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Determining system Hamiltonian from eigenstate measurements without correlation functions

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arXiv 2019年

作者： Hou, Shi-Yao Cao, Ningping Lu, Sirui Shen, Yi Poon, Yiu-Tung Zeng, Bei College of Physics and Electronic Engineering Center for Computational Sciences Sichuan Normal University Chengdu610068 China Center for Quantum Computing Peng Cheng Laboratory Shenzhen518055 China Shenzhen Institute for Quantum Science and Engineering Southern University of Science and Technology Shenzhen518055 China Department of Mathematics & Statistics University of Guelph GuelphONN1G 2W1 Canada Institute for Quantum Computing University of Waterloo WaterlooONN2L 3G1 Canada Department of Physics Tsinghua University Beijing100084 China Department of Statistics and Actuarial Science University of Waterloo WaterlooON Canada Department of Mathematics Iowa State University AmesIA50011 United States Department of Physics Hong Kong University of Science and Technology Clear Water Bay Kowloon Hong Kong

Local Hamiltonians arise naturally in physical systems. Despite its seemingly 'simple' local structure, exotic features such as nonlocal correlations and topological orders exhibit in eigenstates of these systems. Previous studies for recovering local Hamiltonians from measurements on an eigenstate |ψ〉 require information of nonlocal correlation functions. In this work, we develop an algorithm to determine local Hamiltonians from only local measurements on |ψ〉, by reformulating the task as an unconstrained optimization problem of certain target function of Hamiltonian parameters, with only polynomial number of parameters in terms of system size. We also develop a machine learning-based-method to solve the first-order gradient used in the algorithm. Our method is tested numerically for randomly generated local Hamiltonians and returns promising reconstruction in the desired accuracy. Our result shed light on the fundamental question on how a single eigenstate can encode the full system Hamiltonian, indicating a somewhat surprising answer that only local measurements are enough without additional assumptions, for generic cases. Copyright © 2019, The Authors. All rights reserved.

关键词： Hamiltonians

Verification benchmarks for single-phase flow in three-dimensional fractured porous media

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arXiv 2020年

作者： Berre, Inga Boon, Wietse M. Flemisch, Bernd Fumagalli, Alessio Gläser, Dennis Keilegavlen, Eirik Scotti, Anna Stefansson, Ivar Tatomir, Alexandru Brenner, Konstantin Burbulla, Samuel Devloo, Philippe Duran, Omar Favino, Marco Hennicker, Julian Lee, I-Hsien Lipnikov, Konstantin Masson, Roland Mosthaf, Klaus Chiara Nestola, Maria Giuseppina Ni, Chuen-Fa Nikitin, Kirill Schädle, Philipp Svyatskiy, Daniil Yanbarisov, Ruslan Zulianp, Patrick Department of Mathematics University of Bergen Allégaten 41 Bergen5007 Norway Department of Mathematics KTH Royal Institute of Technology Lindstedtsvägen 25 Stockholm11428 Sweden Department of Hydromechanics and Modelling of Hydrosystems University of Stuttgart Pfaffenwaldring 61 Stuttgart70569 Germany Laboratory for Modeling and Scientific Computing MOX Politecnico di Milano p.za Leonardo da Vinci 32 Milano20133 Italy Department of Applied Geology Geosciences Center University of Göttingen Goldschmidtstrasse 3 Göttingen37077 Germany Department of Earth Sciences Uppsala University Villavägen 16 UppsalaS-75236 Sweden University of Côte d’Azur CNRS INRIA LJAD Nice France Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany FEC Universidade Estadual de Campinas Cidade Universitária R. Josiah Willard Gibbs 85 Campinas-SPCEP 13083-839 Brazil Institute of Earth Sciences University of Lausanne Building Geopolis UNIL-Mouline Lausanne1015 Switzerland Section de Mathématiques Université de Genève 2-4 rue du Lièvre CP 64 Genève1211 Switzerland Graduate Institute of Applied Geology National Central University Taiwan Center for Environmental Studies National Central University Taiwan Los Alamos National Laboratory New Mexico United States Department of Environmental Engineering Technical University of Denmark Bygningstorvet Building 115 Kgs. Lyngby2800 Denmark Numerical Simulation in Science Medicine and Engineering Group Institute of Computational Science Università della Svizzera italiana. Via G. Buffi 13 Lugano Ticino6900 Switzerland Marchuk Institute of Numerical Mathematics of Russian Academy of Sciences Moscow Russia ETH Zürich Geothermal Energy and Geofluids Group Institute of Geophysics Zürich8092 Switzerland

Flow in fractured porous media occurs in the earth’s subsurface, in biological tissues, and in man-made materials. Fractures have a dominating influence on flow processes, and the last decade has seen an extensive development of models and numerical methods that explicitly account for their presence. To support these developments, we present a portfolio of four benchmark cases for single-phase flow in three-dimensional fractured porous media. The cases are specifically designed to test the methods’ capabilities in handling various complexities common to the geometrical structures of fracture networks. Based on an open call for participation, results obtained with 17 numerical methods were collected. This paper presents the underlying mathematical model, an overview of the features of the participating numerical methods, and their performance in solving the benchmark cases. Copyright © 2020, The Authors. All rights reserved.

