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RKDG Methods with Multi-resolution WENO Limiters for Solving Steady-State Problems on Triangular Meshes

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Communications on applied mathematics and Computation 2024年第3期6卷 1575-1599页

作者： Jun Zhu Chi-Wang Shu Jianxian Qiu State Key Laboratory of Mechanics and Control of Mechanical Structures and Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing University of Aeronautics and AstronauticsNanjing 210016JiangsuChina Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamen 361005FujianChina

In this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state problems on triangular meshes. A troubled cell indicator extended from structured meshes to unstructured meshes is constructed to identify triangular cells in which the application of the limiting procedures is required. In such troubled cells, the multi-resolution WENO limiting methods are used to the hierarchical L^(2) projection polynomial sequence of the DG solution. Through using the RKDG methods with multi-resolution WENO limiters, the optimal high-order accuracy can be gradually reduced to first-order in the triangular troubled cells, so that the shock wave oscillations can be well suppressed. In steady-state simulations on triangular meshes, the numerical residual converges to near machine zero. The proposed spatial reconstruction methods enhance the robustness of classical DG methods on triangular meshes. The good results of these RKDG methods with multi-resolution WENO limiters are verified by a series of two-dimensional steady-state problems.

关键词： RKDG method Steady-state problem Multi-resolution WENO limiter Triangular mesh Machine zero

Optimal Second-Order Rates for Quantum Soft Covering and Privacy Amplification

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IEEE Transactions on Information Theory 2024年第7期70卷 5077-5091页

作者： Shen, Yu-Chen Gao, Li Cheng, Hao-Chung National Taiwan University Department of Electrical Engineering Taipei10617 Taiwan University of Houston Department of Mathematics HoustonTX77204 United States Graduate Institute of Communication Engineering Department of Mathematics Institute of Applied Mathematical Sciences Center for Quantum Science and Engineering National Taiwan University Department of Electrical Engineering Taipei10617 Taiwan Physics Division National Center for Theoretical Sciences Taipei10617 Taiwan Quantum Computing Center New Taipei City23678 Taiwan

We study quantum soft covering and privacy amplification against quantum side information. The former task aims to approximate a quantum state by sampling from a prior distribution and querying a quantum channel. The latter task aims to extract uniform and independent randomness against quantum adversaries. For both tasks, we use trace distance to measure the closeness between the processed state and the ideal target state. We show that the minimal amount of samples for achieving an \varepsilon -covering is given by the (1-\varepsilon ) -hypothesis testing information (with additional logarithmic additive terms), while the maximal extractable randomness for an \varepsilon -secret extractor is characterized by the conditional (1-\varepsilon ) -hypothesis testing entropy. When performing independent and identical repetitions of the tasks, our one-shot characterizations lead to tight asymptotic expansions of the above-mentioned operational quantities. We establish their second-order rates given by the quantum mutual information variance and the quantum conditional information variance, respectively. Moreover, our results extend to the moderate deviation regime, which are the optimal asymptotic rates when the trace distances vanish at sub-exponential speed. Our proof technique is direct analysis of trace distance without smoothing. © 1963-2012 IEEE.

关键词： Hash functions

Efficient and Accurate Numerical Methods Using the Accelerated Spectral Deferred Correction for Solving Fractional Differential Equations

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Numerical mathematics(Theory,Methods and Applications) 2022年第4期15卷 876-902页

作者： Xuejuan Chen Zhiping Mao George Em Karniadakis School of Science Jimei UniversityXiamen 361021China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamenChina Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically,we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm(GMRES).We first derive the correction matrix of the SDC method for FDEs and analyze the convergence region of the SDC *** then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear ODEs as well as fractional phase field models,demonstrating that the accelerated SDC method is much more efficient than the original SDC method,especially for stiff ***,we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh,leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational ***,for long-time integration of FDEs,using the geometric mesh leads to great computational savings as the total number of degrees of freedom required is relatively small.

