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Analysis of the second order bdf scheme with variable steps for the molecular beam epitaxial model without slope selection

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arXiv 2020年

作者： Liao, Hong-Lin Song, Xuehua Tang, Tao Zhou, Tao Department of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing211106 China Department of Mathematics Nanjing University of Aeronautics and Astronautics 211101 China Department of Mathematics and International Center for Mathematics Southern University of Science and Technology ShenzhenGuangdong China Division of Science and Technology BNU-HKBU United International College ZhuhaiGuangdong China NCMIS & LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

In this work, we are concerned with the stability and convergence analysis of the second order BDF (BDF2) scheme with variable steps for the molecular beam epitaxial model without slope selection. We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint. Then we show that it preserves an energy dissipation law if the adjacent time-step ratios rk := τk/τk−1 2 norm stability and rigorous error estimates are established, under the same step-ratios constraint that ensuring the energy stability., i.e., 0 k MSC Codes 65M06, 65M12 Copyright © 2020, The Authors. All rights reserved.

关键词： Number theory

Energy-preserving exponential integrable numerical method for stochastic cubic wave equation with additive noise

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arXiv 2019年

作者： Cui, Jianbo Hong, Jialin Ji, Lihai Sun, Liying School of Mathematics Georgia Institute of Technology AtlantaGA30332 United States Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Institute of Applied Physics and Computational Mathematics Beijing100094 China

In this paper, we present an energy-preserving exponentially integrable numerical method for stochastic wave equation with cubic nonlinearity and additive noise. We first apply the spectral Galerkin method to discretizing the original equation and show that this spatial discretization possesses an energy evolution law and certain exponential integrability property. Then the exponential integrability property of the exact solution is deduced by proving the strong convergence of the semi-discretization. To propose a full discrete numerical method which could inherit both the energy evolution law and the exponential integrability, we use the splitting technique and averaged vector field method in the temporal direction. Combining these structure-preserving properties with regularity estimates of the exact and the numerical solutions, we obtain the strong convergence rate of the numerical method. Numerical experiments coincide with these theoretical results. Copyright © 2019, The Authors. All rights reserved.

关键词： Stochastic systems

Nonuniform 3D finite difference elastic wave simulation on staggered grids

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arXiv 2020年

作者： Gao, Longfei Ghattas, Omar Keyes, David Oden Institute for Computational Engineering and Sciences The University of Texas at Austin AustinTX78712 United States Oden Institute for Computational Engineering and Sciences Jackson School of Geosciences Department of Mechanical Engineering The University of Texas at Austin AustinTX78712 United States Applied Mathematics and Computational Science Program Extreme Computing Research Center King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

We present an approach to simulate the 3D isotropic elastic wave propagation using nonuniform finite difference discretization on staggered grids. Specifically, we consider simulation domains composed of layers of uniform grids with different grid spacings, separated by nonconforming interfaces. We demonstrate that this layer-wise finite difference discretization has the potential to significantly reduce the simulation cost, compared to its fully uniform counterpart. Stability of such a discretization is achieved by using specially designed difference operators, which are variants of the standard difference operators with adaptations near boundaries or interfaces, and penalty terms, which are appended to the discretized wave system to weakly impose boundary or interface conditions. Combined with specially designed interpolation operators, the discretized wave system is shown to preserve the energy conserving property of the continuous elastic wave equation, and a fortiori ensure the stability of the simulation. Numerical examples are presented to demonstrate the efficacy of the proposed simulation approach. © 2020, CC BY.

