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Structured Decompositions and Efficient Algorithms 2008

作者： Dahlke, Stephan Daubechies, Ingrid Elad, M. Kutyniok, Gitta Teschke, Gerd Philipps-Universität Marburg FB12 Mathematik und Informatik Hans-Meerwein Straße Lahnberge Marburg35032 Germany Princeton University Program in Applied and Computational Mathematics Fine Hall Washington Road PrincetonNJ08544-1000 United States Technion Computer Science Department Haifa Technion City32000 Israel Universität Osnabrück Institute of Mathematics Osnabrück49069 Germany Hochschule Neubrandenburg University of Applied Sciences Institute for Computational Mathematics in Science and Technology Brodaer Str. 2 Neubrandenburg17033 Germany

New emerging technologies such as high-precision sensors or new MRI machines drive us towards a challenging quest for new, more effective, and more daring mathematical models and algorithms. Therefore, in the last few years researchers have started to investigate different methods to efficiently represent or extract relevant information from complex, high dimensional and/or multimodal data. Efficiently in this context means a representation that is linked to the features or characteristics of interest, thereby typically providing a sparse expansion of such. Besides the construction of new and advanced ansatz systems the central question is how to design algorithms that are able to treat complex and high dimensional data and that efficiently perform a suitable approximation of the signal. One of the main challenges is to design new sparse approximation algorithms that would ideally combine, with an adjustable tradeoff, two properties: a provably good 'quality' of the resulting decomposition under mild assumptions on the analyzed sparse signal, and numerically efficient design. The topic is driven by applications as well as by theoretical questions. Therefore, the aim of this seminar was to bring together a good mixture of scientists with different backgrounds in order to discuss recent progress as well as new challenging perspectives. In particular, it was intended to strengthen the interaction of mathematicians and computer scientists. © 2009 Dagstuhl Seminar Proceedings. All Rights Reserved.

关键词： Clustering algorithms

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Coarse graining the dynamics of heterogeneous oscillators in networks with spectral gaps

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Physical Review E 2011年第3期84卷 036708-036708页

作者： Karthikeyan Rajendran Ioannis G. Kevrekidis Department of Chemical and Biological Engineering Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics (PACM) Princeton University Princeton New Jersey 08544 USA

We present a computer-assisted approach to coarse graining the evolutionary dynamics of a system of nonidentical oscillators coupled through a (fixed) network structure. The existence of a spectral gap for the coupling network graph Laplacian suggests that the graph dynamics may quickly become low dimensional. Our first choice of coarse variables consists of the components of the oscillator states—their (complex) phase angles—along the leading eigenvectors of this Laplacian. We then use the equation-free framework, circumventing the derivation of explicit coarse-grained equations, to perform computational tasks such as coarse projective integration, coarse fixed-point, and coarse limit-cycle computations. In a second step, we explore an approach to incorporating oscillator heterogeneity in the coarse-graining process. The approach is based on the observation of fast-developing correlations between oscillator state and oscillator intrinsic properties and establishes a connection with tools developed in the context of uncertainty quantification.

关键词： GRAINING COMPUTER-aided design LAPLACIAN operator EIGENVECTORS HARMONIC oscillators

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Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning

arXiv

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arXiv 2020年

作者： Weinan, E. Han, Jiequn Jentzen, Arnulf Department of Mathematics Princeton University United States Program in Applied and Computational Mathematics Princeton University United States Faculty of Mathematics and Computer Science University of Münster Germany

In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the Deep BSDE method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near *** Codes 65C05, 65K10, 65M75, 90C06 Copyright © 2020, The Authors. All rights reserved.

关键词： Monte Carlo methods

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Applications of mean field games in financial engineering and economic theory

arXiv

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arXiv 2020年

作者： Carmona, René DEPARTMENT OF OPERATIONS RESEARCH & FINANCIAL ENGINEERING BENDHEIM CENTER FOR FINANCE PROGRAM IN APPLIED & COMPUTATIONAL MATHEMATICS PRINCETON UNIVERSITY United States

This is an expanded version of the lecture given at the AMS Short Course on Mean Field Games, on January 13, 2020 in Denver CO. The assignment was to discuss applications of Mean Field Games in finance and economics. I need to admit upfront that several of the examples reviewed in this chapter were already discussed in book form. Still, they are here accompanied with discussions of, and references to, works which appeared over the last three years. Moreover, several completely new sections are added to show how recent developments in financial engineering and economics can benefit from being viewed through the lens of the Mean Field Game paradigm. The new financial engineering applications deal with bitcoin mining and the energy markets, while the new economic applications concern models offering a smooth transition between macro-economics and finance, and contract *** Codes 60, 91 © 2020, CC BY.

关键词： Finance

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Price of anarchy for mean field games

arXiv

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arXiv 2018年

作者： Carmona, René Graves, Christy V. Tan, Zongjun Operations Research and Financial Engineering Princeton University Partially United States Program in Applied and Computational Mathematics Princeton University Partially

The price of anarchy, originally introduced to quantify the inefficiency of selfish behavior in routing games, is extended to mean field games. The price of anarchy is defined as the ratio of a worst case social cost computed for a mean field game equilibrium to the optimal social cost as computed by a central planner. We illustrate properties of such a price of anarchy on linear quadratic extended mean field games, for which explicit computations are possible. A sufficient and necessary condition to have no price of anarchy is presented. Various asymptotic behaviors of the price of anarchy are proved for limiting behaviors of the coefficients in the model and numerics are *** Codes 60K35 Copyright © 2018, The Authors. All rights reserved.

