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Multigoal-oriented error estimates for non-linear problems

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arXiv 2018年

作者： Endtmayer, B. Langer, U. Wick, T. Doctoral Program on Computational Mathematics Johannes Kepler University Altenbergerstr. 69 LinzA-4040 Austria Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenbergerstr. 69 LinzA-4040 Austria Institut für Angewandte Mathematik Leibniz Universität Hannover Welfengarten 1 Hannover30167 Germany

In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton's method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled. Copyright © 2018, The Authors. All rights reserved.

关键词： Newton-Raphson method

Publisher Correction: Comprehensive molecular characterization of mitochondrial genomes in human cancers

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Nature genetics 2023年第5期55卷 893页

作者： Yuan Yuan Young Seok Ju Youngwook Kim Jun Li Yumeng Wang Christopher J Yoon Yang Yang Inigo Martincorena Chad J Creighton John N Weinstein Yanxun Xu Leng Han Hyung-Lae Kim Hidewaki Nakagawa Keunchil Park Peter J Campbell Han Liang Department of Bioinformatics and Computational Biology The University of Texas MD Anderson Cancer Center Houston TX USA. Cancer Genome Project Wellcome Trust Sanger Institute Hinxton UK. Graduate School of Medical Science and Engineering Korea Advanced Institute of Science and Technology Daejeon Korea. Department of Health Science and Technology Samsung Advanced Institute for Health Science and Technology Sungkyunkwan University School of Medicine Seoul Korea. Quantitative and Computational Biosciences Graduate Program Baylor College of Medicine Houston TX USA. Division of Biostatistics The University of Texas Health Science Center at Houston School of Public Health Houston TX USA. Department of Medicine and Dan L. Duncan Cancer Center Division of Biostatistics Baylor College of Medicine Houston TX USA. Department of Systems Biology The University of Texas MD Anderson Cancer Center Houston TX USA. Department of Applied Mathematics and Statistics Johns Hopkins University Baltimore MD USA. Department of Biochemistry and Molecular Biology The University of Texas Health Science Center at Houston McGovern Medical School Houston TX USA. Department of Biochemistry Ewha Womans University School of Medicine Seoul Korea. Laboratory for Cancer Genomics RIKEN Center for Integrative Medical Sciences Yokohama Japan. Division of Hematology/Oncology Samsung Medical Center Sungkyunkwan University School of Medicine Seoul Korea. kpark@skku.edu. Cancer Genome Project Wellcome Trust Sanger Institute Hinxton UK. pc8@sanger.ac.uk. Cambridge University Hospitals NHS Foundation Trust Cambridge UK. pc8@sanger.ac.uk. Department of Bioinformatics and Computational Biology The University of Texas MD Anderson Cancer Center Houston TX USA. hliang1@***. Quantitative and Computational Biosciences Graduate Program Baylor College of Medicine Houston TX USA. hliang1@***. Department of Systems Biology The University of Texas MD Anderson Cancer Center Houston TX USA. hliang1@mdanderson

Continuously heterogeneous hyper-objects in cryo-EM and 3-D movies of many temporal dimensions

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arXiv 2017年

作者： Lederman, Roy R. Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Department of Mathematics Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States

Single particle cryo-electron microscopy (EM) is an increasingly popular method for determining the 3-D structure of macromolecules from noisy 2-D images of single macromolecules whose orientations and positions are random and unknown. One of the great opportunities in cryo-EM is to recover the structure of macromolecules in heterogeneous samples, where multiple types or multiple conformations are mixed together. Indeed, in recent years, many tools have been introduced for the analysis of multiple discrete classes of molecules mixed together in a cryo-EM experiment. However, many interesting structures have a continuum of conformations which do not fit discrete models nicely;the analysis of such continuously heterogeneous models has remained a more elusive goal. In this manuscript we propose to represent heterogeneous molecules and similar structures as higher dimensional objects. We generalize the basic operations used in many existing reconstruction algorithms, making our approach generic in the sense that, in principle, existing algorithms can be adapted to reconstruct those higher dimensional objects. As proof of concept, we present a prototype of a new algorithm which we use to solve simulated reconstruction problems. Copyright © 2017, The Authors. All rights reserved.

