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arXiv 2017年

作者： Li, Qianxiao Dietrich, Felix Bollt, Erik M. Kevrekidis, Ioannis G. Institute of High Performance Computing Singapore Singapore Faculty of Mathematics Technical University of Munich Germany Department of Mathematics Clarkson University United States Department of Chemical and Biological Engineering Program in Applied and Computational Mathematics Princeton University United States

Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables that spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky PDE as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems. Copyright © 2017, The Authors. All rights reserved.

关键词： Machine learning

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Inversion problems for fourier transforms of particle distributions

arXiv

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arXiv 2018年

作者： Kim, Jaeuk Zhang, Ge Stillinger, Frank H. Torquato, Salvatore Department of Physics Princeton University PrincetonNJ08544 United States Department of Physics and Astronomy University of Pennsylvania PhiladelphiaPA19104-6396 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Collective coordinates in a many-particle system are complex Fourier components of the particle density n(x) ≡ PNj=1 δ(x − rj), and often provide useful physical insights. However, given collective coordinates, it is desirable to infer the particle coordinates via inverse transformations. In principle, a sufficiently large set of collective coordinates are equivalent to particle coordinates, but the nonlinear relation between collective and particle coordinates makes the inversion procedure highly nontrivial. Given a "target" configuration in one-dimensional Euclidean space, we investigate the minimal set of its collective coordinates that can be uniquely inverted into particle coordinates. For this purpose, we treat a finite number M of the real and/or the imaginary parts of collective coordinates of the target configuration as constraints, and then reconstruct "solution" configurations whose collective coordinates satisfy these constraints. Both theoretical and numerical investigations reveal that the number of numerically distinct solutions depends sensitively on the chosen collective-coordinate constraints and target configurations. From detailed analysis, we conclude that collective coordinates at the dN2 e smallest wavevectors is the minimal set of constraints for unique inversion, where d·e represents the ceiling function. This result provides useful groundwork to the inverse transform of collective coordinates in higher-dimensional systems. Copyright © 2018, The Authors. All rights reserved.

关键词： Inverse transforms

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Sharp rates of convergence for accumulated spectrograms

arXiv

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arXiv 2017年

作者： Abreu, Luis Daniel Pereira, Joao Romero, Jose Luis Acoustics Research Institute Austrian Academy of Science Wohllebengasse 12-14 ViennaA-1040 Austria Program in Applied and Computational Mathematics Princeton University NJ08544 United States

We investigate an inverse problem in time-frequency localization: the approximation of the symbol of a time-frequency localization operator from partial spectral information by the method of accumulated spectrograms (the sum of the spectrograms corresponding to large eigenvalues). We derive a sharp bound for the rate of convergence of the accumulated spectrogram, improving on recent *** Codes 65F18, 94A12, 65N21, 42C25 Copyright © 2017, The Authors. All rights reserved.

关键词： Spectrographs

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Multireference alignment is easier with an aperiodic translation distribution

arXiv

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arXiv 2017年

作者： Abbe, Emmanuel Bendory, Tamir Leeb, William Pereira, João M. Sharon, Nir Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Electrical Engineering Department Princeton University PrincetonNJ United States Department of Mathematics Princeton University PrincetonNJ United States

In the multireference alignment model, a signal is observed by the action of a random circular translation and the addition of Gaussian noise. The goal is to recover the signal's orbit by accessing multiple independent observations. Of particular interest is the sample complexity, i.e., the number of observations/samples needed in terms of the signal-to-noise ratio (the signal energy divided by the noise variance) in order to drive the mean-square error (MSE) to zero. Previous work showed that if the translations are drawn from the uniform distribution, then, in the low SNR regime, the sample complexity of the problem scales as ω(1/SNR3). In this work, using a generalization of the Chapman-Robbins bound for orbits and expansions of the X2divergence at low SNR, we show that in the same regime the sample complexity for any aperiodic translation distribution scales as ω(1/SNR2). This rate is achieved by a simple spectral algorithm. We propose two additional algorithms based on non-convex optimization and expectation-maximization. We also draw a connection between the multireference alignment problem and the spiked covariance model. Copyright © 2017, The Authors. All rights reserved.

