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arXiv 2018年

作者： Daus, Esther S. Jin, Shi Liu, Liu Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstrasse 8-10 Wien1040 Austria School of Mathematical Sciences Institute of Natural Sciences Shanghai Jiao Tong University Shanghai200240 China Institute for Computational Engineering and Sciences University of Texas at Austin AustinTX78705 United States

In [L. Liu and S. Jin, Multiscale Model. Simult., 16, 1085-1114, 2018], spectral convergence and long-Time decay of the numerical solution to-wards the global equilibrium of the stochastic Galerkin approximation for the Boltzmann equation with random inputs in the initial data and collision kernel for hard potentials and Maxwellian molecules under Grad's angular cutoff were established using the hypocoercive properties of the collisional kinetic model. One assumption for the random perturbation of the collision kernel is that the perturbation is in the order of the Knudsen number, which can be very small in the fluid dynamical regime. In this article, we remove this smallness assumption, and establish the same results but now for random perturbations of the collision kernel that can be of order one. The new analysis relies on the establishment of a spectral gap for the numerical collision *** Codes 35Q20, 65M70 Copyright © 2018, The Authors. All rights reserved.

关键词： Stochastic systems

Tangible reduction in learning sample complexity with large classical samples and small quantum system

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arXiv 2019年

作者： Song, Wooyeong Wieśniak, Marcin Liu, Nana Pawlowski, Marcin Lee, Jinhyoung Kim, Jaewan Bang, Jeongho Department of Physics Hanyang University Seoul04763 Korea Republic of Seoul02792 Korea Republic of Institute of Theoretical Physics and Astrophysics Faculty of Mathematics Physics and Informatics University of Gdańsk Gdańsk80-308 Poland International Centre for Theory of Quantum Technologies University of Gdańsk Gdańsk80-308 Poland Institute of Natural Science Shanghai Jiao Tong University Shanghai200240 China Ministry of Education Key Laboratory in Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai200240 China University of Michigan-Shanghai Jiao Tong University Joint Institute Shanghai200240 China School of Computational Sciences Korea Institute for Advanced Study Seoul02455 Korea Republic of Electronics and Telecommunications Research Institute Daejeon34129 Korea Republic of

Quantum computation requires large classical datasets to be embedded into quantum states in order to exploit quantum parallelism. However, this embedding requires considerable resources in general. It would therefore be desirable to avoid it, if possible, for noisy intermediate-scale quantum (NISQ) implementation. Accordingly, we consider a classical-quantum hybrid architecture, which allows large classical input data, with a relatively small-scale quantum system. This hybrid architecture is used to implement a sampling oracle. It is shown that in the presence of noise in the hybrid oracle, the effects of internal noise can cancel each other out and thereby improve the query success rate. It is also shown that such an immunity of the hybrid oracle to noise directly and tangibly reduces the sample complexity in the framework of computational learning theory. This NISQ-compatible learning advantage is attributed to the oracle’s ability to handle large input features. Copyright © 2019, The Authors. All rights reserved.

关键词： Quantum optics

Mixed frameworks and structure preserving integration for coupled electro-elastodynamics

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PAMM 2019年第1期19卷

作者： Marlon Franke Rogelio Ortigosa Alexander Janz Antonio J. Gil Peter Betsch Karlsruhe Institute of Technology Institute of Mechanics Otto-Ammann-Platz 9 76131 Karlsruhe Germany Technical University of Cartagena Computational Mechanics & Scientific Computing Group 30202 Cartagena Spain Swansea University Zienkiewicz Centre for Computational Engineering Fabian Way Swansea SA1 8EN United Kingdom

In the present contribution new approaches for the design of structure preserving time integrators for nonlinear coupled problems are proposed. Polyconvexity inspired energy functionals are obtained by using the rediscovered tensor cross product which greatly simplifies the algebra, see [1]. In this connection an extended kinematic set, consisting of the right Cauchy-Green tensor, its co-factor and its Jacobian, is introduced. On this basis coupled problems like e.g. non-linear thermo-elastodynamics, see [2] or electro-elastodynamics, see [3], can be considered. Furthermore the formulations are readily extendible for mixed Hu-Washizu type formulations where the extended kinematic set is introduced as unknown field. In particular in [1] an elegant cascade system of kinematic constraints was introduced for elastodynamics, crucial for the satisfaction of the required conservation properties of a new family of energy momentum (EM) consistent time integrators. The objective of the present contribution is the introduction of new mixed variational principles for EM consistent time integrators in electro-elastodynamics, hence bridging the gap between the previous works [3] and [1], opening the possibility to a variety of new finite element implementations, see [4]. The following characteristics of the proposed EM consistent time integrator make it very appealing: (i) the new family of time integrators can be shown to be thermodynamically consistent and second order accurate; (ii) piecewise discontinuous interpolation of the mixed fields is carried out in order to obtain a computational cost comparable to that of standard displacement, electric potential formulations. Eventually, the superior numerical performance of the proposed formulation is demonstrated.