关键词： Inverse problems

Manifold learning for organizing unstructured sets of process observations

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arXiv 2018年

作者： Dietrich, Felix Kooshkbaghi, Mahdi Bollt, Erik M. Kevrekidis, Ioannis G. Department of Chemical and Biomolecular Engineering Department of Applied Mathematics and Statistics Johns Hopkins University BaltimoreMD21218 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Department of Electrical and Computer Engineering Clarkson Center for Complex Systems Science Clarkson University PotsdamNY13699-5815 United States

Data mining is routinely used to organize ensembles of short temporal observations so as to reconstruct useful, low-dimensional realizations of an underlying dynamical system. In this paper, we use manifold learning to organize unstructured ensembles of observations ("trials") of a system's response surface. We have no control over where every trial starts;and during each trial operating conditions are varied by turning "agnostic" knobs, which change system parameters in a systematic but unknown way. As one (or more) knobs "turn" we record (possibly partial) observations of the system response. We demonstrate how such partial and disorganized observation ensembles can be integrated into coherent response surfaces whose dimension and parametrization can be systematically recovered in a data-driven fashion. The approach can be justified through the Whitney and Takens embedding theorems, allowing reconstruction of manifolds/attractors through different types of observations. We demonstrate our approach by organizing unstructured observations of response surfaces, including the reconstruction of a cusp bifurcation surface for Hydrogen combustion in a Continuous Stirred Tank Reactor. Finally, we demonstrate how this observation-based reconstruction naturally leads to informative transport maps between input parameter space and output/state variable *** codes 37M20 Copyright © 2018, The Authors. All rights reserved.

关键词： Dynamical systems

Rational design of stealthy hyperuniform two-phase media with tunable order

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Physical Review E 2018年第2期97卷 023311-023311页

作者： Robert A. DiStasio, Jr. Ge Zhang Frank H. Stillinger Salvatore Torquato Department of Chemistry and Chemical Biology Cornell University Ithaca New York 14853 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA 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

Disordered stealthy hyperuniform materials are exotic amorphous states of matter that have attracted recent attention because of their novel structural characteristics (hidden order at large length scales) and physical properties, including desirable photonic and transport properties. It is therefore useful to devise algorithms that enable one to design a wide class of such amorphous configurations at will. In this paper, we present several algorithms enabling the systematic identification and generation of discrete (digitized) stealthy hyperuniform patterns with a tunable degree of order, paving the way towards the rational design of disordered materials endowed with novel thermodynamic and physical properties. To quantify the degree of order or disorder of the stealthy systems, we utilize the discrete version of the τ order metric, which accounts for the underlying spatial correlations that exist across all relevant length scales in a given digitized two-phase (or, equivalently, a two-spin state) system of interest. Our results impinge on a myriad of fields, ranging from physics, materials science and engineering, visual perception, and information theory to modern data science.

关键词： Pattern formation

Multifunctional Hyperuniform Cellular Networks: Optimality, Anisotropy and Disorder

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arXiv 2018年

作者： Torquato, S. Chen, D. Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States

Disordered hyperuniform heterogeneous materials are new, exotic amorphous states of matter that behave like crystals in the manner in which they suppress volume-fraction fluctuations at large length scales, and yet are statistically isotropic with no Bragg peaks. It has recently been shown that disordered hyperuniform dielectric two-dimensional cellular network solids possess complete photonic band gaps comparable in size to photonic crystals, while at the same time maintaining statistical isotropy, enabling waveguide geometries not possible with photonic crystals. Motivated by these developments, we explore other functionalities of various two-dimensional ordered and disordered hyperuniform cellular networks, including their effective thermal or electrical conductivities and elastic moduli. We establish the multifunctionality of a class of such low-density networks by demonstrating that they maximize or virtually maximize the effective conductivities and elastic moduli. This is accomplished using the machinery of homogenization theory, including optimal bounds and cross-property bounds, and statistical mechanics. We rigorously prove that anisotropic networks consisting of sets of intersecting parallel channels in the low-density limit, ordered or disordered, possess optimal effective conductivity tensors. For a variety of different disordered networks, we show that when short-range and long-range order increases, there is an increase in both the effective conductivity and elastic moduli of the network. Moreover, we demonstrate that the effective conductivity and elastic moduli of various disordered networks derived from disordered "stealthy" hyperuniform point patterns possess virtually optimal values. We note that the optimal networks for conductivity are also optimal for the fluid permeability associated with slow viscous flow through the channels as well as the mean survival time associated with diffusion-controlled reactions in the channels. In summary, we have