关键词： Maximum principle

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Group synchronization on grids

arXiv

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arXiv 2017年

作者： Abbe, Emmanuel Massoulie, Laurent Montanari, Andrea Sly, Allan Srivastava, Nikhil Program in Applied and Computational Mathematics EE Department Princeton University Inria MSR-Inria Joint Centre Department of Electrical Engineering Department of Sttistics Stanford University Department of Mathematics Princeton University Department of Mathematics University of California Berkeley United States

Group synchronization requires to estimate unknown elements (θv)v∈V of a compact group G associated to the vertices of a graph G = (V, E), using noisy observations of the group differences associated to the edges. This model is relevant to a variety of applications ranging from structure from motion in computer vision to graph localization and positioning, to certain families of community detection problems. We focus on the case in which the graph G is the d-dimensional grid. Since the unknowns θv are only determined up to a global action of the group, we consider the following weak recovery question. Can we determine the group difference θ−u1θv between far apart vertices u, v better than by random guessing? We prove that weak recovery is possible (provided the noise is small enough) for d ≥ 3 and, for certain finite groups, for d ≥ 2. Viceversa, for some continuous groups, we prove that weak recovery is impossible for d = 2. Finally, for strong enough noise, weak recovery is always impossible. Copyright © 2017, The Authors. All rights reserved.

关键词： Recovery

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On singularity formation in a Hele-Shaw model

arXiv

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arXiv 2017年

作者： Constantin, Peter Elgindi, Tarek Nguyen, Huy Vicol, Vlad Department of Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Uc San Diego San DiegoCA92093 United States Program In Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We discuss a lubrication approximation model of the interface between two immiscible fluids in a Hele-Shaw cell, derived in [CDG+93] and widely studied since. The model consists of a single one dimensional evolution equation for the thickness 2h = 2h(x, t) of a thin neck of fluid, ∂th + ∂x(h ∂x3h) = 0, for x ∈ (−1, 1) and t ≥ 0. The boundary conditions fix the neck height and the pressure jump: h(±1, t) = 1, ∂x2h(±1, t) = P > 0. We prove that starting from smooth and positive h, as long as h(x, t) > 0, for x ∈ [−1, 1], t ∈ [0, T ], no singularity can arise in the solution up to time T. As a consequence, we prove for any P > 2 and any smooth and positive initial datum that the solution pinches off in either finite or infinite time, i.e., inf[−1,1]×[0,T∗) h = 0, for some T∗ ∈ (0, ∞]. These facts have been long anticipated on the basis of numerical and theoretical studies. August 30, 2017.35Q35, 35Q86 Copyright © 2017, The Authors. All rights reserved.

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Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings

arXiv

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arXiv 2018年

作者： Oguz, Erdal C. Socolar, Joshua E.S. Steinhardt, Paul J. Torquato, Salvatore School of Mechanical Engineering Sackler Center for Computational Molecular and Materials Science Tel Aviv University Tel Aviv Israel Physics Department Duke University DurhamNC27708 United States Department of Physics Princeton Center for Theoretical Science Princeton University PrincetonNJ08544 United States Center for Cosmology and Particle Physics Department of Physics New York University New YorkNY10003 United States Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States

We consider the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long wavelength fluctuations in a broad class of one-dimensional substitution tilings. We present a simple argument that predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and confirm with numerical computations that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra, and limit-periodic tilings. Tilings with quasiperiodic or singular continuous spectra can be constructed with α arbitrarily close to any given value between −1 and 3. Limit-periodic tilings can be constructed with α between −1 and 1 or with Fourier intensities that approach zero faster than any power law. Copyright © 2018, The Authors. All rights reserved.

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Quantifying Gerrymandering in North Carolina

arXiv

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arXiv 2018年

作者： Herschlag, Gregory Kang, Han Sung Luo, Justin Graves, Christy Vaughn Bangia, Sachet Ravier, Robert Mattingly, Jonathan C. Department of Mathematics Duke University DurhamNC27708 United States Department of Biomedical Engineering Duke University DurhamNC27708 United States Department of Computer Science Duke University DurhamNC27708 United States Department of Electrical and Computer Engineering Duke University DurhamNC27708 United States Department of Economics Duke University DurhamNC27708 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Department of Statistical Science Duke University DurhamNC27708 United States

Using an ensemble of redistricting plans, we evaluate whether a given political districting faithfully represents the geo-political landscape. Redistricting plans are sampled by a Monte Carlo algorithm from a probability distribution that adheres to realistic and non-partisan criteria. Using the sampled redistricting plans and historical voting data, we produce an ensemble of elections that reveal geo-political structure within the state. We showcase our methods on the two most recent districtings of NC for the U.S. House of Representatives, as well as a plan drawn by a bipartisan redistricting panel. We find the two state enacted plans are highly atypical outliers whereas the bipartisan plan accurately represents the ensemble both in partisan outcome and in the fine scale structure of district-level results. Copyright © 2018, The Authors. All rights reserved.

