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Compressing the memory variables in constant-Q viscoelastic wave propagation via an improved sum-of-exponentials approximation

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JOURNAL OF COMPUTATIONAL PHYSICS 2024年 518卷

作者： Guo, Xu Jiang, Shidong Xiong, Yunfeng Zhang, Jiwei Shandong Univ Geotech & Struct Engn Ctr Jinan 250061 Shandong Peoples R China Simons Fdn Flatiron Inst Ctr Computat Math New York NY 10010 USA Beijing Normal Univ Sch Math Sci Beijing 100875 Peoples R China Beijing Normal Univ Lab Math & Complex Syst Minist Educ Beijing 100875 Peoples R China Wuhan Univ Sch Math & Stat Wuhan 430072 Hubei Peoples R China Wuhan Univ Hubei Key Lab Computat Sci Wuhan 430072 Hubei Peoples R China

Earth introduces strong attenuation and dispersion to propagating waves. The time-fractional wave equation with very small fractional exponent, based on Kjartansson's constant-Q theory, is widely recognized in the field of geophysics as a reliable model for frequency-independent Q anelastic behavior. Nonetheless, the numerical resolution of this equation poses considerable challenges due to the requirement of storing a complete time history of wavefields. To address this computational challenge, we present a novel approach: a nearly optimal sum-of-exponentials (SOE) approximation to the Caputo fractional derivative with very small fractional exponent, utilizing the machinery of generalized Gaussian quadrature. This method minimizes the number of memory variables needed to approximate the power attenuation law within a specified error tolerance. We establish a mathematical equivalence between this SOE approximation and the continuous fractional stress-strain relationship, relating it to the generalized Maxwell body model. Furthermore, we prove an improved SOE approximation error bound to thoroughly assess the ability of rheological models to replicate the power attenuation law. Numerical simulations on constant-Q viscoacoustic equation in 3D homogeneous media and variable-order P- and S- viscoelastic wave equations in 3D inhomogeneous media are performed. These simulations demonstrate that our proposed technique accurately captures changes in amplitude and phase resulting from material anelasticity. This advancement provides a significant step towards the practical usage of the time-fractional wave equation in seismic inversion.

关键词： Viscoelasticity Fractional derivative sum-of-exponentials approximation Generalized Gaussian quadrature Fast algorithm Rheological model

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An Euler-Maruyama method for Caputo-Hadamard fractional stochastic differential equations on exponential meshes and its fast approximation

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NUMERICAL ALGORITHMS 2025年第2期99卷 573-601页

作者： Li, Min Guo, Shangjiang Hu, Peng Song, Haiyan China Univ Geosci Sch Math & Phys Wuhan 430074 Peoples R China China Univ Geosci Ctr Math Sci Wuhan 430074 Peoples R China China Univ Geosci Sch Math & Phys Wuhan 430074 Peoples R China NingboTech Univ Sch Comp & Data Engn Ningbo 315100 Peoples R China

This paper studies the numerical solutions of Caputo-Hadamard fractional stochastic differential equations. Firstly, we construct an Euler-Maruyama (EM) scheme for the equations, and the corresponding convergence rate is investigated. Secondly, we propose a fast EM scheme based on the sum-of-exponentials approximation to decrease the computational cost of the EM scheme. More concretely, the fast EM scheme reduces the computational cost from O(N2) to O(N log2 N) and the storage from O(N) to O(log2 N) when the final time T approximate to e, where N is the total number of time steps. Moreover, considering the statistical errors from Monte Carlo path approximations, multilevelMonte Carlo (MLMC) techniques are utilized to reduce computational complexity. In particular, for a prescribed accuracy epsilon > 0, the EM scheme and the fast EM scheme, integrated with the MLMC technique, respectively reduce the standard EM scheme's computational complexity from O(epsilon-2- 2/ alpha ) to O(epsilon- 2 /alpha ) and the fast EM scheme's complexity to O(epsilon- 1 /alpha |log epsilon| 3), where 0 < alpha = alpha - 1 /2 < 1/ 2 . Finally, numerical examples are included to verify the theoretical results and demonstrate the performance of our methods.