关键词： Elastic moduli

End-to-end symmetry preserving inter-atomic potential energy model for finite and extended systems

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arXiv 2018年

作者： Zhang, Linfeng Han, Jiequn Wang, Han Saidi, Wissam A. Car, Roberto Weinan, E. Program in Applied and Computational Mathematics Princeton University United States Institute of Applied Physics and Computational Mathematics China CAEP Software Center for High Performance Numerical Simulation China Department of Mechanical Engineering and Materials Science University of Pittsburgh United States Department of Chemistry and Department of Physics Princeton University United States Princeton Institute for the Science and Technology of Materials Princeton University United States Department of Mathematics Princeton University United States Beijing Institute of Big Data Research China

Machine learning models are changing the paradigm of molecular modeling, which is a fundamental tool for material science, chemistry, and computational biology. Of particular interest is the inter-atomic potential energy surface (PES). Here we develop Deep Potential - Smooth Edition (DeepPot-SE), an end-to-end machine learning-based PES model, which is able to efficiently represent the PES of a wide variety of systems with the accuracy of ab initio quantum mechanics models. By construction, DeepPot-SE is extensive and continuously differentiable, scales linearly with system size, and preserves all the natural symmetries of the system. Further, we show that DeepPot-SE describes finite and extended systems including organic molecules, metals, semiconductors, and insulators with high fidelity. Copyright © 2018, The Authors. All rights reserved.

关键词： Potential energy

Uncovering multiscale order in the prime numbers via scattering

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arXiv 2018年

作者： Torquato, .S. Zhang, .G. De Courcy-Ireland, M. Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Mathematics Princeton University PrincetonNJ08544 United States

The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which we call effectively limit-periodic. In particular, the primes in this regime are hyperuniform. This is shown analytically using the structure factor S(k), proportional to the scattering intensity from a many-particle system. Remarkably, the structure factor for primes is characterized by dense Bragg peaks, like a quasicrystal, but positioned at certain rational wavenumbers, like a limit-periodic point pattern. We identify a transition between ordered and disordered prime regimes that depends on the intervals studied. Our analysis leads to an algorithm that enables one to predict primes with high accuracy. Effective limit-periodicity deserves future investigation in physics, independent of its link to the primes. Copyright © 2018, The Authors. All rights reserved.

关键词：

HIDDEN MULTISCALE ORDER IN THE PRIMES

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arXiv 2018年

作者： Torquato, Salvatore Zhang, Ge Courcy-Ireland, Matthew De Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Mathematics Princeton University PrincetonNJ08544 United States

We study the pair correlations between prime numbers in an interval M ≤ p ≤ M + L with M → ∞, L/M → β > 0. By analyzing the structure factor, we prove, conditionally on the Hardy-Littlewood conjecture on prime pairs, that the primes are characterized by unanticipated multiscale order. Specifically, their limiting structure factor is that of a union of an infinite number of periodic systems and is characterized by dense set of Dirac delta functions. Primes in dyadic intervals are the first examples of what we call effectively limit-periodic point configurations. This behavior implies anomalously suppressed density fluctuations compared to uncorrelated (Poisson) systems at large length scales, which is now known as hyperuniformity. Using a scalar order metric τ calculated from the structure factor, we identify a transition between the order exhibited when L is comparable to M and the uncorrelated behavior when L is only logarithmic in M. Our analysis for the structure factor leads to an algorithm to reconstruct primes in a dyadic interval with high accuracy. Copyright © 2018, The Authors. All rights reserved.

关键词： Delta functions