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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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TEXAS HOLD'EM ALGORITHMS FOR DISTRIBUTED COMPRESSIVE SENSING

TEXAS HOLD'EM ALGORITHMS FOR DISTRIBUTED COMPRESSIVE SENSING

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IEEE International Conference on Acoustics, Speech, and Signal Processing

作者： Stephen R. Schnelle Jason N. Laska Chinmay Hegde Marco F. Duarte Mark A. Davenport Richard G. Baraniuk Department of Electrical and Computer Engineering Rice University Houston TX 77005 Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544

ISBN: (纸本)9781424442959

This paper develops a new class of algorithms for signal recovery in the distributed compressive sensing (DCS) framework. DCS exploits both intra-signal and inter-signal correlations through the concept of joint sparsity to further reduce the number of measurements required for recovery. DCS is well-suited for sensor network applications due to its universality, computational asymmetry, tolerance to quantization and noise, and robustness to measurement loss. In this paper we propose recovery algorithms for the sparse common and innovation joint sparsity model. Our approach leads to a class of efficient algorithms, the Texas Hold 'Em algorithms, which are scalable both in terms of communication bandwidth and computational complexity.

关键词： Signal reconstruction multisensor systems data compression Multisensor systems Signal reconstruction Data compression New classes algorithms Texas Signal restoration Quantization Scalability

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KRONECKER PRODUCT MATRICES FOR COMPRESSIVE SENSING

KRONECKER PRODUCT MATRICES FOR COMPRESSIVE SENSING

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IEEE International Conference on Acoustics, Speech, and Signal Processing

作者： Marco F. Duarte Richard G. Baraniuk Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 Department of Electrical and Computer Engineering Rice University Houston TX 77005

ISBN: (纸本)9781424442959

Compressive sensing (CS) is an emerging approach for acquisition of signals having a sparse or compressible representation in some basis. While CS literature has mostly focused on problems involving 1-D and 2-D signals, many important applications involve signals that are multidimensional. We propose the use of Kronecker product matrices in CS for two purposes. First, we can use such matrices as sparsifying bases that jointly model the different types of structure present in the signal. Second, the measurement matrices used in distributed measurement settings can be easily expressed as Kronecker products. This new formulation enables the derivation of analytical bounds for sparse approximation and CS recovery of multidimensional signals.

关键词： Data compression multidimensional signal processing signal reconstruction

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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

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Frame coherence and sparse signal processing

Frame coherence and sparse signal processing

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IEEE International Symposium on Information Theory

作者： Dustin G. Mixon Waheed U. Bajwa Robert Calderbank Program in Applied and Computational Mathematics Princeton University Princeton NJ USA Department of Electrical and Computer Engineering Duke University Durham NC USA

The sparse signal processing literature often uses random sensing matrices to obtain performance guarantees. Unfortunately, in the real world, sensing matrices do not always come from random processes. It is therefore desirable to evaluate whether an arbitrary matrix, or frame, is suitable for sensing sparse signals. To this end, the present paper investigates two parameters that measure the coherence of a frame: worst-case and average coherence. We first provide several examples of frames that have small spectral norm, worst-case coherence, and average coherence. Next, we present a new lower bound on worst-case coherence and compare it to the Welch bound. Later, we propose an algorithm that decreases the average coherence of a frame without changing its spectral norm or worst-case coherence. Finally, we use worst-case and average coherence, as opposed to the Restricted Isometry Property, to garner near-optimal probabilistic guarantees on both sparse signal detection and reconstruction in the presence of noise. This contrasts with recent results that only guarantee noiseless signal recovery from arbitrary frames, and which further assume independence across the nonzero entries of the signal-in a sense, requiring small average coherence replaces the need for such an assumption.

关键词： Coherence Signal processing algorithms Noise Harmonic analysis Noise measurement Geometry

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Robust estimation of rotations from relative measurements by maximum likelihood

Robust estimation of rotations from relative measurements by...

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IEEE Annual Conference on Decision and Control

作者： Nicolas Boumal Amit Singer P.-A. Absil Department of Mathematical Engineering ICTEAM Institute Universite catholique de Louvain Belgium Program in Applied and Computational Mathematics Princeton University NJ USA

ISBN: (纸本)9781467357159

We estimate unknown rotation matrices R_i from a set of measurements of relative rotations R_iR_j~T. Measurements are strongly affected by noise such that a small fraction of them are well concentrated around the true relative rotations while the majority of measurements are outliers bearing little or no information. We propose a maximum likelihood estimator (MLE) that explicitly acknowledges this noise model, yielding a robust estimation algorithm. The MLE is computed via Riemannian trust-region optimization using the Manopt toolbox. Comparisons of the MLE with Cramer-Rao bounds suggest the estimator is asymptotically efficient.

关键词： Maximum likelihood estimation Noise Synchronization Rotation measurement Noise measurement Vectors Optimization

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Designed interaction potentials via inverse methods for self-assembly

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Physical Review E 2006年第1期73卷 011406-011406页

作者： Mikael Rechtsman Frank Stillinger Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics and PRISM Princeton New Jersey 08544 USA

We formulate statistical-mechanical inverse methods in order to determine optimized interparticle interactions that spontaneously produce target many-particle configurations. Motivated by advances that give experimentalists greater and greater control over colloidal interaction potentials, we propose and discuss two computational algorithms that search for optimal potentials for self-assembly of a given target configuration. The first optimizes the potential near the ground state and the second near the melting point. We begin by applying these techniques to assembling open structures in two dimensions (square and honeycomb lattices) using only circularly symmetric pair interaction potentials; we demonstrate that the algorithms do indeed cause self-assembly of the target lattice. Our approach is distinguished from previous work in that we consider (i) lattice sums, (ii) mechanical stability (phonon spectra), and (iii) annealed Monte Carlo simulations. We also devise circularly symmetric potentials that yield chainlike structures as well as systems of clusters.