关键词： Probability distributions

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Author Correction: Genomic footprints of activated telomere maintenance mechanisms in cancer (Dec, 10.1038/s41467-019-13824-9, 2022)

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NATURE COMMUNICATIONS 2022年第1期13卷 1-1页

作者： Sieverling, Lina Hong, Chen Koser, Sandra D. Ginsbach, Philip Kleinheinz, Kortine Hutter, Barbara Braun, Delia M. Cortes-Ciriano, Isidro Xi, Ruibin Kabbe, Rolf Park, Peter J. Eils, Roland Schlesner, Matthias Brors, Benedikt Rippe, Karsten Jones, David T. W. Feuerbach, Lars Division of Applied Bioinformatics German Cancer Research Center (DKFZ) 69120 Heidelberg Germany Faculty of Biosciences Heidelberg University 69120 Heidelberg Germany Division of Applied Bioinformatics German Cancer Research Center (DKFZ) Heidelberg Germany Faculty of Biosciences Heidelberg University Heidelberg Germany Division of Theoretical Bioinformatics German Cancer Research Center (DKFZ) 69120 Heidelberg Germany Department for Bioinformatics and Functional Genomics Institute for Pharmacy and Molecular Biotechnology (IPMB) and BioQuant 69120 Heidelberg Germany Division of Theoretical Bioinformatics German Cancer Research Center (DKFZ) Heidelberg Germany Institute of Pharmacy and Molecular Biotechnology and BioQuant Heidelberg University Heidelberg Germany German Cancer Consortium (DKTK) Heidelberg Germany National Center for Tumor Diseases (NCT) Heidelberg Heidelberg Germany Heidelberg Center for Personalized Oncology (DKFZ-HIPO) German Cancer Research Center (DKFZ) Heidelberg Germany German Cancer Consortium (DKTK) German Cancer Research Center (DKFZ) Heidelberg Germany Division of Chromatin Networks German Cancer Research Center (DKFZ) and BioQuant 69120 Heidelberg Germany Department of Biomedical Informatics Harvard Medical School Boston Massachusetts 02115 USA Department of Chemistry Centre for Molecular Science Informatics University of Cambridge Cambridge CB2 1EW UK Ludwig Center at Harvard Medical School Boston Massachusetts 02115 USA Department of Biomedical Informatics Harvard Medical School Boston MA USA Department of Chemistry Centre for Molecular Science Informatics University of Cambridge Cambridge UK Ludwig Center at Harvard Medical School Boston MA USA School of Mathematical Sciences and Center for Statistical Science Peking University Beijing 100871 China Heidelberg University Heidelberg Germany New BIH Digital Health Center Berlin Institute of Health (BIH) and Charité - Universitätsmedizin Berlin Berlin Germany Bioi

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On signal reconstruction from frog measurements

arXiv

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arXiv 2017年

作者： Bendory, Tamir Edidin, Dan Eldar, Yonina C. Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Department of Mathematics University of Missouri ColumbiaMO United States The Andrew and Erna Viterbi Faculty of Electrical Engineering Technion – Israel Institute of Technology Haifa Israel

Phase retrieval refers to recovering a signal from its Fourier magnitude. This problem arises naturally in many scientific applications, such as ultra-short laser pulse characterization and diffraction imaging. Unfortunately, phase retrieval is ill-posed for almost all one-dimensional signals. In order to characterize a laser pulse and overcome the ill-posedness, it is common to use a technique called Frequency-Resolved Optical Gating (FROG). In FROG, the measured data, referred to as FROG trace, is the Fourier magnitude of the product of the underlying signal with several translated versions of itself. The FROG trace results in a system of phaseless quartic Fourier measurements. In this paper, we prove that it suffices to consider only three translations of the signal to determine almost all bandlimited signals, up to trivial ambiguities. In practice, one usually also has access to the signal’s Fourier magnitude. We show that in this case only two translations suffice. Our results significantly improve upon earlier work. Copyright © 2017, The Authors. All rights reserved.

关键词： Signal reconstruction

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