关键词： Elastic waves

Stable recovery of entangled weights: Towards robust identification of deep neural networks from minimal samples

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arXiv 2021年

作者： Fiedler, Christian Fornasier, Massimo Klock, Timo Rauchensteiner, Michael Intelligent Control Systems Group Max Planck Institute for Intelligent Systems Heisenbergstr. 3 Stuttgart70569 Germany Institute for Data Science in Mechanical Engineering RWTH-Aachen University Dennewartstraße 27 Aachen50868 Germany Department of Mathematics Bolzmannstraße 3 85748 Garching Germany Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway Department of Mathematics University of San Diego California San Diego United States

In this paper we approach the problem of unique and stable identifiability of generic deep artificial neural networks with pyramidal shape and smooth activation functions from a finite number of input-output samples. More specifically we introduce the so-called entangled weights, which compose weights of successive layers intertwined with suitable diagonal and invertible matrices depending on the activation functions and their shifts. We prove that entangled weights are completely and stably approximated by an efficient and robust algorithm as soon as O(D2 × m) nonadaptive input-output samples of the network are collected, where D is the input dimension and m is the number of neurons of the network. Moreover, we empirically observe that the approach applies to networks with up to O(D × mL) neurons, where mL is the number of output neurons at layer L. Provided knowledge of layer assignments of entangled weights and of remaining scaling and shift parameters, which may be further heuristically obtained by least squares, the entangled weights identify the network completely and uniquely. To highlight the relevance of the theoretical result of stable recovery of entangled weights, we present numerical experiments, which demonstrate that multilayered networks with generic weights can be robustly identified and therefore uniformly approximated by the presented algorithmic pipeline. In contrast backpropagation cannot generalize stably very well in this setting, being always limited by relatively large uniform error. In terms of practical impact, our study shows that we can relate input-output information uniquely and stably to network parameters, providing a form of explainability. Moreover, our method paves the way for compression of overparametrized networks and for the training of minimal complexity *** Codes 65D15, 68T07, 90C26 Copyright © 2021, The Authors. All rights reserved.

关键词： Deep neural networks

An aggregation model of cockroaches with fast-or-slow motion dichotomy

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arXiv 2021年

作者： Eliaš, Ján Izuhara, Hirofumi Mimura, Masayasu Tang, Bao Quoc Institute of Mathematics and Scientific Computing University of Graz Heinrichstrasse 36 Graz8010 Austria Faculty of Engineering University of Miyazaki 1-1 Gakuen Kibanadai Nishi Miyazaki889-2192 Japan Graduate School of Integrated Sciences for Life Hiroshima University 1-3-1 Kagamiyama Hiroshima Higashi-Hiroshima City739-8526 Japan

We propose a mathematical model, namely a reaction-diffusion system, to describe social behaviour of cockroaches. An essential new aspect in our model is that the dispersion behaviour due to overcrowding effect is taken into account as a counterpart to commonly studied aggregation. This consideration leads to an intriguing new phenomenon which has not been observed in the literature. Namely, due to the competition between aggregation towards areas of higher concentration of pheromone and dispersion avoiding overcrowded areas, the cockroaches aggregate more at the transition area of pheromone. Moreover, we also consider the fast reaction limit where the switching rate between active and inactive subpopulations tends to infinity. By utilising improved duality and energy methods, together with the regularisation of heat operator, we prove that the weak solution of the reaction-diffusion system converges to that of a reaction-cross-diffusion system. © 2021, CC0.

关键词： Dispersions

Analysis of GWTC-3 with fully precessing numerical relativity surrogate models

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arXiv 2023年

作者： Islam, Tousif Vajpeyi, Avi Shaik, Feroz H. Haster, Carl-Johan Varma, Vijay Field, Scott E. Lange, Jacob O’Shaughnessy, Richard Smith, Rory Department of Mathematics University of Massachusetts DartmouthMA02747 United States Center for Scientific Computing and Data Science Research University of Massachusetts DartmouthMA02747 United States Department of Statistics The University of Auckland New Zealand School of Physics and Astronomy Monash University VIC3800 Australia OzGrav The ARC Centre of Excellence for Gravitational Wave Discovery ClaytonVIC3800 Australia Department of Physics and Astronomy University of Nevada Las Vegas 4505 South Maryland Parkway Las VegasNV89154 United States Nevada Center for Astrophysics University of Nevada Las VegasNV89154 United States LIGO Laboratory Massachusetts Institute of Technology CambridgeMA02139 United States Department of Physics Kavli Institute for Astrophysics and Space Research Massachusetts Institute of Technology 77 Massachusetts Ave CambridgeMA02139 United States Max Planck Institute for Gravitational Physics Albert Einstein Institute PotsdamD-14476 Germany Center for Computational Relativity and Gravitation Rochester Institute of Technology RochesterNY14623 United States University of Texas AustinTX78712 United States

