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RECURSIVE SAMPLING OF PLANAR GRAPHS AND FRACTAL PROPERTIES OF A TWO-DIMENSIONAL QUANTUM GRAVITY

International Journal of Modern Physics C 1990年第1期01卷 165-179页

作者： MICHAEL E. AGISHTEIN ALEXANDER A. MIGDAL Physics Department and Program in Applied and Computational Mathematics Princeton University Fine Hall Princeton NJ 08544 USA

We describe a new approach to the Monte-Carlo simulations of two-dimensional gravity. Standard dynamical triangulation technique was combined with results of direct enumeration of the cubic graphs. As a result we were able to build large (128K vertices) statistically independent random graphs directly. The quantitative correspondence between our results and those obtained by standard methods has been observed. The algorithm proved to be so efficient that we were able to conduct all the simulations, which usually require the most powerful computers, on an Iris workstation. An opportunity to generate large random graphs allowed us to observe that the internal geometry of random surfaces is more complicated than simple fractals. External geometry also proved to be rather peculiar.

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A priori estimates for classification problems using neural networks

arXiv

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arXiv 2020年

作者： Weinan, E. Wojtowytsch, Stephan Program for Applied and Computational Mathematics Department of Mathematics Princeton University PrincetonNJ08544 United States Program for Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider binary and multi-class classification problems using hypothesis classes of neural networks. For a given hypothesis class, we use Rademacher complexity estimates and direct approximation theorems to obtain a priori error estimates for regularized loss *** Codes 68T07, 60-08 Copyright © 2020, The Authors. All rights reserved.

关键词： Classification (of information)

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Computing Optimal Interfacial Structure of Modulated Phases

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Communications in computational Physics 2017年第1期21卷 1-15页

作者： Jie Xu Chu Wang An-Chang Shi Pingwen Zhang LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China Program in Applied and Computational Mathematics Princeton UniversityPrincetonNew Jersey 08544USA Department of Physics and Astronomy McMaster UniversityHamiltonOntario L8S4M1Canada

We propose a general framework of computing interfacial structures between two modulated *** we propose to use a computational box consisting of two half spaces,each occupied by a modulated phase with given position and *** boundary conditions and basis functions are chosen to be commensurate with the bulk *** observe that the ordered nature of modulated structures stabilizes the interface,which enables us to obtain optimal interfacial structures by searching local minima of the free energy *** framework is applied to the Landau-Brazovskii model to investigate interfaces between modulated phases with different relative positions and *** types of novel complex interfacial structures emerge from the calculations.

关键词： Interface modulated phase metastable state compatibility Landau-Brazovskiimodel

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Three-dimensional dynamics and transition to turbulence in the wake of bluff objects

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Journal of Fluid Mechanics 1992年第1期238卷 1-30页

作者： Karniadakis, George Em Triantafyllou, George S. Department of Mechanical and Aerospace Engineering and Program of Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States The Levich Institute and Department of Mechanical Engineering The City College of New York New York NY 10031 United States

The wakes of bluff objects and in particular of circular cylinders are known to undergo a ‘fast ’ transition, from a laminar two-dimensional state a t Reynolds number 200 to a turbulent state a t Reynolds number 400. The process has been documented in several eXperimental mvestigations, but the underlying physical mechanisms have remained largely unknown so far. In this paper, the transition process is investigated numerically, through direct simulation of the NavierStokes equations at representative Reynolds numbers, up to 500. A high-order timeaccurate, miXed spectral/spectral element technique is used. It is shown that the wake first becomes three-dimensional, as a result of a secondary instability of the two-dimensional vorteX street. This secondary instability appears at a Reynolds number close to 200. For slightly supercritical Reynolds numbers, a harmonic state develops, in which the flow oscillates at its fundamental frequency (Strouhal number) around a spanwise modulated time-average flow. In the near wake the modulation wavelength of the time-average flow is half of the spanwise wavelength of the perturbation flow, consistently with linear instability theory. The vorteX filaments have a spanwise wavy shape in the near wake, and form rib-like structures further downstream. At higher Reynolds numbers the three-dimensional flow oscillation undergoes a period-doubling bifurcation, in which the flow alternates between two different states. Phase-space analysis of the flow shows that the basic limit cycle has branched into two connected limit cycles. In physical space the period doubling appears as the shedding of two distinct types of vorteX filaments. Further increases of the Reynolds number result in a cascade of period-doubling bifurcations, which create a chaotic state in the flow at a Reynolds number of about 500. The flow is characterized by broadband power spectra, and the appearance intermittent phenomena. It is concluded that the wake undergoes transit