关键词： Caputo-Hadamard fractional stochastic differential equations EM scheme sum-of-exponentials approximation Fast EM scheme Multilevel Monte Carlo simulation Strong convergence

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Efficient sum-of-exponentials approximations for the heat kernel and their applications

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2015年第3期41卷 529-551页

作者： Jiang, Shidong Greengard, Leslie Wang, Shaobo New Jersey Inst Technol Dept Math Sci Newark NJ 07102 USA NYU Courant Inst Math Sci New York NY 10012 USA

In this paper, we show that efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. In one space dimension, the heat kernel admits an approximation involving a number of terms that is of the order for any and delta a parts per thousand currency signta parts per thousand currency signT, where oee- is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only terms for fixed accuracy oee-. These approximations can be used to accelerate integral equation-based methods for boundary value problems governed by the heat equation in complex geometry. The resulting algorithms are nearly optimal. For N (S) points in the spatial discretization and N (T) time steps, the cost is in terms of both memory and CPU time for fixed accuracy oee-. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.

关键词： Heat kernels sum-of-exponentials approximation Inverse laplace transform Heat potentials

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A FAST HIGH ORDER METHOD FOR THE TIME-FRACTIONAL DIFFUSION EQUATION

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2019年第6期57卷 2829-2849页

作者： Zhu, Hongyi Xu, Chuanju Xiamen Univ Sch Math Sci Xiamen 361005 Peoples R China Xiamen Univ Fujian Prov Key Lab Math Modeling & High Performa Xiamen 361005 Peoples R China

In this paper, we present a fast (3 - alpha)-order numerical method for the Caputo fractional derivative based on the L2 scheme and the sum-of-exponentials (SOE) approximation to the convolution kernel involved in the fractional derivative. This work can be regarded as a continuation of previous works reported by one of the authors in [C.W. Lv and C.J. Xu, SIAM T. Sci. Comput., 38 (2016), pp. A2699-A2724], which constructed and analyzed a (3-alpha)-order L2 time stepping scheme for the time-fractional diffusion equation. It is now extended to take into account the fast SOE evaluation method, which allows us to reduce the storage and overall computational cost from O(N-T) and O(N-T(2)) for the L2 scheme to O(N-epsilon) and O(NTN epsilon), respectively, with N-T being the number of time steps and N-epsilon being the number of fast evaluation terms. The proposed method is then used for the time-fractional diffusion equation in bounded domains. The stability as well as the accuracy of the resulting scheme are rigorously analyzed. Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method. Finally, we extend the discussion to a graded mesh to make the scheme more suitable for problems having weakly singular solutions at the initial time.

关键词： time-fractional diffusion equation time stepping scheme sum-of-exponentials approximation error estimate

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Roundoff error problems in interpolation methods for time-fractional problems

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APPLIED NUMERICAL MATHEMATICS 2024年 203卷 202-224页

作者： Quan, Chaoyu Wang, Shijie Wu, Xu Chinese Univ Hong Kong Sch Sci & Engn Shenzhen 518172 Guangdong Peoples R China Southern Univ Sci & Technol SUSTech Int Ctr Math Dept Math Shenzhen Peoples R China Hong Kong Polytech Univ Dept Appl Math Hong Kong Peoples R China

Roundoff errors often disrupt interpolation methods for time -fractional equations, potentially causing suboptimal convergence or even failure. These issues primarily result from catastrophic cancellations. To address this, we introduce a novel framework for computing coefficients in standard and fast interpolation methods on nonuniform meshes. We propose 6 -cancellation and associated threshold conditions to prevent such cancellations. If the thresholds aren't met, a Taylor expansion technique can be applied. Numerical experiments demonstrate our method's accuracy, on par with the Gauss-Kronrod quadrature, but significantly more efficient, allowing for extensive simulations with hundreds of thousands of time steps.