The third Gravitational-Wave Transient Catalog (GWTC-3) contains 90 binary coalescence candidates detected by the LIGO-Virgo-KAGRA Collaboration (LVK). We provide a re-analysis of binary black hole (BBH) events using a recently developed numerical relativity (NR) waveform surrogate model, NRSur7dq4, that includes all ≤ 4 spin-weighted spherical harmonic modes as well as the complete physical effects of precession. Properties of the remnant black holes’ (BH’s) mass, spin vector, and kick vector are found using an associated remnant surrogate model NRSur7dq4Remnant. Both NRSur7dq4 and NRSur7dq4Remnant models have errors comparable to numerical relativity simulations and allow for high-accuracy parameter estimates. We restrict our analysis to 47 BBH events that fall within the regime of validity of NRSur7dq4 (mass ratios greater than 1/6 and total masses greater than 60M☉). While for most of these events our results match the LVK analyses that were obtained using the semi-analytical models such as IMRPhenomXPHM and SEOBNRv4PHM, we find that for more than 20% of events the NRSur7dq4 model recovers noticeably different measurements of black hole properties like the masses and spins, as well as extrinsic properties like the binary inclination and distance. For instance, GW150914_095045 exhibits noticeable differences in spin precession and spin magnitude measurements. Other notable findings include one event (GW191109_010717) that constrains the effective spin χeff to be negative at a 99.3% credible level and two events (GW191109_010717 and GW200129_065458) with well-constrained kick velocities. Furthermore, compared to the models used in the LVK analyses, NRSur7dq4 recovers a larger signal-to-noise ratio and/or Bayes factors for several events. While these findings have important astrophysical implications, some of the more interesting events are affected by transient detector noise, which can be challenging to remove from short-duration signals. Copyright © 2023, The Au

关键词： Black holes

Collective Neutrino Oscillations in Three Flavors on Qubit and Qutrit Processors

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arXiv 2025年

作者： Spagnoli, Luca Goss, Noah Roggero, Alessandro Rrapaj, Ermal Cervia, Michael J. Patwardhan, Amol V. Naik, Ravi K. Balantekin, A. Baha Younis, Ed Santiago, David I. Siddiqi, Irfan Aldaihan, Sheakha Physics Department University of Trento Via Sommarive 14 TrentoI-38123 Italy INFN-TIFPA Trento Institute of Fundamental Physics and Applications Trento Italy Quantum Nanoelectronics Laboratory Department of Physics University of California at Berkeley BerkeleyCA94720 United States Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory BerkeleyCA94720 United States National Energy Research Scientific Computing Center Lawrence Berkeley National Laboratory BerkeleyCA94720 United States Department of Physics University of California BerkeleyCA94720 United States RIKEN iTHEMS Saitama Wako351-0198 Japan Department of Physics University of Washington SeattleWA98195 United States School of Physics and Astronomy University of Minnesota MinneapolisMN55455 United States Department of Physics New York Institute of Technology New YorkNY10023 United States Department of Physics University of Wisconsin MadisonWI53706 United States Department of Physics Gettysburg College GettysburgPA17325 United States

Collective neutrino flavor oscillations are of primary importance in understanding the dynamic evolution of core-collapse supernovae and subsequent terrestrial detection, but also among the most challenging aspects of numerical simulations. This situation is complicated by the quantum many-body nature of the problem due to neutrino-neutrino interactions which demands a quantum treatment. An additional complication is the presence of three flavors, which often is approximated by the electron flavor and a heavy lepton flavor. In this work, we provide both qubit and qutrit encodings for all three flavors, and develop optimized quantum circuits for the time evolution and analyze the Trotter error. We conclude our study with a hardware experiment of a system of two neutrinos with superconducting hardware: the IBM Torino device for qubits and AQT device for qutrits. We find that error mitigation greatly helps in obtaining a signal consistent with simulations. While hardware results are comparable at this stage, we expect the qutrit setup to be more convenient for large-scale simulations since it does not suffer from probability leakage into nonphsycial qubit space, unlike the qubit setup. Copyright © 2025, The Authors. All rights reserved.

