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Automatic Variationally Stable Analysis for Finite Element Computations: Transient Convection-Diffusion Problems

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arXiv 2020年

作者： Valseth, Eirik Behnoudfar, Pouria Dawson, Clint Romkes, Albert Oden Institute for Computational Engineering and Sciences University of Texas at Austin AustinTX78712 United States The Department of Data Science The Norwegian University of Life Science Drøbakveien 31 Ås1433 Norway Department of Scientific Computing and Numerical Analysis Simula Research Laboratory Kristian Augusts gate 23 Oslo0164 Norway PerthWA6152 Australia Department of Mechanical Engineering South Dakota School of Mines & Technology Rapid CitySD57701 United States

We present an application of stable finite element (FE) approximations of convection-diffusion initial boundary value problems (IBVPs) using a weighted least squares FE method, the automatic variationally stable finite element (AVS-FE) method [1]. The transient convection-diffusion problem leads to issues in classical FE methods as the differential operator can be considered a singular perturbation in both space and time. The stability property of the AVS-FE method, allows us significant flexibility in the construction of FE approximations in both space and time. Thus, in this paper, we take two distinct approaches to the FE discretization of the convection-diffusion problem: i) considering a space-time approach in which the temporal discretization is established using finite elements, and ii) a method of lines approach in which we employ the AVS-FE method in space whereas the temporal domain is discretized using the generalized-α method. We also consider another space-time technique in which the temporal direction is partitioned, thereby leading to finite space-time"slices" in an attempt to reduce the computational cost of the space-time discretizations. We present numerical verifications for these approaches, including numerical asymptotic convergence studies highlighting optimal convergence properties. Furthermore, in the spirit of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan [2–6], the AVS-FE method also leads to readily available a posteriori error estimates through a Riesz representer of the residual of the AVS-FE approximations. Hence, the norm of the resulting local restrictions of these estimates serve as error indicators in both space and time for which we present multiple numerical verifications in mesh adaptive *** Codes 65M60 65M12 65M20 65M50 Copyright © 2020, The Authors. All rights reserved.

关键词： Finite element method

Randomized Compiling for Scalable Quantum computing on a Noisy Superconducting Quantum Processor

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Physical Review X 2021年第4期11卷 041039-041039页

作者： Akel Hashim Ravi K. Naik Alexis Morvan Jean-Loup Ville Bradley Mitchell John Mark Kreikebaum Marc Davis Ethan Smith Costin Iancu Kevin P. O’Brien Ian Hincks Joel J. Wallman Joseph Emerson Irfan Siddiqi Quantum Nanoelectronics Laboratory Department of Physics University of California at Berkeley Berkeley California 94720 USA Graduate Group in Applied Science and Technology University of California at Berkeley Berkeley California 94720 USA Computational Research Division Lawrence Berkeley National Lab Berkeley California 94720 USA Materials Sciences Division Lawrence Berkeley National Lab Berkeley California 94720 USA Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology Cambridge Massachusetts 02142 USA Quantum Benchmark Inc. Kitchener Ontario N2H 5G5 Canada Keysight Technologies Canada Kanata Ontario K2K 2W5 Canada Institute for Quantum Computing and Department of Applied Mathematics University of Waterloo Waterloo Ontario N2L 3G1 Canada

The successful implementation of algorithms on quantum processors relies on the accurate control of quantum bits (qubits) to perform logic gate operations. In this era of noisy intermediate-scale quantum (NISQ) computing, systematic miscalibrations, drift, and crosstalk in the control of qubits can lead to a coherent form of error that has no classical analog. Coherent errors severely limit the performance of quantum algorithms in an unpredictable manner, and mitigating their impact is necessary for realizing reliable quantum computations. Moreover, the average error rates measured by randomized benchmarking and related protocols are not sensitive to the full impact of coherent errors and therefore do not reliably predict the global performance of quantum algorithms, leaving us unprepared to validate the accuracy of future large-scale quantum computations. Randomized compiling is a protocol designed to overcome these performance limitations by converting coherent errors into stochastic noise, dramatically reducing unpredictable errors in quantum algorithms and enabling accurate predictions of algorithmic performance from error rates measured via cycle benchmarking. In this work, we demonstrate significant performance gains under randomized compiling for the four-qubit quantum Fourier transform algorithm and for random circuits of variable depth on a superconducting quantum processor. Additionally, we accurately predict algorithm performance using experimentally measured error rates. Our results demonstrate that randomized compiling can be utilized to leverage and predict the capabilities of modern-day noisy quantum processors, paving the way forward for scalable quantum computing.

