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Phase unwinding, or invariant subspace decompositions of hardy spaces

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arXiv 2017年

作者： Coifman, Ronald R. Peyrie, Jacques Department of Mathematics Program in Applied Mathematics Yale University New HavenCT06510 United States Laboratoire de Mathématiques d'Orsay Univ. Paris-Sud CNRS Université Paris-Saclay Orsay91405 France

We consider orthogonal decompositions of invariant subspaces of Hardy spaces, these relate to the Blaschke based phase unwinding decompositions [1, 2, 7]. We prove convergence in Lp. In particular we build an explicit multiscale wavelet basis. We also obtain an explicit unwindinig decomposition for the singular inner function, exp 2iπ/*** Codes 30B50, 30A10, 42C40, 65T99 Copyright © 2017, The Authors. All rights reserved.

关键词： Functions

Structural Effects and Aggregation in a Social-Network Model of Natural Resource Consumption

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IFAC-PapersOnLine 2017年第1期50卷 7675-7680页

作者： Manzoor T. Rovenskaya E. Muhammad A. Department of Electrical Engineering Lahore University of Management Sciences (LUMS) Lahore Pakistan Center for Water Informatics & Technology (WIT) Lahore University of Management Sciences (LUMS) Pakistan Advanced Systems Analysis Program International Institute for Applied Systems Analysis (IIASA) Laxenburg Austria Faculty of Computational Mathematics and Cybernetics Lomonosov Moscow State University Moscow Russian Federation

In this paper we consider a networked system of natural-resource consumption, where the agents are governed by a recently reported model of consumer psychology Dynamics of the consumption for each agent are influenced by the state of the resource and consumption of her neighbors. The process is parameterized by the psychological characteristics of each agent. This study extends the original model to incorporate the underlying network topology and explores the effects of aggregation via densely-connected communities present in the network. The exercise yields new interpretations of the predictions made by the original model in context of the influence network. We present aggregation mechanisms first for certain canonical structures of the consuming population and later on for a class of non-canonically structured populations. In the end we present an approximate aggregation scheme for populations where only some inaccurate information of the consumer characteristics is available. © 2017

关键词： Control in economics identification of environmental systems Modeling Multi-agent systems Natural resources management Social Networks

Unsupervised extraction of phenotypes from cancer clinical notes for association studies

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arXiv 2019年

作者： Stark, Stefan G. Hyland, Stephanie L. Pradier, Melanie F. Lehmann, Kjong-Van Wicki, Andreas Perez-Cruz, Fernando Vogt, Julia E. Rätsch, Gunnar Computational Biology Program Memorial Sloan Kettering Cancer Center New York United States Tri-Institutional Ph.D. Program in Computational Biology and Medicine Weill Cornell Medicine New York United States Department of Computer Science ETH Zürich Zürich Switzerland Medical Informatics Group University Hospital Zürich Zürich Switzerland Swiss Institute for Bioinformatics Zurich Switzerland Department of Biology ETH Zürich Zürich Switzerland Department of Signal Processing and Information Theory University Carlos III in Madrid Leganés Spain School of Engineering and Applied Sciences Harvard University CambridgeMA United States Department of Biomedicine University of Basel Basel Switzerland Tumorzentrum University Hospital Basel Basel Switzerland Swiss Data Science Center ETH Zürich and EPFL Lausanne Switzerland Department of Mathematics and Computer Science University of Basel Basel Switzerland