关键词： Numerical methods

A singularity-based eigenfunction decomposition for Kohn-Sham equations

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Science China mathematics 2016年第8期59卷 1623-1634页

作者： ZHANG Xin ZHOU Aihui SWIEE Southwest China Research Institute of Electronic Equipment LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems Science Chinese Academy of Sciences

We show that the eigenfunctions of Kohn-Sham equations can be decomposed as ■ = F ψ, where F depends on the Coulomb potential only and is locally Lipschitz, while ψ has better regularity than ■.

关键词： decomposition eigenfunction Kohn-Sham equation full-potential regularity singularity

Viscosity Variability Impact on 2D Laminar and Turbulent Poiseuille Velocity Profiles;Characteristic-Based Split (CBS) Stabilization

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Viscosity Variability Impact on 2D Laminar and Turbulent Poi...

International Conference on mathematics and Computers in Sciences and in Industry (MCSI)

作者： Antonio Pasculli Dept. of Engineering Geology (INGEO) and Member of the INDAM Research group GNCS (National Institute of Advanced Mathematics National Group of Scientific Computing) University G. D’Annunzio Chieti-Pescara Italy

ISBN: (纸本)9781538675014

The erosion of riverbeds and riverbanks depends, among other causes, both on the velocity fields and on their gradient near their boundaries, with the generation of shear stresses. The presence of sediments modifies the viscosity and, accordingly, modifies the profiles, particularly near the edges right where they are generated. Therefore, in this work, the distortion of the velocity profiles due to an imposed spatial variability of viscosity, was studied applying the computational Fluid Dynamics (CFD). In particular, as test cases, laminar and turbulent Plane Poiseuille flows, were selected. For simplicity, it was assumed that the sediment distribution and therefore the viscosity distribution was not influenced by the mixing due to velocity field. That is, the equilibrium configuration was determined as a consequence of a spatially variable distribution of viscosity. The 2D Navier-Stokes equations, in steady state conditions, were numerically solved exploiting a research software developed and discussed by the author [1]. The turbulence was considered through the RANS (Reynolds Averaged Navier Stokes) approach. The two equations k - ε models were employed. The turbulence phenomena near solid boundaries was simulated by the means of Wall-Functions. Spatial discretization was carried out using the Finite Element Method (FEM). A structured meshing with h like adaptability was developed. Then, in order to avoid velocities and pressure instabilities, the Characteristic-based split algorithm (CBS) was applied, while, in order to correctly consider incompressibility, by a numerical point of view, the Method of Artificial Compressibility (AC) was selected. Accordingly, the related CBS-AC three steps algorithm was implemented [1]. Then, some parametric numerical experiments were performed, considering a semi-implicit, approach. As was to be expected, the velocity profiles, for both laminar and turbulent were influenced by the viscosity distribution. The discussion of the ov

关键词： Mathematical model Handheld computers Finite element analysis Production Kinetic energy Viscosity

A Barzilai-Borwein conjugate gradient method

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Science China mathematics 2016年第8期59卷 1511-1524页

作者： DAI YuHong KOU CaiXia LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems Science Chinese Academy of Sciences School of Science Beijing University of Posts and Telecommunications

The linear conjugate gradient method is an optimal method for convex quadratic minimization due to the Krylov subspace minimization property. The proposition of limited-memory BFGS method and Barzilai-Borwein gradient method, however, heavily restricted the use of conjugate gradient method for largescale nonlinear optimization. This is, to the great extent, due to the requirement of a relatively exact line search at each iteration and the loss of conjugacy property of the search directions in various occasions. On the contrary, the limited-memory BFGS method and the Barzilai-Bowein gradient method share the so-called asymptotical one stepsize per line-search property, namely, the trial stepsize in the method will asymptotically be accepted by the line search when the iteration is close to the solution. This paper will focus on the analysis of the subspace minimization conjugate gradient method by Yuan and Stoer(1995). Specifically, if choosing the parameter in the method by combining the Barzilai-Borwein idea, we will be able to provide some efficient Barzilai-Borwein conjugate gradient(BBCG) methods. The initial numerical experiments show that one of the variants, BBCG3, is specially efficient among many others without line searches. This variant of the BBCG method might enjoy the asymptotical one stepsize per line-search property and become a strong candidate for large-scale nonlinear optimization.

关键词： conjugate gradient method subspace minimization Barzilai-Bowein gradient method line search descent property global convergence

MINIMAL CONSISTENT FINITE ELEMENT SPACE FOR THE BIHARMONIC EQUATION ON QUADRILATERAL GRIDS

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arXiv 2017年

作者： Zhang, Shuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China

In this paper, a finite element space is presented on quadrilateral grids which can provide consistent discretization for the biharmonic equations. The space consists of piecewise quadratic polynomials and is of minim...

