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Communications in Theoretical Physics 2014年第6期61卷

作者： QIAN Xu SONG Song-He LI Wei-Bin Department of Mathematics and Systems Science National University of Defense Technology Program in Applied and Computational Mathematics Princeton University State Key Laboratory of High Performance Computing National University of Defense Technology

Considering the coupled nonlinear Schr?dinger system with multiply components, we provide a novel framework for constructing energy-preserving algorithms. In detail, based on the high order compact finite difference method, Fourier pseudospectral method and wavelet collocation method for spatial discretizations, a series of high accurate conservative algorithms are presented. The proposed algorithms can preserve the corresponding discrete charge and energy conservation laws exactly, which would guarantee their numerical stabilities during long time computations. Furthermore, several analogous multi-symplectic algorithms are constructed as comparison. Numerical experiments for the unstable plane waves will show the advantages of the proposed algorithms over long time and verify the theoretical analysis.

关键词： conservative algorithm coupled nonlinear Schrödinger equation multiply components unstable wave

An analysis of reconstruction noise from undersampled 4D flow MRI

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arXiv 2022年

作者： Partin, Lauren Schiavazzi, Daniele E. Long, Carlos A. Sing Department Of Applied And Computational Mathematics And Statistics University Of Notre Dame Notre DameIN United States Institute For Mathematical And Computational Engineering Institute For Biological And Medical Engineering Pontificia Universidad Católica De Chile Santiago Chile ANID - Millennium Science Initiative Program Millennium Nucleus Center For The Discovery Of Structures In Complex Data Chile ANID - Millennium Science Initiative Program Millennium Nucleus Center For Cardiovascular Magnetic Resonance Chile

Novel Magnetic Resonance (MR) imaging modalities can quantify hemodynamics but require long acquisition times, precluding its widespread use for early diagnosis of cardiovascular disease. To reduce the acquisition times, reconstruction methods from undersampled measurements are routinely used, that leverage representations designed to increase image compressibility. Reconstructed anatomical and hemodynamic images may present visual artifacts. Although some of these artifact are essentially reconstruction errors, and thus a consequence of undersampling, others may be due to measurement noise or the random choice of the sampled frequencies. Said otherwise, a reconstructed image becomes a random variable, and both its bias and its covariance can lead to visual artifacts;the latter leads to spatial correlations that may be misconstrued for visual information. Although the nature of the former has been studied in the literature, the latter has not received as much attention. In this study, we investigate the theoretical properties of the random perturbations arising from the reconstruction process, and perform a number of numerical experiments on simulated and MR aortic flow. Our results show that the correlation length remains limited to two to three pixels when a Gaussian undersampling pattern is combined with recovery algorithms based on 1-norm minimization. However, the correlation length may increase significantly for other undersampling patterns, higher undersampling factors (i.e., 8× or 16× compression), and different reconstruction methods. © 2022, CC BY-NC-ND.

关键词： Magnetic resonance

Erratum to: An Incompressible 2D Didactic Model with Singularity and Explicit Solutions of the 2D Boussinesq Equations

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Journal of Mathematical Fluid Mechanics 2014年第3期16卷 481-481页

作者： Dongho Chae Peter Constantin Jiahong Wu Department of Mathematics College of Natural Science Chung-Ang University Seoul South Korea Program in Applied and Computational Mathematics Princeton University Princeton USA Department of Mathematics Oklahoma State University Stillwater USA Department of Mathematics Chung-Ang University Seoul Republic of Korea

Global and local reduced models for interacting, heterogeneous agents

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arXiv 2021年

作者： Thiem, Thomas N. Kemeth, Felix P. Bertalan, Tom Laing, Carlo R. Kevrekidis, Ioannis G. Department of Chemical and Biological Engineering Princeton University United States Department of Chemical and Biomolecular Engineering Whiting School of Engineering Johns Hopkins University United States Department of Mechanical Engineering Massachusetts Institute of Technology United States School of Natural and Computational Sciences Massey University New Zealand Department of Applied Mathematics and Statistics Johns Hopkins University United States

Large collections of coupled, heterogeneous agents can manifest complex dynamical behavior presenting difficulties for simulation and analysis. However, if the collective dynamics lie on a low-dimensional manifold then the original agent-based model may be approximated with a simplified surrogate model on and near the low-dimensional space where the dynamics live. Analytically identifying such simplified models can be challenging or impossible, but here we present a data-driven coarse-graining methodology for discovering such reduced models. We consider two types of reduced models: globally-based models which use global information and predict dynamics using information from the whole ensemble, and locally-based models that use local information, that is, information from just a subset of agents close (close in heterogeneity space, not physical space) to an agent, to predict the dynamics of an agent. For both approaches we are able to learn laws governing the behavior of the reduced system on the low-dimensional manifold directly from time series of states from the agent-based system. These laws take the form of either a system of ordinary differential equations (ODEs), for the globally-based approach, or a partial differential equation (PDE) in the locally-based case. For each technique we employ a specialized artificial neural network integrator that has been templated on an Euler time stepper (i.e. a ResNet) to learn the laws of the reduced model. As part of our methodology, we utilize the proper orthogonal decomposition (POD) to identify the low-dimensional space of the dynamics. Our globally-based technique uses the resulting POD basis to define a set of coordinates for the agent states in this space, and then seeks to learn the time evolution of these coordinates as a system of ODEs. For the locally-based technique, we propose a methodology for learning a partial differential equation representation of the agents;the PDE law depends on the state variables and