The recent adoption of Electronic Health Records (EHRs) by healthcare providers has introduced an important source of data that provides detailed and highly specific insights into patient phenotypes over large cohorts. These datasets, in combination with machine learning and statistical approaches, generate new opportunities for research and clinical care. However, many methods require the patient representations to be in structured formats, while the information in the EHR is often locked in unstructured text designed for human readability. In this work, we develop the methodology to automatically extract clinical features from clinical narratives from large EHR corpora without the need for prior knowledge. We consider medical terms and sentences appearing in clinical narratives as atomic information units. We propose an efficient clustering strategy suitable for the analysis of large text corpora and utilize the clusters to represent information about the patient compactly. Additionally, we define the sentences on ontologic and natural language vocabularies to automatically detect pertinent combinations of concepts present in the corpus, even when an ontology is not available. To demonstrate the utility of our approach, we perform an association study of clinical features with somatic mutation profiles from 4,007 cancer patients and their tumors. We apply the proposed algorithm to a dataset consisting of .65 thousand documents with a total of .3.2 million sentences. After correcting for cancer type and other confounding factors, we identify a total of 340 significant statistical associations between the presence of somatic mutations and clinical features. We annotated these associations according to their novelty and we report several known associations. We also propose 37 plausible, testable hypothesis for associations where the underlying biological mechanism does not appear to be known. These results illustrate that the automated discovery of clinical features

关键词： Diseases

Dynamical Sampling with Random Noise

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Dynamical Sampling with Random Noise

International Conference on Sampling Theory and Applications

作者： Akram Aldroubi Longxiu Huang Ilya Krishtal Roy Lederman Department of Mathematics Vanderbilt University Department of Mathematics Northern Illinois University Program in Applied and Computational Mathematics Princeton University

ISBN: (纸本)9781538615669

In this paper we consider a system of dynamical sampling, i.e. sampling a signal f that evolves in time under the action of an evolution operator A. We discuss the error in the recovery of the original signal when the samples are corrupted by additive, independent and identically distributed (i.i.d) noise. We focus on the study of the mean squared error E(‖ε_n‖_2~2) between the original signal and the reconstructed signal obtained by solving a least squares problem. In the theoretical part, we give a formula for E(‖ε_n‖_2~2) and prove that E(‖ε_n‖_2~2) decreases as the number of the samples increases. In addition, we discuss several numerical experiments that verify the theoretical results.

关键词： original signal random noise time action formula Identically distributed Mean square error signals

Solving High-Dimensional Partial Differential Equations Using Deep Learning

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arXiv 2017年

作者： Han, Jiequn Jentzen, Arnulf Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics ETH Zürich Rämistrasse 101 Zürich8092 Switzerland Department of Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black-Scholes equation, the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up new possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their inter-relationships. Copyright © 2017, The Authors. All rights reserved.

关键词： Partial differential equations

Limited angle acousto-electrical tomography

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arXiv 2017年

作者： Hubmer, Simon Knudsen, Kim Li, Changyou Sherina, Ekaterina Johannes Kepler University Linz Doctoral Program Computational Mathematics Altenbergerstraße 69 LinzA-4040 Austria Technical University of Denmark Department of Applied Mathematics and Computer Science Asmussens Allé Kongens Lyngby2800 Denmark Technical University of Denmark Department of Applied Mathematics and Computer Science Asmussens Allé Kongens Lyngby2800 Denmark School of Electronics and Information Northwestern Polytechnical University Xian China Technical University of Denmark Department of Applied Mathematics and Computer Science Asmussens Allé Kongens Lyngby2800 Denmark

This paper considers the reconstruction problem in Acousto-Electrical Tomography, i.e., the problem of estimating a spatially varying conductivity in a bounded domain from measurements of the internal power densities resulting from different prescribed boundary conditions. Particular emphasis is placed on the limited angle scenario, in which the boundary conditions are supported only on a part of the boundary. The reconstruction problem is formulated as an optimization problem in a Hilbert space setting and solved using Landweber iteration. The resulting algorithm is implemented numerically in two spatial dimensions and tested on simulated data. The results quantify the intuition that features close to the measurement boundary are stably reconstructed and features further away are less well reconstructed. Finally, the ill-posedness of the limited angle problem is quantified numerically using the singular value decomposition of the corresponding linearized problem.65J22, 35R30, 65M32 Copyright © 2017, The Authors. All rights reserved.