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Reinforced dynamics for enhanced sampling in large atomic and molecular systems

arXiv

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arXiv 2017年

作者： Zhang, Linfeng Wang, Han Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 China CAEP Software Center for High Performance Numerical Simulation Huayuan Road 6 Beijing100088 China Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China

A new approach for efficiently exploring the configuration space and computing the free energy of large atomic and molecular systems is proposed, motivated by an analogy with reinforcement learning. There are two major components in this new approach. Like metadynamics, it allows for an efficient exploration of the configuration space by adding an adaptively computed biasing potential to the original dynamics. Like deep reinforcement learning, this biasing potential is trained on the fly using deep neural networks, with data collected judiciously from the exploration and an uncertainty indicator from the neural network model playing the role of the reward function. Parameterization using neural networks makes it feasible to handle cases with a large set of collective variables. This has the potential advantage that selecting precisely the right set of collective variables has now become less critical for capturing the structural transformations of the system. The method is illustrated by studying the full-atom, explicit solvent models of alanine dipeptide and tripeptide, as well as the system of a polyalanine-10 molecule with 20 collective variables. Copyright © 2017, The Authors. All rights reserved.

关键词： Wave functions

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Generalization error of GAN from the discriminator's perspective

arXiv

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arXiv 2021年

作者： Yang, Hongkang Weinan, E. Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics Princeton University United States Beijing Institute of Big Data Research China

The generative adversarial network (GAN) is a well-known model for learning high-dimensional distributions, but the mechanism for its generalization ability is not understood. In particular, GAN is vulnerable to the memorization phenomenon, the eventual convergence to the empirical distribution. We consider a simplified GAN model with the generator replaced by a density, and analyze how the discriminator contributes to generalization. We show that with early stopping, the generalization error measured by Wasserstein metric escapes from the curse of dimensionality, despite that in the long term, memorization is inevitable. In addition, we present a hardness of learning result for *** Codes 68T07, 62G07, 60-08 Copyright © 2021, The Authors. All rights reserved.

关键词： Probability distributions

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QUANTITATIVE SPECTRAL ANALYSIS OF ELECTROMAGNETIC SCATTERING. II: EVOLUTION SEMIGROUPS AND NON-PERTURBATIVE SOLUTIONS

arXiv

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arXiv 2021年

作者： Zhou, Yajun Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Peking University Beijing100871 China

We carry out quantitative studies on the Green operator Ĝ associated with the Born equation, an integral equation that models electromagnetic scattering, building the strong stability of the evolution semigroup {exp(−iτĜ)|τ ≥0} on polynomial compactness and the Arendt–Batty–Lyubich–Vũ theorem. The strongly-stable evolution semigroup inspires our proposal of a non-perturbative method to solve the light scattering problem and improve the Born *** Codes 35Q61, 45E99, 47B38, 47D06, 78A45 Copyright © 2021, The Authors. All rights reserved.

关键词： Spectrum analysis

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NUMERICAL ANALYSIS OF ELLIPTIC HEMIVARIATIONAL INEQUALITIES FOR SEMIPERMEABLE MEDIA

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Journal of computational mathematics 2019年第4期37卷 506-523页

作者： Weimin Han Ziping Huang Cheng Wang Wei Xu Program in Applied Mathematical and Computational Sciences (A MCS)& Department of Mathematics University of Iowa Iowa City IA 522/2 USA School of Mathematical Sciences Tongji University Shanghai 200092 China Tongji Zhejiang College J taxing 314051 China School of Mathematical Sciences Tojigji University Shanghai 200092 China School of Mathematical Sciences Tongji University Shanghai 200092 China Tongji Zhejiang College Jiaxing 314051 China

In this paper, we consider elliptic hemivariational inequalities arising in applications in semipermeable media. In its general form, the model includes both interior and boundary semipermeability terms. Detailed study is given on the hemivariational inequality in the case of isotropic and homogeneous semipermeable media. Solution existence and uniqueness of the problem are explored. Convergence of the Galerkin method is shown under the basic solution regularity available from the existence result. An optimal order error estimate is derived for the linear finite element solution under suitable solution regularity assumptions. The results can be readily extended to the study of more general hemivariational inequalities for non-isotropic and heterogeneous semipermeable media with interior semipermeability and/or boundary semiperrneability. Numerical examples are presented to show the performance of the finite element approximations;in particular, the theoretically predicted optimal first order convergence in H' norm of the linear element solutions is clearly observed.

关键词： Hemivariational inequality Interior semipermeability Boundary semipermeability Finite element method Error estimate

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Complexity measures for neural networks with general activation functions using path-based norms

arXiv

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arXiv 2020年

作者： Li, Zhong Ma, Chao Wu, Lei School of Mathematical Sciences Peking University China Department of Mathematics Stanford University United States Program in Applied and Computational Mathematics Princeton University United States

A simple approach is proposed to obtain complexity controls for neural networks with general activation functions. The approach is motivated by approximating the general activation functions with one-dimensional ReLU networks, which reduces the problem to the complexity controls of ReLU networks. Specifically, we consider two-layer networks and deep residual networks, for which path-based norms are derived to control complexities. We also provide preliminary analyses of the function spaces induced by these norms and a priori estimates of the corresponding regularized estimators. Copyright © 2020, The Authors. All rights reserved.

关键词： Chemical activation

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