linear-scaled, computationally efficient, density-partitioned, multi-KEDF scheme opens the door to large-scale simulations of molecules, semiconductors, and insulators with OFDFT.

关键词： PHYSICAL sciences research MOLECULES SEMICONDUCTORS BENCHMARKING (Management) CRYSTALS SURFACE energy COVALENT bonds (Chemistry)

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Dense sphere packings from optimized correlation functions

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Physical Review E 2009年第3期79卷 031123-031123页

作者： Adam B. Hopkins Frank H. Stillinger Salvatore Torquato Department of Chemistry Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA and School of Natural Sciences Institute for Advanced Study Princeton New Jersey 08544 USA

Elementary smooth functions (beyond contact) are employed to construct pair correlation functions that mimic jammed disordered sphere packings. Using the g2-invariant optimization method of Torquato and Stillinger [J. Phys. Chem. B 106, 8354 (2002)], parameters in these functions are optimized under necessary realizability conditions to maximize the packing fraction ϕ and average number of contacts per sphere Z. A pair correlation function that incorporates the salient features of a disordered packing and that is smooth beyond contact is shown to permit a ϕ of 0.6850: this value represents a 45% reduction in the difference between the maximum for congruent hard spheres in three dimensions, π/18≈0.7405 and 0.64, the approximate fraction associated with maximally random jammed packings in three dimensions. We show that, surprisingly, the continued addition of elementary functions consisting of smooth sinusoids decaying as r−4 permits packing fractions approaching π/18. A translational order metric is used to discriminate between degrees of order in the packings presented. We find that to achieve higher packing fractions, the degree of order must increase, which is consistent with the results of a previous study [Torquato et al., Phys. Rev. Lett. 84, 2064 (2000)].

关键词： Packing

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Local two-sample testing over graphs and point-clouds by random-walk distributions

arXiv

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arXiv 2020年

作者： Landa, Boris Qu, Rihao Chang, Joseph Kluger, Yuval Program in Applied Mathematics Yale University Department of Pathology Yale University School of Medicine Department of Statistics and Data Science Yale University Interdepartmental Program in Computational Biology and Bioinformatics Yale University Department of Immunology Yale School of Medicine

Rejecting the null hypothesis in two-sample testing is a fundamental tool for scientific discovery. Yet, aside from concluding that two samples do not come from the same probability distribution, it is often of interest to characterize how the two distributions differ. Given samples from two densities f1and f0, we consider the task of localizing occurrences of the inequality f1> f0. To avoid the challenges associated with high-dimensional space, we propose a general hypothesis testing framework where hypotheses are formulated adaptively to the data by conditioning on the combined sample from the two densities. We then investigate a special case of this framework where the notion of locality is captured by a random walk on a weighted graph constructed over this combined sample. We derive a tractable testing procedure for this case employing a type of scan statistic, and provide non-asymptotic lower bounds on the power and accuracy of our test to detect whether f1> f0in a local sense. Furthermore, we characterize the test's consistency according to a certain problem-hardness parameter, and show that our test achieves the minimax detection rate for this parameter. We conduct numerical experiments to validate our method, and demonstrate our approach on two real-world applications: detecting and localizing arsenic well contamination across the United States, and analyzing two-sample single-cell RNA sequencing data from melanoma *** Codes 62G10 Copyright © 2020, The Authors. All rights reserved.

关键词： Random processes

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Modeling subgrid-scale force and divergence of heat flux of compressible isotropic turbulence by artificial neural network

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Physical Review Fluids 2019年第10期4卷 104605-104605页

作者： Chenyue Xie Ke Li Chao Ma Jianchun Wang Shenzhen Key Laboratory of Complex Aerospace Flows Center for Complex Flows and Soft Matter Research Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Shenzhen 518055 People's Republic of China Institute of Computational Mathematics and Scientific Engineering Computing Chinese Academy of Sciences Beijing 100190 People's Republic of China The Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

In this paper, the subgrid-scale (SGS) force and the divergence of SGS heat flux of compressible isotropic turbulence are modeled directly by an artificial neural network (ANN), which serves as a data-driven SGS modeling tool for large-eddy simulations (LESs). The unclosed SGS force and divergence of SGS heat flux are modeled based on the local stencil geometry with Galilean invariance. The input features include the first-order and second-order derivatives of filtered velocity and temperature, filtered density, and its first-order derivative. It is shown that the proposed ANN-F7 model shows an advantage over the gradient model in the a priori test. Specifically, the ANN-F7 model gives larger correlation coefficients and smaller relative errors than the gradient model. In an a posteriori analysis, the ANN-F7 model performs better than the dynamic Smagorinsky model (DSM) and dynamic mixed model (DMM) in the prediction of the statistical properties of flow fields at the Taylor microscale Reynolds number Reλ ranging from 180 to 250. The DSM and DMM models lead to the typical tilted spectral distribution of velocity, where low wave numbers are too energy rich, while those near the cutoff are damped too strongly. In contrast, it is shown that the velocity spectrum predicted by the ANN-F7 model almost overlaps with the filtered direct numerical simulation data. Besides, the ANN-F7 model reconstructs the probability density functions of SGS force and divergence of SGS heat flux much better than the DSM and DMM models. An artificial neural network with reasonable physical input features can deepen our understanding of turbulence modeling.

关键词： Compressible flows Turbulence Artificial neural networks

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