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On energy dissipation theory and numerical stability for time-fractional phase field equations

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arXiv 2018年

作者： Tang, Tao Yu, Haijun Zhou, Tao Department of Mathematics Southern University of Science and Technology Shenzhen China NCMIS LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing China

For the time-fractional phase field models, the corresponding energy dissipation law has not been settled on both the continuous level and the discrete level. In this work, we shall address this open issue. More precisely, we prove for the first time that the time-fractional phase field models indeed admit an energy dissipation law of an integral type. In the discrete level, we propose a class of finite difference schemes that can inherit the theoretical energy stability. Our discussion covers the time-fractional gradient systems, including the time-fractional Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional molecular beam epitaxy models. Numerical examples are presented to confirm the theoretical results. Moreover, a numerical study of the coarsening rate of random initial states depending on the fractional parameter α reveals that there are several coarsening stages for both time-fractional Cahn-Hilliard equation and timefractional molecular beam epitaxy model, while there exists a -α/3 power law coarsening *** Codes 65M12, 65M06, 35Q99, 74A50 Copyright © 2018, The Authors. All rights reserved.

关键词： Energy dissipation

Graph Ranking Guarantees for Numerical Approximations to Katz Centrality

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Procedia Computer Science 2017年 108卷 68-78页

作者： Eisha Nathan Geoffrey Sanders James Fairbanks Van Emden Henson David A. Bader School of Computational Science and Engineering Georgia Institute of Technology Center for Applied Scientific Computing Lawrence Livermore National Laboratory Georgia Tech Research Institute

Graphs and networks are prevalent in modeling relational datasets from many fields of research. By using iterative solvers to approximate graph measures (specifically Katz Centrality), we can obtain a ranking vector consisting of a number for each vertex in the graph identifying its relative importance. We use the residual to accurately estimate how much of the ranking from an approximate solution matches the ranking given by the exact solution. Using probabilistic matrix norms and applying numerical analysis to the computation of Katz Centrality, we obtain bounds on the accuracy of the approximation compared to the exact solution with respect to the highly ranked nodes. This relates the numerical accuracy of the linear solver to the data analysis accuracy of finding the correct ranking. In particular, we answer the question of which pairwise rankings are reliable given an approximate solution to the linear system. Experiments on many real-world networks up to several million vertices and several hundred million edges validate our theory and show that we are able to accurately estimate large portions of the approximation. By analyzing convergence error, we develop confidence in the ranking schemes of data mining.

关键词： graphs data analysis numerical accuracy katz centrality ranking

Extending Perfect Spatial Hashing to Index Tuple-based Graphs Representing Super Carbon Nanotubes

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Procedia Computer Science 2017年 108卷 435-444页

作者： Michael Burger Giang Nam Nguyen Christian Bischof Graduate School of Computational Engineering Dolivostr. 15 Darmstadt Germany Institute for Scientific Computing Mornewegstr. 30 Darmstadt Germany

In this paper, we demonstrate how to extend perfect spatial hashing (PSH) in order to hash multidimensional scientific data. As a use case we employ the problem domain of indexing nodes in a graph that represents Super Carbon Nanotubes (SCNTs). The goal of PSH is to hash multidimensional data without collisions. Since PSH results from the research on computer graphics, its principles and methods have only been tested on 2- and 3-dimensional problems. In our case, we need to hash up to 28 dimensions. In contrast to the original applications of PSH, we do not focus on GPUs as target hardware but on an efficient CPU implementation. Thus, this paper highlights the extensions to the original algorithm to make it suitable for higher dimensions. Comparing the compression and performance results of the new PSH based graphs and a structure-tailored custom data structure in our parallelized SCNT simulation software, we find that PSH in some cases achieves better compression by a factor of 1.7 while only increasing the total runtime by several percent. In particular, after our extension, PSH can also be employed to index sparse multidimensional scientific data from other domains where PSH can avoid additional index-structures like KD- or R-trees.