关键词： Boundary conditions

Cluster synchronization of diffusively-coupled nonlinear systems: A contraction based approach

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arXiv 2017年

作者： Aminzare, Zahra Dey, Biswadip Davison, Elizabeth N. Leonard, Naomi Ehrich Program in Applied and Computational Mathematics Princeton University NJ08544 United States Department of Mechanical and Aerospace Engineering Princeton University NJ08540 United States

Finding the conditions that foster synchronization in networked oscillatory systems is critical to understanding a wide range of biological and mechanical systems. However, the conditions proved in the literature for synchronization in nonlinear systems with linear coupling, such as has been used to model neuronal networks, are in general not strict enough to accurately determine the system behavior. We leverage contraction theory to derive new sufficient conditions for cluster synchronization in terms of the network structure, for a network where the intrinsic nonlinear dynamics of each node may differ. Our result requires that network connections satisfy a cluster-input-equivalence condition, and we explore the influence of this requirement on network dynamics. For application to networks of nodes with neuronal spiking dynamics, we show that our new sufficient condition is tighter than those found in previous analyses which used nonsmooth Lyapunov functions. Improving the analytical conditions for when cluster synchronization will occur based on network configuration is a significant step toward facilitating understanding and control of complex oscillatory systems. Copyright © 2017, The Authors. All rights reserved.

关键词： Synchronization

Local-order metric for condensed-phase environments

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Physical Review B 2018年第6期97卷 064105-064105页

作者： Fausto Martelli Hsin-Yu Ko Erdal C. Oğuz Roberto Car Department of Chemistry Princeton University Princeton New Jersey 08544 USA Institut für Theoretische Physik II Heinrich-Heine-Universität D-40225 Düsseldorf Germany Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

We introduce a local order metric (LOM) that measures the degree of order in the neighborhood of an atomic or molecular site in a condensed medium. The LOM maximizes the overlap between the spatial distribution of sites belonging to that neighborhood and the corresponding distribution in a suitable reference system. The LOM takes a value tending to zero for completely disordered environments and tending to one for environments that perfectly match the reference. The site-averaged LOM and its standard deviation define two scalar order parameters, S and δS, that characterize with excellent resolution crystals, liquids, and amorphous materials. We show with molecular dynamics simulations that S, δS, and the LOM provide very insightful information in the study of structural transformations, such as those occurring when ice spontaneously nucleates from supercooled water or when a supercooled water sample becomes amorphous upon progressive cooling.

关键词： Glass transition Nucleation Order parameters Phase transitions

Classical many-particle systems with unique disordered ground states

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Physical Review E 2017年第4期96卷 042146-042146页

作者： G. Zhang F. H. Stillinger S. Torquato Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Classical ground states (global energy-minimizing configurations) of many-particle systems are typically unique crystalline structures, implying zero enumeration entropy of distinct patterns (aside from trivial symmetry operations). By contrast, the few previously known disordered classical ground states of many-particle systems are all high-entropy (highly degenerate) states. Here we show computationally that our recently proposed “perfect-glass” many-particle model [Sci. Rep. 6, 36963 (2016)] possesses disordered classical ground states with a zero entropy: a highly counterintuitive situation . For all of the system sizes, parameters, and space dimensions that we have numerically investigated, the disordered ground states are unique such that they can always be superposed onto each other or their mirror image. At low energies, the density of states obtained from simulations matches those calculated from the harmonic approximation near a single ground state, further confirming ground-state uniqueness. Our discovery provides singular examples in which entropy and disorder are at odds with one another. The zero-entropy ground states provide a unique perspective on the celebrated Kauzmann-entropy crisis in which the extrapolated entropy of a supercooled liquid drops below that of the crystal. We expect that our disordered unique patterns to be of value in fields beyond glass physics, including applications in cryptography as pseudorandom functions with tunable computational complexity.

关键词： Nonequilibrium statistical mechanics Pattern formation