关键词： Quantum algorithms Quantum benchmarking Quantum control Quantum information with solid state qubits

Conductivity in the Square Lattice Hubbard Model at High Temperatures: Importance of Vertex Corrections

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Physical Review Letters 2019年第3期123卷 036601-036601页

作者： J. Vučičević J. Kokalj R. Žitko N. Wentzell D. Tanasković J. Mravlje Scientific Computing Laboratory Center for the Study of Complex Systems Institute of Physics Belgrade University of Belgrade Pregrevica 118 11080 Belgrade Serbia University of Ljubljana Faculty of Civil and Geodetic Engineering Jamova 2 Ljubljana Slovenia Jozef Stefan Institute Jamova 39 SI-1000 Ljubljana Slovenia University of Ljubljana Faculty of Mathematics and Physics Jadranska 19 Ljubljana Slovenia Center for Computational Quantum Physics Simons Foundation Flatiron Institute New York New York 10010 USA

Recent experiments on cold atoms in optical lattices allow for a quantitative comparison of the measurements to the conductivity calculations in the square lattice Hubbard model. However, the available calculations do not give consistent results, and the question of the exact solution for the conductivity in the Hubbard model remained open. In this Letter, we employ several complementary state-of-the-art numerical methods to disentangle various contributions to conductivity and identify the best available result to be compared to experiment. We find that, at relevant (high) temperatures, the self-energy is practically local, yet the vertex corrections remain rather important, contrary to expectations. The finite-size effects are small even at the lattice size 4×4, and the corresponding Lanczos diagonalization result is, therefore, close to the exact result in the thermodynamic limit.

关键词： Electrical conductivity 2-dimensional systems Mott insulators Strongly correlated systems Dynamical mean field theory Exact diagonalization Hubbard model Lattice models in condensed matter Monte Carlo methods Numerical Renormalization Group

VARIATIONAL STRUCTURE AND TWO-DIMENSIONAL SUBSONIC JET FLOWS FOR COMPRESSIBLE EULER SYSTEM WITH GENERAL INCOMING FLOWS

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arXiv 2020年

作者： Li, Yan Shi, Wenhui Tang, Lan Xie, Chunjing School of Mathematical Sciences Shanghai Jiao Tong University 800 Dongchuan Road Shanghai200240 China School of Mathematics Monash University 9 Rainforest Walk ClaytonVIC3800 Australia School of Mathematics and Statistics Central China Normal University Hubei Wuhan430079 China School of mathematical Sciences Institute of Natural Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing CMA-Shanghai Shanghai Jiao Tong University 800 Dongchuan Road Shanghai China

In this paper, we proved the well-posedness theory of compressible subsonic jet flows for two-dimensional steady Euler system with general incoming horizontal velocity as long as the flux is larger than a critical value. One of the key observations is that the stream function formulation for two-dimensional compressible steady Euler system enjoys a variational structure even when the flows have nontrivial vorticity, so that the jet problem can be reformulated as a domain variation problem. This variational structure helps to adapt the framework developed by Alt, Caffarelli, and Friedman to study the jet problem, which is a Bernoulli type free boundary problem. A major technical point to analyze the jet flows is that the inhomogeneous terms in the rescaled equation near the free boundary are always small, even when the vorticity of the flows is big. Copyright © 2020, The Authors. All rights reserved.