关键词： Stiff problem generalized minimal residual geometric mesh refinement long time evolution fractional phase field models

computing the many-body Green's function with adaptive variational quantum dynamics

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arXiv 2023年

作者： Gomes, Niladri Williams-Young, David B. de Jong, Wibe A. Applied Mathematics and Computing Sciences Division Lawrence Berkeley National Laboratory BerkeleyCA94720 United States

We present a method to compute the many-body real time Green’s function using an adaptive variational quantum dynamics simulation approach. The real-time Green’s function involves time evolution of a quantum state with one additional electron with respect to the ground state wavefunction that is first expressed as a linear linear combination of state vectors. The real time evolution and the Green’s function is obtained by combining the dynamics of the individual statevectors in the linear combination. The use of the adaptive protocol enables us to generate compact ansätze on-the-fly while running the simulation. In order to improve the convergence of spectral features Padé approximants are applied to obtain the Fourier transform of the Green’s function. We demonstrate the evaluation of the Green’s function on a IBM Q quantum computer. As a part of our error mitigation strategy, we develop a resolution enhancing method that we successfully apply on the noisy data from the real quantum hardware. Copyright © 2023, The Authors. All rights reserved.

关键词： Quantum computers

Moment-Based Multi-Resolution HWENO Scheme for Hyperbolic Conservation Laws

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Communications in Computational Physics 2022年第7期32卷 364-400页

作者： Jiayin Li Chi-Wang Shu Jianxian Qiu School of Mathematical Sciences Xiamen UniversityXiamenFujian 361005P.R.China Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathe-matical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamenFujian 361005P.R.China

In this paper,a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory(HWENO)scheme is designed for hyperbolic conservation *** main idea of this scheme is derived from our previous work[***.,446(2021)110653],in which the integral averages of the function and its first order derivative are used to reconstruct both the function and its first order derivative values at the ***,in this paper,only the function values at the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed by using the information of the zeroth and first order *** addition,an extra modification procedure is used to modify those first order moments in the troubledcells,which leads to an improvement of stability and an enhancement of resolution near *** obtain the same order of accuracy,the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the general HWENO scheme and is more compact than the generalWENO ***,the linear weights are not unique and are independent of the node position,and the CFL number can still be 0.6whether for the one or two dimensional case,which has to be 0.2 in the two dimensional case for other HWENO *** numerical examples are given to demonstrate the stability and resolution of such moment-based multi-resolution HWENO scheme.

关键词： Moment-based scheme multi-resolution scheme HWENO scheme hyperbolic conservation laws KXRCF troubled-cell indicator HLLC-flux

SUPERFAST DIRECT INVERSION OF THE NONUNIFORM DISCRETE FOURIER TRANSFORM VIA HIERARCHICALLY SEMI-SEPARABLE LEAST SQUARES

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arXiv 2024年

作者： Wilber, Heather Epperly, Ethan N. Barnett, Alex H. Department of Applied Mathematics University of Washington SeattleWA98105 United States Division of Computing and Mathematical Sciences California Institute of Technology PasadenaCA91125 United States Center for Computational Mathematics Flatiron Institute New YorkNY10010 United States

A direct solver is introduced for solving overdetermined linear systems involving nonuniform discrete Fourier transform matrices. Such matrices can be transformed into a Cauchy-like form that has hierarchical low rank structure. The rank structure of this matrix is explained, and it is shown that the ranks of the relevant submatrices grow only logarithmically with the number of columns of the matrix. A fast rank-structured hierarchical approximation method based on this analysis is developed, along with a hierarchical least-squares solver for these and related systems. This result is a direct method for inverting nonuniform discrete transforms with a complexity that is usually nearly linear with respect to the degrees of freedom in the problem. This solver is benchmarked against various iterative and direct solvers in the setting of inverting the one-dimensional type-II (or forward) transform, for a range of condition numbers and problem sizes (up to 4 × 106 by 2 × 106). These experiments demonstrate that this method is especially useful for large problems with multiple right-hand *** Codes 65T50, 65F55, 65F20 Copyright © 2024, The Authors. All rights reserved.