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A multi-level mixed element scheme of the two dimensional Helmholtz transmission eigenvalue problem

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arXiv 2017年

作者： Xi, Yingxia Ji, Xia Zhang, Shuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China

In this paper, we present a multi-level mixed element scheme for the Helmholtz transmission eigenvalue problem on polygonal domains that are not necessarily able to be covered by rectangle grids. We first construct an equivalent linear mixed formulation of the transmission eigenvalue problem and then discretize it with Lagrangian finite elements of low regularities. The proposed scheme admits a natural nested discretization, based on which we construct a multi-level scheme. Optimal convergence rate and optimal computational cost can be obtained with the scheme.65N25, 65N30, 47B07 Copyright © 2017, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

Stable mixed element schemes for plate models on multiply-connected domains

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arXiv 2017年

作者： Zhang, Shuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China

In this paper, we study the mixed element schemes of the Reissner-Mindlin plate model and the Kirchhoff plate model in multiply-connected domains. By a regular decomposition of H0(rot, Ω) and a Helmholtz decomposition of its dual, we develop mixed formulations of the models which are equivalent to the primal ones respectively and which are uniformly stable. A framework of designing uniformly stable finite element schemes is presented, and a specific example is given. Copyright © 2017, The Authors. All rights reserved.

关键词： Finite element method

Learning from learning machines: A new generation of AI technology to meet the needs of science

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arXiv 2021年

作者： Pion-Tonachini, Luca Bouchard, Kristofer Martin, Hector Garcia Peisert, Sean Holtz, W. Bradley Aswani, Anil Dwivedi, Dipankar Wainwright, Haruko Pilania, Ghanshyam Nachman, Benjamin Marrone, Babetta L. Falco, Nicola Prabhat Arnold, Daniel Wolf-Yadlin, Alejandro Powers, Sarah Climer, Sharlee Jackson, Quinn Carlson, Ty Sohn, Michael Zwart, Petrus Kumar, Neeraj Justice, Amy Tomlin, Claire Jacobson, Daniel Micklem, Gos Gkoutos, Georgios V. Bickel, Peter J. Cazier, Jean-Baptiste Müller, Juliane Webb-Robertson, Bobbie-Jo Stevens, Rick Anderson, Mark Kreutz-Delgado, Ken Mahoney, Michael W. Brown, James B. Pattern Computer Inc. Friday HarborWA98250 United States Biosciences Area Lawrence Berkeley National Lab BerkeleyCA94803 United States Computational Research Division Lawrence Berkeley National Lab BerkeleyCA94720 United States Helen Wils Neuroscience Institute Redwood Center for Theoretical Neuroscience Uc Berkeley BerkeleyCA94720 United States Doe Agile BioFoundry Lawrence Berkeley National Laboratory BerkeleyCA94803 United States Joint BioEnergy Institute Lawrence Berkeley National Laboratory BerkeleyCA94803 United States Bcam Basque Center for Applied Mathematics Bilbao48009 Spain Computer Science University of California Davis DavisCA95616 United States Cenic La MiradaCA90638 United States Health Informatics University of California Davis School of Medicine SacramentoCA95817 United States Berkeley Institute for Data Science University of California Berkeley BerkeleyCA94720 United States Industrial Engineering and Operations Research University of California Berkeley BerkeleyCA94720 United States Environmental & Earth Sciences Area Lawrence Berkeley National Laboratory BerkeleyCA94803 United States Nuclear Engineering University of California Berkeley BerkeleyCA94720 United States Materials Science and Technology Division Los Alamos National Laboratory Los AlamosNM87545 United States Physics Division Lawrence Berkeley National Lab BerkeleyCA94720 United States Bioscience Division Los Alamos National Laboratory Los AlamosNM87545 United States Earth and Environmental Sciences Area Lawrence Berkeley National Lab BerkeleyCA94803 United States Energy Technologies Area Lawrence Berkeley National Lab BerkeleyCA94803 United States Computing and Computational Sciences Oak Ridge National Laboratory Oak RidgeTN37831 United States Industrial & Systems Engineering The University of Tennessee KnoxvilleTN37996 United States Department of Computer Science University of Missouri-Saint Louis St. LouisMO63121 United

We outline emerging opportunities and challenges to enhance the utility of AI for scientific discovery. The distinct goals of AI for industry versus the goals of AI for science create tension between identifying patterns in data versus discovering patterns in the world from data. If we address the fundamental challenges associated with "bridging the gap" between domain-driven scientific models and data-driven AI learning machines, then we expect that these AI models can transform hypothesis generation, scientific discovery, and the scientific process itself. © 2021, CC BY.

关键词： Learning systems