关键词： Dynamics

On the Lagrangian-Eulerian Coupling in the Immersed Finite Element/Difference Method

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arXiv 2021年

作者： Lee, Jae H. Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States Department of Mechanical Engineering Johns Hopkins University BaltimoreMD United States Institute for Computational Medicine Johns Hopkins University BaltimoreMD United States Departments of Mathematics Applied Physical Sciences and Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Computational Medicine Program University of North Carolina School of Medicine Chapel HillNC United States McAllister Heart Institute University of North Carolina School of Medicine Chapel HillNC United States

The immersed boundary (IB) method is a non-body conforming approach to fluid-structure interaction (FSI) that uses an Eulerian description of the momentum, viscosity, and incompressibility of a coupled fluid-structure system and a Lagrangian description of the deformations, stresses, and resultant forces of the immersed structure. Integral transforms with Dirac delta function kernels couple the Eulerian and Lagrangian variables, and in practice, discretizations of these integral transforms use regularized delta function kernels. Many different kernel functions have been proposed, but prior numerical work investigating the impact of the choice of kernel function on the accuracy of the methodology has often been limited to simplified test cases or Stokes flow conditions that may not reflect the method’s performance in applications, particularly at intermediate-to-high Reynolds numbers, or under different loading conditions. This work systematically studies the effect of the choice of regularized delta function in several fluid-structure interaction benchmark tests using the immersed finite element/difference (IFED) method, which is an extension of the IB method that uses a finite element structural discretizations combined with a Cartesian grid finite difference method for the incompressible Navier-Stokes equations. Whereas the conventional IB method spreads forces from the nodes of the structural mesh and interpolates velocities to those nodes, the IFED formulation evaluates the regularized delta function on a collection of interaction points that can be chosen to be denser than the nodes of the Lagrangian mesh. This opens the possibility of using structural discretizations with wide node spacings that would produce gaps in the Eulerian force in nodally coupled schemes (e.g., if the node spacing is comparable to or broader than the support of the regularized delta function). Earlier work with this methodology suggested that such coarse structural meshes can yield imp

关键词： Fluid structure interaction

Robust Long-Term Growth Rate of Expected Utility for Leveraged ETFs

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arXiv 2023年

作者： Leung, Tim Park, Hyungbin Yeo, Heejun Department of Applied Mathematics Computational Finance & Risk Management Program University of Washington SeattleWA98195 United States Department of Mathematical Sciences Research Institute of Mathematics Seoul National University 1 Gwanak-ro Gwanak-gu Seoul08826 Korea Republic of Department of Mathematical Sciences Seoul National University 1 Gwanak-ro Gwanak-gu Seoul08826 Korea Republic of

This paper analyzes the robust long-term growth rate of expected utility and expected return from holding a leveraged exchange-traded fund (LETF). When the Markovian model parameters in the reference asset are uncertain, the robust long-term growth rate is derived by analyzing the worst-case parameters among an uncertainty set. We compute the growth rate and describe the optimal leverage ratio maximizing the robust long-term growth rate. To achieve this, the worst-case parameters are analyzed by the comparison principle, and the growth rate of the worst-case is computed using the martingale extraction method. The robust long-term growth rates are obtained explicitly under a number of models for the reference asset, including the geometric Brownian motion (GBM), Cox–Ingersoll–Ross (CIR), 3/2, and Heston and 3/2 stochastic volatility models. Additionally, we demonstrate the impact of stochastic interest rates, such as the Vasicek and inverse GARCH short rate models. This paper is an extended work of Leung and Park [2017]. Copyright © 2023, The Authors. All rights reserved.

关键词： Growth rate

Constraining reionization with the first measurement of the cross-correlation between the CMB optical-depth fluctuations and the Compton y-map

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Physical Review D 2021年第6期104卷 063514-063514页

作者： Toshiya Namikawa Anirban Roy Blake D. Sherwin Nicholas Battaglia David N. Spergel Department of Applied Mathematics and Theoretical Physics University of Cambridge Wilberforce Road Cambridge CB3 0WA United Kingdom Department of Astronomy Cornell University Ithaca New York 14853 USA Kavli Institute for Cosmology University of Cambridge Madingley Road Cambridge CB3 OHA United Kingdom Center for Computational Astrophysics Flatiron Institute 162 Fifth Avenue New York New York 10010 USA Department of Astrophysical Sciences Princeton University Peyton Hall Princeton New Jersey 08544 USA