关键词： Discrete Fourier transforms

Leveraging Interpolation Models and Error Bounds for Verifiable Scientific Machine Learning

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arXiv 2024年

作者： Chang, Tyler Gillette, Andrew Maulik, Romit Mathematics and Computer Science Division Argonne National Laboratory United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory United States Information Sciences and Technology Department The Pennsylvania State University United States

Effective verification and validation techniques for modern scientific machine learning workflows are challenging to devise. Statistical methods are abundant and easily deployed, but often rely on speculative assumptions about the data and methods involved. Error bounds for classical interpolation techniques can provide mathematically rigorous estimates of accuracy, but often are difficult or impractical to determine computationally. In this work, we present a best-of-both-worlds approach to verifiable scientific machine learning by demonstrating that (1) multiple standard interpolation techniques have informative error bounds that can be computed or estimated efficiently;(2) comparative performance among distinct interpolants can aid in validation goals;(3) deploying interpolation methods on latent spaces generated by deep learning techniques enables some interpretability for black-box models. We present a detailed case study of our approach for predicting lift-drag ratios from airfoil images. Code developed for this work is available in a public Github repository. © 2024, CC BY-NC-SA.

关键词： Interpolation

Tensor neural networks for high-dimensional Fokker-Planck equations

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arXiv 2024年

作者： Wang, Taorui Hu, Zheyuan Kawaguchi, Kenji Zhang, Zhongqiang Karniadakis, George Em Department of Mathematical Sciences Worcester Polytechnic Institute WorcesterMA United States Department of Computer Science National University of Singapore 119077 Singapore Division of Applied Mathematics Brown University ProvidenceRI02912 United States Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory RichlandWA United States

We solve high-dimensional steady-state Fokker-Planck equations on the whole space by applying tensor neural networks. The tensor networks are a linear combination of tensor products of one-dimensional feedforward networks or a linear combination of several selected radial basis functions. The use of tensor feedforward networks allows us to efficiently exploit auto-differentiation (in physical variables) in major Python packages while using radial basis functions can fully avoid auto-differentiation, which is rather expensive in high dimensions. We then use the physics-informed neural networks and stochastic gradient descent methods to learn the tensor networks. One essential step is to determine a proper bounded domain or numerical support for the Fokker-Planck equation. To better train the tensor radial basis function networks, we impose some constraints on parameters, which lead to relatively high accuracy. We demonstrate numerically that the tensor neural networks in physics-informed machine learning are efficient for steady-state Fokker-Planck equations from two to ten dimensions. Copyright © 2024, The Authors. All rights reserved.

关键词： Radial basis function networks

The ESPRIT algorithm under high noise: Optimal error scaling and noisy super-resolution

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arXiv 2024年

作者： Ding, Zhiyan Epperly, Ethan N. Lin, Lin Zhang, Ruizhe Department of Mathematics University of California Berkeley United States Department of Computing and Mathematical Sciences California Institute of Technology PasadenaCA United States Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory United States Challenge Institute for Quantum Computation University of California Berkeley United States Simons Institute for the Theory of Computing United States

Subspace-based signal processing techniques, such as the Estimation of Signal Parameters via Rotational Invariant Techniques (ESPRIT) algorithm, are popular methods for spectral estimation. These algorithms can achieve the so-called super-resolution scaling under low noise conditions, surpassing the well-known Nyquist limit. However, the performance of these algorithms under high-noise conditions is not as well understood. Existing state-of-the-art analysis indicates that ESPRIT and related algorithms can be resilient even for signals where each observation is corrupted by statistically independent, mean-zero noise of size O(1), but these analyses only show that the error ∊ decays at a slow rate ∊ = O͂(n−1/2) with respect to the cutoff frequency n (i.e., the maximum frequency of the measurements). In this work, we prove that under certain assumptions, the ESPRIT algorithm can attain a significantly improved error scaling ∊ = O͂(n−3/2), exhibiting noisy super-resolution scaling beyond the Nyquist limit ∊ = O(n−1) given by the Nyquist-–Shannon sampling theorem. We further establish a theoretical lower bound and show that this scaling is optimal. Our analysis introduces novel matrix perturbation results, which could be of independent interest. © 2024, CC BY.

关键词： Cutoff frequency