关键词： perfect spatial hashing data indexing super carbon nanotubes simulation

A fully discrete energy stable scheme for a phase-field moving contact line model with variabledensities and viscosities

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arXiv 2018年

作者： Zhu, Guangpu Chen, Huangxin Sun, Shuyu Yao, Jun Qingdao266580 China Sch. of Math. Sci. and Fujian Prov. Key Lab. on Math. Modeling and High Perf. Scientific Computing Xiamen University Xiamen361005 China Computational Transport Phenomena Laboratory Division of Physical Science and Engineering King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this work, we propose a fully discrete energy stable scheme for the phase-field moving contact line model with variable densities and viscosities. The mathematical model consists of a Cahn-Hilliard equation, a Navier-Stokes equation and the generalized Navier boundary condition. A scalar auxiliary variable is introduced to transform the nonlinear potential into an equivalent form, allowing the nonlinear potential to be treated effectively and semi-explicitly. A pressure stabilization method is used to decouple the computation of velocity and pressure. Some subtle implicit-explicit treatments are adopted to deal with convention and stress terms. An artificial term is added to balance the explicit nonlinear term originating from the surface energy at fluid-solid interface. We establisharigorous proof of energy stability for the proposed time-marching *** a finite difference methodon staggered grids isusedto spatially discretize the constructed time-marchingscheme. We prove that the fully discrete scheme satisfiesthe discrete energy dissipationlaw. Numerical results demonstrate accuracy and energy stability of the proposed scheme. Using our numerical scheme, we analyze theevolution of kinetic energy, mixing energy and surface energy at fluid-solid interfaceduring the bubble rising process. Three-dimensional droplet spreading is also investigated onachemically patterned surface. Our numerical simulation accurately predicts the expected energy evolutions and it successfully reproduces expected phenomenathat an oil droplet contractsinwardson a hydrophobiczone and spreadsoutwards quickly on a hydrophiliczone. Copyright © 2018, The Authors. All rights reserved.

关键词： Finite difference method

Two-scale sparse finite element approximations

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Science China mathematics 2016年第4期59卷 789-808页

作者： LIU Fang ZHU JinWei School of Statistics and Mathematics Central University of Finance and Economics The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and ScientificEngineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of Sciences

To reduce computational cost,we study some two-scale finite element approximations on sparse grids for elliptic partial differential equations of second order in a general *** any tensor product domain ?R^d with d = 2,3,we construct the two-scale finite element approximations for both boundary value and eigenvalue problems by using a Boolean sum of some existing finite element approximations on a coarse grid and some univariate fine grids and hence they are cheaper *** applications,we obtain some new efficient finite element discretizations for the two classes of problem:The new two-scale finite element approximation on a sparse grid not only has the less degrees of freedom but also achieves a good accuracy of approximation.

关键词： combination discretization eigenvalue finite element postprocessing two-scale

A New Fully Polynomial Time Approximation Scheme for the Interval Subset Sum Problem

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arXiv 2017年

作者： Diao, Rui Liu, Ya-Feng Dai, Yu-Hong State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

The interval subset sum problem (ISSP) is a generalization of the well-known subset sum problem. Given a set of intervals {[ai,1, ai,2]}ni=1 and a target integer T, the ISSP is to find a set of integers, at most one from each interval, such that their sum best approximates the target T but cannot exceed it. In this paper, we first study the computational complexity of the ISSP. We show that the ISSP is relatively easy to solve compared to the 0-1 Knapsack problem (KP). We also identify several subclasses of the ISSP which are polynomial time solvable (with high probability), albeit the problem is generally NP-hard. Then, we propose a new fully polynomial time approximation scheme (FPTAS) for solving the general ISSP problem. The time and space complexities of the proposed scheme are O (n max {1/ǫ, log n}) and O (n + 1/ǫ), respectively, where ǫ is the relative approximation error. To the best of our knowledge, the proposed scheme has almost the same time complexity but a significantly lower space complexity compared to the best known scheme. Both the correctness and efficiency of the proposed scheme are validated by numerical simulations. In particular, the proposed scheme successfully solves ISSP instances with n = 100, 000 and ǫ = 0.1% within one second. Copyright © 2017, The Authors. All rights reserved.

关键词： computational complexity

The Cauchy two-matrix model, C-Toda lattice and CKP hierarchy

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arXiv 2018年

作者： Li, Chunxia Li, Shi-Hao School of Mathematical Sciences Capital Normal University Beijing100048 China Lsec Institute of Computational Mathematics and Scientific Engineering Computing Amss Chinese Academy of Sciences P.O.Box 2719 Beijing100190 China Department of Mathematical Sciences University of the Chinese Academy of Sciences Beijing China

This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hierarchy with the help of orthogonal polynomials theory and Toda-type equations. Starting from the symmetric reduction of Cauchy biorthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing time flows. Then, matrix integral solutions to the C-Toda lattice are extended to give solutions to the CKP hierarchy which reveals the time-dependent partition function of the Cauchy two-matrix model is nothing but the τ -function of the CKP hiearchy. At last, the connection between the Cauchy two-matrix model and Bures ensemble is established from the point of view of integrable *** Codes 37K10, 33C47, 15A15, 41A21 Copyright © 2018, The Authors. All rights reserved.

关键词： Polynomials