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Novel Conservative Methods for Schrödinger Equations with Variable Coefficients over Long Time

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Communications in computational Physics 2014年第3期15卷 692-711页

作者： Xu Qian Yaming Chen Songhe Song Department of Mathematics and Systems Science and State Key Laboratory of High Performance ComputingNational University of Defense TechnologyChangsha 410073P.R.China Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA School of Mathematical Sciences Queen Mary University of LondonLondon E14NSUK

In this paper,we propose a wavelet collocation splitting(WCS)method,and a Fourier pseudospectral splitting(FPSS)method as comparison,for solving onedimensional and two-dimensional Schrödinger equations with variable coefficients in quantum *** two methods can preserve the intrinsic properties of original problems as much as *** splitting technique increases the computational ***,the error estimation and some conservative properties are *** is proved to preserve the charge conservation *** global energy and momentum conservation laws can be preserved under several *** experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

关键词： Schrödinger equation wavelet collocation method splitting technique conservative property

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Deep potentials for materials science

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Materials Futures 2022年第2期1卷 89-115页

作者： Tongqi Wen Linfeng Zhang Han Wang Weinan E David J Srolovitz Department of Mechanical Engineering The University of Hong KongHong KongHong Kong Special Administrative Region of China DP Technology BeijingPeople’s Republic of China AI for Science Institute BeijingPeople’s Republic of China Laboratory of Computational Physics Institute of Applied Physics and Computational MathematicsBeijingPeople’s Republic of China HEDPS CAPTCollege of EngineeringPeking UniversityBeijingPeople’s Republic of China School of Mathematical Sciences Peking UniversityBeijingPeople’s Republic of China Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJUnited States of America International Digital Economy Academy(IDEA) ShenzhenPeople’s Republic of China

To fill the gap between accurate(and expensive)ab initio calculations and efficient atomistic simulations based on empirical interatomic potentials,a new class of descriptions of atomic interactions has emerged and been widely applied;*** learning potentials(MLPs).One recently developed type of MLP is the deep potential(DP)*** this review,we provide an introduction to DP methods in computational materials *** theory underlying the DP method is presented along with a step-by-step introduction to their development and *** also review materials applications of DPs in a wide range of materials *** DP Library provides a platform for the development of DPs and a database of extant *** discuss the accuracy and efficiency of DPs compared with ab initio methods and empirical potentials.

关键词： deep potential atomistic simulation machine learning potential neural network

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Locality and low-dimensions in the prediction of natural experience from fMRI

Locality and low-dimensions in the prediction of natural exp...

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21st Annual Conference on Neural Information Processing Systems, NIPS 2007

作者： Meyer, François G. Stephens, Greg J. Center for the Study of Brain Mind and Behavior Program in Applied and Computational Mathematics Princeton University United States Center for the Study of Brain Mind and Behavior Department of Physics Princeton University United States

ISBN: (纸本)160560352X

Functional Magnetic Resonance Imaging (fMRI) provides dynamical access into the complex functioning of the human brain, detailing the hemodynamic activity of thousands of voxels during hundreds of sequential time points. One approach towards illuminating the connection between fMRI and cognitive function is through decoding;how do the time series of voxel activities combine to provide information about internal and external experience? Here we seek models of fMRI decoding which are balanced between the simplicity of their interpretation and the effectiveness of their prediction. We use signals from a subject immersed in virtual reality to compare global and local methods of prediction applying both linear and nonlinear techniques of dimensionality reduction. We find that the prediction of complex stimuli is remarkably low-dimensional, saturating with less than 100 features. In particular, we build effective models based on the decorrelated components of cognitive activity in the classically-defined Brodmann areas. For some of the stimuli, the top predictive areas were surprisingly transparent, including Wernicke's area for verbal instructions, visual cortex for facial and body features, and visual-temporal regions for velocity. Direct sensory experience resulted in the most robust predictions, with the highest correlation (c ∼ 0:8) between the predicted and experienced time series of verbal instructions. Techniques based on non-linear dimensionality reduction (Laplacian eigenmaps) performed similarly. The interpretability and relative simplicity of our approach provides a conceptual basis upon which to build more sophisticated techniques for fMRI decoding and offers a window into cognitive function during dynamic, natural experience.