关键词： Vorticity

Quantum solvers for plane-wave hamiltonians: Abridging virtual spaces through the optimization of pairwise correlations

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arXiv 2020年

作者： Bylaska, Eric J. Song, Duo Bauman, Nicholas P. Kowalski, Karol Claudino, Daniel Humble, Travis S. Pacific Northwest National Laboratory RichlandWA99352 United States Quantum Computing Institute Oak Ridge National Laboratory Oak RidgeTN37831 United States Computer Science and Mathematics Oak Ridge National Laboratory Oak RidgeTN37831 United States Computational Sciences and Engineering Oak Ridge National Laboratory Oak RidgeTN37831 United States

For many-body methods such as MCSCF and CASSCF, in which the number of one-electron orbitals are optimized and independent of basis set used, there are no problems with using plane-wave basis sets. However, for methods currently used in quantum computing such as select configuration interaction (CI) and coupled cluster (CC) methods, it is necessary to have a virtual space that is able to capture a significant amount of electron-electron correlation in the system. The virtual orbitals in a pseudopotential plane-wave Hartree–Fock calculation, because of Coulomb repulsion, are often scattering states that interact very weakly with the filled orbitals. As a result, very little correlation energy is captured from them. The use of virtual spaces derived from the one-electron operators have also been tried, and while some correlation is captured, the amount is quite low. To overcome these limitations, we have been developing new classes of algorithms to define virtual spaces by optimizing orbitals from small pairwise CI Hamiltonians, which we term as correlation optimized virtual orbitals with the abbreviation COVOs. With these procedures we have been able to derive virtual spaces, containing only a few orbitals, that are able to capture a significant amount of correlation. Besides, using these derived basis sets for quantum computing calculations targeting full CI (FCI) quality-results, they can also be used in other many-body approaches, including CC and Møller–Plesset perturbation theories, and open up the door to many-body calculations for pseudopotential plane-wave basis set methods. For the H2 molecule, we were able to obtain good agreement with FCI/cc-pVTZ results with just 4 virtual orbitals, for both FCI and quantum simulations. Copyright © 2020, The Authors. All rights reserved.

关键词： Molecules

An Adaptive Lagrangian Algorithm for Optimal Portfolio Deleveraging with Cross-Impact

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Journal of Systems Science & Complexity 2017年第5期30卷 1121-1135页

作者： XU Fengmin SUN Min DAI Yuhong School of Economics and Finance Xi'an Jiaotong UniversityXi'an 710049China School of Mathematics and Statistics Xi'an Jiaotong UniversityXi'an 710049China The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Science//Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

This paper considers the problem of optimal portfolio deleveraging, which is a crucial problem in finance. Taking the permanent and temporary price cross-impact into account, the authors establish a quadratic program with box constraints and a singly quadratic constraint. Under some assumptions, the authors give an optimal trading priority and show that the optimal solution must be achieved when the quadratic constraint is active. Further, the authors propose an adaptive Lagrangian algorithm for the model, where a piecewise quadratic root-finding method is used to find the Lagrangian multiplier. The convergence of the algorithm is established. The authors also present some numerical results, which show the usefulness of the algorithm and validate the optimal trading priority.

关键词： Adaptive Lagrangian algorithm deleveraging price cross-impact

Meson spectroscopy from spectral densities in lattice gauge theories

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arXiv 2024年

作者： Bennett, Ed Del Debbio, Luigi Forzano, Niccoló Hill, Ryan C. Hong, Deog Ki Hsiao, Ho Lee, Jong-Wan David Lin, C.-J. Lucini, Biagio Lupo, Alessandro Piai, Maurizio Vadacchino, Davide Zierler, Fabian Swansea Academy of Advanced Computing Swansea University Bay Campus Fabian Way SwanseaSA1 8EN United Kingdom Higgs Centre for Theoretical Physics School of Physics and Astronomy The University of Edinburgh Peter Guthrie Tait Road EdinburghEH9 3FD United Kingdom Department of Physics Faculty of Science and Engineering Swansea University Singleton Park SwanseaSA2 8PP United Kingdom School of Physics and Astronomy The University of Edinburgh EdinburghEH9 3FD United Kingdom Department of Physics Pusan National University Busan46241 Korea Republic of Extreme Physics Institute Pusan National University Busan46241 Korea Republic of Institute of Physics National Yang Ming Chiao Tung University 1001 Ta-Hsueh Road Hsinchu30010 Taiwan Daejeon34126 Korea Republic of Centre for High Energy Physics Chung-Yuan Christian University Chung-Li32023 Taiwan Centre for Theoretical and Computational Physics National Yang Ming Chiao Tung University 1001 Ta-Hsueh Road Hsinchu30010 Taiwan Department of Mathematics Faculty of Science and Engineering Swansea University Bay Campus Fabian Way SwanseaSA1 8EN United Kingdom Aix-Marseille Université Université de Toulon CNRS CPT Marseille France Centre for Mathematical Sciences University of Plymouth PlymouthPL4 8AA United Kingdom