关键词： THERMAL-EXPANSION MICROSTRUCTURES SYSTEM PORES MODEL

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Image Recovery from Rotational And Translational Invariants

Image Recovery from Rotational And Translational Invariants

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IEEE International Conference on Acoustics, Speech and Signal Processing

作者： Nicholas F. Marshall Ti-Yen Lan Tamir Bendory Amit Singer Department of Mathematics Princeton University Princeton NJ USA The Program in Applied and Computational Mathematics Princeton University Princeton NJ USA School of Electrical Engineering Tel Aviv University Tel Aviv Israel

ISBN: (数字)9781509066315

ISBN: (纸本)9781509066322

We introduce a framework for recovering an image from its rotationally and translationally invariant features based on autocorrelation analysis. This work is an instance of the multi-target detection statistical model, which is mainly used to study the mathematical and computational properties of single-particle reconstruction using cryo-electron microscopy (cryo-EM) at low signal-to-noise ratios. We demonstrate with synthetic numerical experiments that an image can be reconstructed from rotational and translational invariants and show that the reconstruction is robust to noise. These results constitute an important step towards the goal of structure determination of small biomolecules using cryo-EM.

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Elastic and Thermodynamic Properties of Complex Mg-Al Intermetallic Compounds via Orbital-Free Density Functional Theory

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Physical Review applied 2016年第6期5卷 064021-064021页

作者： Houlong Zhuang Mohan Chen Emily A. Carter Department of Mechanical and Aerospace Engineering Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics and Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544 USA

Magnesium-aluminum (Mg-Al) alloys are important metal alloys with a wide range of engineering applications. We investigate the elastic and thermodynamic properties of Mg, Al, and four stoichiometric Mg-Al compounds including Mg17Al12, Mg13Al14, and Mg23Al30, and MgAl2 with orbital-free density-functional theory (OFDFT). We first calculate the lattice constants, zero-temperature formation energy, and independent elastic constants of these six materials and compare the results to those computed via Kohn-Sham DFT (KSDFT) benchmarks. We obtain excellent agreement between these two methods. Our calculated elastic constants of hexagonal close-packed Mg and face-centered-cubic Al are also consistent with available experimental data. We next compute their phonon spectra using the force constants extracted from the very fast OFDFT calculations, because such calculations are computationally challenging using KSDFT. This is especially the case for the Mg23Al30 compound, whose 3×3×3 supercell consists of 1431 atoms. We finally employ the quasiharmonic approximation to investigate temperature-dependent thermodynamic properties, including formation energies, heat capacities, and thermal expansion of the four Mg-Al intermetallic compounds. The calculated heat capacity and thermal expansion of both Mg and Al agree well with experimental data. We additionally find that Mg13Al14 and MgAl2 are both unstable, consistent with their absence from the equilibrium Mg-Al phase diagram. Our work demonstrates that OFDFT is an efficient and accurate quantum-mechanical computational tool for predicting elastic and thermodynamic properties of complicated Mg-Al alloys and also should be applicable to many other engineering alloys.

关键词： Elastic deformation Elastic modulus First-principles calculations Specific heat Thermal expansion Alloys

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LARGE SCALE DYNAMICS OF PRECIPITATION FRONTS IN THE TROPICAL ATMOSPHERE: A NOVEL RELAXATION LIMIT ¤

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Communications in Mathematical Sciences 2004年第4期2卷 591-626页

作者： Frierson, Dargan M. W. Majda, Andrew J. Pauluis, Olivier M. Program in Applied and Computational Mathematics Princeton University Fine HallWashington Rd Princeton 08544 NJ United States Department of Mathematics and Center for Atmosphere Ocean Science Courant Institute of Mathematical Sciences New York University 251 Mercer St New York 10012 NY United States Center for Atmosphere Ocean Science Courant Institute of Mathematical Sciences New York University 251 Mercer St. New York 10012 NY United States

A simplified set of equations is derived systematically below for the interaction of large scale flow fields and precipitation in the tropical atmosphere. These equations, the Tropical Climate Model, have the form of a shallow water equation and an equation for moisture coupled through a strongly nonlinear source term. This source term, the precipitation, is of relaxation type in one region of state space for the temperature and moisture, and vanishes identically elsewhere in the state space of these variables. In addition, the equations are coupled nonlinearly to the equations for barotropic incompressible flow. Several mathematical features of this system are developed below including energy principles for solutions and their first derivatives independent of relaxation time. With these estimates, the formal infinitely fast relaxation limit converges to a novel hyperbolic free boundary problem for the motion of precipitation fronts from a large scale dynamical perspective. Elementary exact solutions of the limiting dynamics involving precipitation fronts are developed below and include three families of waves: fast drying fronts as well as slow and fast moistening fronts. The last two families of waves violate Lax's Shock Inequalities;nevertheless, numerical experiments presented below confirm their robust realizability with realistic finite relaxation times. From the viewpoint of tropical atmospheric dynamics, the theory developed here provides a new perspective on the fashion in which the prominent large scale regions of moisture in the tropics associated with deep convection can move and interact with large scale dynamics in the quasi-equilibrium approximation. © 2004 International Press

关键词： Equatorial waves Hyperbolic free boundary problems Hyperbolic pde Moisture Nonlinear relaxation equations Tropical atmospheric dynamics Tropical convection

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