关键词： Forecasting

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Improving the robustness of the immersed interface method through regularized velocity reconstruction

arXiv

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arXiv 2025年

作者： Sun, Qi Kolahdouz, Ebrahim M. Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States Department of 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

Robust, broadly applicable fluid-structure interaction (FSI) algorithms remain a challenge for computational mechanics. Efforts in this area are driven by the need to enhance predictive accuracy and efficiency in FSI simulations, align with experimental observations, and unravel complex multiscale and multiphysics phenomena, while addressing challenges in developing more robust and efficient methodologies. In previous work, we introduced an immersed interface method (IIM) for discrete surfaces and an extension based on an immersed Lagrangia-Eulerian (ILE) coupling strategy for modeling FSI involving complex geometries. The ability of the method to sharply resolve stress discontinuities induced by singular immersed boundary forces in the presence of low-regularity geometrical representations makes it a compelling choice for three-dimensional modeling of complex geometries in diverse engineering applications. Although the IIM we previously introduced offers many desirable advantages, it also imposes a restrictive mesh factor ratio, requiring the surface mesh to be coarser than the fluid grid to ensure stability. This is because when the mesh factor ratio constraint is not satisfied, parts of the structure motion are not controlled by the discrete FSI system. This constraint can significantly increase computational costs, particularly in applications involving multiscale geometries with highly localized complexity or fine-scale features. To address this limitation, we devise a stabilization strategy for the velocity restriction operator inspired by Tikhonov regularization. This study demonstrates that using a stabilized velocity restriction operator in IIM enables a broader range of structure-to-fluid grid-size ratios without compromising accuracy or altering the flow dynamics. This advancement significantly broadens the applicability of the method to real-world FSI problems involving complex geometries and dynamic conditions, offering a robust and computationally effi

关键词： Mesh generation

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Non-Local Patch Regression: Robust Image Denoising in Patch Space

Non-Local Patch Regression: Robust Image Denoising in Patch ...

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IEEE International Conference on Acoustics, Speech, and Signal Processing

作者： Kunal N. Chaudhury Amit Singer Program in Applied and Computational Mathematics (PACM) Princeton University PACM and Department of Mathematics Princeton University

ISBN: (纸本)9781479903573

It was recently demonstrated in [13] that the denoising performance of Non-Local Means (NLM) can be improved at large noise levels by replacing the mean by the robust Euclidean median. Numerical experiments on synthetic and natural images showed that the latter consistently performed better than NLM beyond a certain noise level, and significantly so for images with sharp edges. The Euclidean mean and median can be put into a common regression (on the patch space) framework, in which the l~2 norm of the residuals is considered in the former, while the l~1 norm is considered in the latter. The natural question then is what happens if we consider l~p (0 < p < 1) regression? We investigate this possibility in this paper.

关键词： Image denoising Non-local means Non-local Euclidean medians Edges Inlier-outlier model Robustness Sparsity Non-convex optimization Iteratively reweighted least squares

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An immersed interface method for incompressible flows and geometries with sharp features

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Journal of computational Physics 2025年 537卷

作者： Michael J. Facci Ebrahim M. Kolahdouz Boyce E. Griffith Department of Mathematics University of North Carolina Chapel Hill NC USA Flatiron Institute Simons Foundation New York NY USA Department of Biomedical Engineering University of North Carolina Chapel Hill NC USA Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel Hill NC USA Computational Medicine Program University of North Carolina School of Medicine Chapel Hill NC USA McAllister Heart Institute University of North Carolina School of Medicine Chapel Hill NC USA

The immersed interface method (IIM) for models of fluid flow and fluid-structure interaction imposes jump conditions that capture stress discontinuities generated by forces that are concentrated along immersed boundaries. Most prior work using the IIM for fluid dynamics applications has focused on smooth interfaces, but boundaries with sharp features such as corners and edges can appear in practical analyses, particularly on engineered structures. The present study builds on our work to integrate finite element-type representations of interface geometries with the IIM. Initial realizations of this approach used a continuous Galerkin (CG) finite element discretization for the boundary, but as we show herein, these approaches generate large errors near sharp geometrical features. To overcome this difficulty, this study introduces an IIM approach using a discontinuous Galerkin (DG) representation of the jump conditions. Numerical examples explore the impacts of different interface representations on accuracy for both smooth and sharp boundaries, particularly flows interacting with fixed interface configurations. We demonstrate that using a DG approach provides accuracy that is comparable to the CG method for smooth cases. Further, we identify a time step size restriction for the CG representation that is directly related to the sharpness of the geometry. In contrast, time step size restrictions imposed by DG representations are demonstrated to be nearly insensitive to the presence of sharp features.

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