Spectral densities encode non-perturbative information that enters the calculation of a plethora of physical observables in strongly coupled field theories. Phenomenological applications encompass aspects of standard-model hadronic physics, observable at current colliders, as well as correlation functions characterizing new physics proposals, testable in future experiments. By making use of numerical data produced in a Sp(4) lattice gauge theory with matter transforming in an admixture of fundamental and 2-index antisymmetric representations of the gauge group, we perform a systematic study to demonstrate the effectiveness of recent technological progress in the reconstruction of spectral densities. To this purpose, we write and test new software packages that use energy-smeared spectral densities to analyze the mass spectrum of mesons. We assess the effectiveness of different smearing kernels and optimize the smearing parameters to the characteristics of available lattice ensembles. We generate new ensembles for the theory in consideration, with lattices that have a longer extent in the time direction with respect to the spatial ones. We run our tests on these ensembles, obtaining new results about the spectrum of light mesons and their excitations. We make available our algorithm and software for the extraction of spectral densities, that can be applied to theories with other gauge groups, including the theory of strong interactions (QCD) governing hadronic physics in the standard model. © 2024, CC BY.

关键词： Bosons

Convergence Analysis of an Unconditionally Energy Stable Linear Crank-Nicolson Scheme for the Cahn-Hilliard Equation

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数学研究 2018年第1期51卷 89-114页

作者： Lin Wang Haijun Yu School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 China NCMIS & LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing 100190 China

Efficient and unconditionally stable high order time marching schemes are very important but not easy to construct for nonlinear phase dynamics. In this paper, we propose and analysis an efficient stabilized linear Crank-Nicolson scheme for the Cahn-Hilliard equation with provable unconditional stability. In this scheme the non-linear bulk force are treated explicitly with two second-order linear stabilization terms. The semi-discretized equation is a linear elliptic system with constant coefficients, thus robust and efficient solution procedures are guaranteed. Rigorous error analysis show that, when the time step-size is small enough, the scheme is second order accurate in time with a prefactor controlled by some lower degree polynomial of 1/ε. Here ε is the interface thickness parameter. Numerical results are presented to verify the accuracy and efficiency of the scheme.

关键词： phase field model Cahn-Hilliard equation unconditionally stable stabilized semi-implicit scheme high order time marching

Application of bounded total variation denoising in urban traffic analysis

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arXiv 2018年

作者： Tang, Shanshan Yu, Haijun School of Mathematical Sciences University of Chinese Academy of Sciences LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing100190 China NCMIS LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences

While it is believed that denoising is not always necessary in many big data applications, we show in this paper that denoising is helpful in urban traffic analysis by applying the method of bounded total variation denoising to the urban road traffic prediction and clustering problem. We propose two easy-to-implement methods to estimate the parameter noise strength in the denoising algorithm, and apply the denoising algorithm to GPS-based traffic data from Beijing taxi system. For the traffic prediction problem, numerical experiments show that the predicting accuracy is improved by applying the proposed bounded total variation denoising algorithm. For the clustering problem, we apply a recently developed parameter-free clustering analysis method for more than one hundred urban road segments in Beijing based on their velocity profiles. Much better clustering results are obtained after denoising. Copyright © 2018, The Authors. All rights reserved.

关键词： Cluster analysis