We propose a new reionization probe that uses cosmic microwave background (CMB) observations; the cross-correlation between fluctuations in the CMB optical depth which probes the integrated electron density, δτ, and the Compton y-map which probes the integrated electron pressure. This cross-correlation is much less contaminated than the y-map power spectrum by late-time cluster contributions. In addition, this cross-correlation can constrain the temperature of ionized bubbles while the optical-depth fluctuations and kinetic Sunyaev Zel’dvich effect can not. We measure this new observable using a Planck y-map as well as a map of optical-depth fluctuations that we reconstruct from Planck CMB temperature data. We use our measurements to derive a first CMB only upper limit on the temperature inside ionized bubbles, Tb≲7.0×105 K (2σ). We also present future forecasts, assuming a fiducial model with characteristic reionization bubble size Rb=5 Mpc and Tb=5×104 K. The signal-to-noise ratio of the fiducial cross-correlation using a signal dominated y-map from the Probe of Inflation and Cosmic Origins (PICO) becomes ≃7 with CMB-S4 δτ and ≃13 with CMB-HD δτ. For the fiducial model, we predict that the CMB-HD—PICO cross-correlation should achieve an accurate measurement of the reionization parameters; Tb≃49800−5100+4500 K and Rb≃5.09−0.79+0.66 Mpc. Since the power spectrum of the electron density fluctuations is constrained by the δτ autospectrum, the temperature constraints should be only weakly model dependent on the details of the electron distributions and should be statistically representative of the temperature in ionized bubbles during reionization. This cross-correlation could, therefore, become an important observable for future CMB experiments.

关键词： Cosmic microwave background

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Wire Before You Walk

Wire Before You Walk

Asilomar Conference on Signals, Systems & Computers

作者： Tesfa Asmara Dhananjay Bhaskar Ian Adelstein Smita Krishnaswamy Michael Perlmutter Department of Computer Science Pomona College Claremont CA USA Mathematics and Statistics Pomona College Claremont CA USA Department of Genetics Yale School of Medicine New Haven CT USA Department of Mathematics Yale University New Haven CT USA Department of Computer Science Yale University New Haven CT USA Program for Applied Mathematics Yale University New Haven CT USA Computational Biology and Bioinformatics Program Yale University New Haven CT USA Wu Tsai Institute Yale University New Haven CT USA Department of Mathematics Boise State University Boise ID USA

Node embeddings aim to associate a vector to every vertex of a graph which can then be used for downstream tasks such as clustering, classification, or link prediction. Many popular node embeddings such as no $d$ e2vec and DeepWalk are based upon counting which nodes frequently co-occur in random walks of the graph. In this paper, we show that the performance of such algorithms can be improved by rewiring the edges of the graph through a variety of network indices before running DeepWalk. These rewirings effectively give the random walker an inductive bias and increase the accuracy of a logistic regression classifier applied to the node embedding on several benchmark data sets.

关键词：

Consensus dynamics and coherence in hierarchical small-world networks

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arXiv 2023年

作者： Liao, Yunhua Maama, Mohamed Aziz-Alaoui, M.A. Department of Mathematics Hunan University of Technology and Business Hunan Changsha410205 China Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation Hunan Changsha410205 China Applied Mathematics and Computational Science Program KAUST Thuwal23955-6900 Saudi Arabia Normandie Univ UNIHAVRE LMAH FR-CNRS-3335 ISCN Le Havre76600 France

The hierarchical small-world network is a real-world network. It models well the benefit transmission web of the pyramid selling in China and many other countries. In this paper, by applying the spectral graph theory, we study three important aspects of the consensus problem in the hierarchical small-world network: convergence speed, communication time-delay robustness, and network coherence. Firstly, we explicitly determine the Laplacian eigenvalues of the hierarchical small-world network by making use of its treelike structure. Secondly, we find that the consensus algorithm on the hierarchical small-world network converges faster than that on some well-studied sparse networks, but is less robust to time delay. The closed-form of the first-order and the second-order network coherence are also derived. Our result shows that the hierarchical small-world network has an optimal structure of noisy consensus dynamics. Therefore, we provide a positive answer to two open questions of Yi et al. Finally, we argue that some network structure characteristics, such as large maximum degree, small average path length, and large vertex and edge connectivity, are responsible for the strong robustness with respect to external perturbations. Copyright © 2023, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

“Pricing Dynamic Investment Fund Protection,” Hans U. Gerber and Gérard Pafumi, April 2000

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North American Actuarial Journal 2001年第1期5卷 153-157页

作者： Huang, Ya-Chun Shiu, Elias S.W. Program in Applied Mathematical and Computational Sciences University of Iowa Iowa City IA 52242-1419 United States Department of Statistics and Actuarial Science University of Iowa Iowa City IA 52242-1409 362 Schaeffer Hall United States Actuarial Science Department of Applied Mathematics Hong Kong Polytechnic University Hung Hom Hong Kong