关键词： Neutrons

Lattice studies of Sp(2N) gauge theories using GRID 40

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Lattice studies of Sp(2N) gauge theories using GRID

40th International Symposium on Lattice Field Theory, LATTICE 2023

作者： Bennett, Ed Boyle, Peter Del Debbio, Luigi Forzano, Niccolò Hong, Deog Ki Lee, Jong-Wan Lenz, Julian Lin, C.-J. David Lucini, Biagio Lupo, Alessandro Piai, Maurizio Vadacchino, Davide Swansea Academy of Advanced Computing Swansea University Bay Campus Fabian Way Wales SwanseaSA1 8EN United Kingdom School of Physics and Astronomy University of Edinburgh EdinburghEH9 3FD United Kingdom Physics Department Brookhaven National Laboratory UptonNY11973 United States 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 Department of Physics Pusan National University Busan46241 Korea Republic of Institute for Extreme Physics Pusan National University Busan46241 Korea Republic of Daejeon34126 Korea Republic of Institute of Physics National Yang Ming Chiao Tung University 1001 Ta-Hsueh Road Hsinchu30010 Taiwan 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 Wales SwanseaSA1 8EN United Kingdom Centre for Mathematical Sciences University of Plymouth PlymouthPL4 8AA United Kingdom

Four-dimensional gauge theories based on symplectic Lie groups provide elegant realisations of the microscopic origin of several new physics models. Numerical studies pursued on the lattice provide quantitative information necessary for phenomenological applications. To this purpose, we implemented Sp(2N) gauge theories using Monte Carlo techniques within Grid, a performant framework designed for the numerical study of quantum field theories on the lattice. We show the first results obtained using this library, focusing on the case-study provided by the Sp(4) theory coupled to Nas = 4 Wilson-Dirac fermions transforming in the 2-index antisymmetric representation. In particular, we discuss preliminary tests of the algorithm and we test some of its main functionalities. © Copyright owned by the author(s) under the terms of the Creative Commons.

关键词： Lattice theory

Tensor network quantum virtual machine for simulating quantum circuits at exascale

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arXiv 2021年

作者： Nguyen, Thien Lyakh, Dmitry Dumitrescu, Eugene Clark, David Larkin, Jeff McCaskey, Alexander Quantum Computing Institute Oak Ridge National Laboratory Oak RidgeTN United States Computer Science and Mathematics Division Oak Ridge National Laboratory Oak RidgeTN United States National Center for Computational Sciences Oak Ridge National Laboratory Oak RidgeTN United States Computational Sciences and Engineering Division Oak Ridge National Laboratory Oak RidgeTN United States NVIDIA Corp. Santa ClaraCA United States

The numerical simulation of quantum circuits is an indispensable tool for development, verification and validation of hybrid quantum-classical algorithms on near-term quantum co-processors. The emergence of exascale high-performance computing (HPC) platforms presents new opportunities for pushing the boundaries of quantum circuit simulation. We present a modernized version of the Tensor Network Quantum Virtual Machine (TNQVM) which serves as a quantum circuit simulation backend in the eXtreme-scale ACCelerator (XACC) framework. The new version is based on the general purpose, scalable tensor network processing library, ExaTN, and provides multiple configurable quantum circuit simulators enabling either exact quantum circuit simulation via the full tensor network contraction, or approximate quantum state representations via suitable tensor factorizations. Upon necessity, stochastic noise modeling from real quantum processors is incorporated into the simulations by modeling quantum channels with Kraus tensors. By combining the portable XACC quantum programming frontend and the scalable ExaTN numerical backend we introduce an end-to-end virtual quantum development environment which can scale from laptops to future exascale platforms. We report initial benchmarks of our framework which include a demonstration of the distributed execution, incorporation of quantum decoherence models, and simulation of the random quantum circuits used for the certification of quantum supremacy on the Google Sycamore superconducting architecture. Copyright © 2021, The Authors. All rights reserved.

关键词： Circuit simulation