关键词： Roundoff error Catastrophic cancellation Time-fractional problem sum-of-exponentials approximation

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Space-time finite element method for the distributed-order time fractional reaction diffusion equations

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APPLIED NUMERICAL MATHEMATICS 2020年 152卷 446-465页

作者： Bu, Weiping Ji, Lun Tang, Yifa Zhou, Jie Xiangtan Univ Sch Math & Computat Sci Xiangtan 411105 Hunan Peoples R China Xiangtan Univ Hunan Key Lab Computat & Simulat Sci & Engn Xiangtan 411105 Hunan Peoples R China Chinese Acad Sci Acad Math & Syst Sci ICMSEC LSEC Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China

In this paper, we propose a space-time finite element method for the distributed-order time fractional reaction diffusion equations (DOTFRDEs). First, using the composite trapezoidal rule, composite Simpson's rule and Gauss-Legendre quadrature rule to discretize the distributed-order derivative and employing the finite element method both in space and time, three fully discrete finite element schemes for DOTFRDEs are developed. Second, for the obtained numerical schemes, the existence, uniqueness and stability are discussed. Third, under the hypothesis about the singular behavior of exact solution near t = 0, the convergence of these numerical schemes are investigated in detail based on the graded time mesh. Forth, in order to reduce the storage requirement and computational cost of these numerical schemes, an efficient sum-of-exponentials approximation for the kernel t(-a), alpha is an element of (0, 1) is introduced, and the developed space-time finite element schemes are improved. At last, some numerical tests are given to verify the rationality and effectiveness of our method. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Distributed-order reaction diffusion equation Space-time finite element method Graded time mesh Well-posedness Error estimate sum-of-exponentials approximation

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Fast High-Order Integral Equation Methods for Solving Boundary Value Problems of Two Dimensional Heat Equation in Complex Geometry

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JOURNAL OF SCIENTIFIC COMPUTING 2019年第2期79卷 787-808页

作者： Wang, Shaobo Jiang, Shidong Wang, Jing New Jersey Inst Technol Dept Math Sci Newark NJ 07102 USA Yunnan Univ Sch Informat Sci & Engn Kunming 650091 Yunnan Peoples R China

Efficient high-order integral equation methods have been developed for solving boundary value problems of the heat equation in complex geometry in two dimensions. First, the classical heat potential theory is applied to convert such problems to Volterra integral equations of the second kind via the heat layer potentials, where the unknowns are only on the space-time boundary. However, the heat layer potentials contain convolution integrals in both space and time whose direct evaluation requires NS is the total number of discretization points on the spatial boundary and NT is the total number of time steps. In order to evaluate the heat layer potentials accurately and efficiently, they are split into two partsthe local part containing the temporal integration from t- to t and the history part containing the temporal integration from 0 to t-. The local part can be dealt with efficiently using conventional fast multipole type algorithms. For problems with complex stationary geometry, efficient separated sum-of-exponentials approximations are constructed for the heat kernel and used for the evaluation of the history part. Here all local and history kernels are compressed only once. The resulting algorithm is very efficient with quasilinear complexity in both space and time for both interior and exterior problems. For problems with complex moving geometry, the spectral Fourier approximation is applied for the heat kernel and nonuniform FFT is used to speed up the evaluation of the history part of heat layer potentials. The performance of both algorithms is demonstrated with several numerical examples.

关键词： Heat equation Integral equation methods High-order methods Heat kernels sum-of-exponentials approximation Nonuniform FFT

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A fast Euler-Maruyama method for fractional stochastic differential equations

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2023年第1期69卷 273-291页

作者： Zhang, Jingna Tang, Yifa Huang, Jianfei Chinese Acad Sci Acad Math & Syst Sci LSEC ICMSEC Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China Yangzhou Univ Coll Math Sci Yangzhou 225002 Jiangsu Peoples R China

In this paper, we first construct a fast Euler-Maruyama (EM) method based on the sumof-exponentials approximation for a class of nonlinear fractional stochastic differential equations whose coefficients satisfy a linear growth condition and the Lipschitz continuity. Then the strong convergence of this fast EM method is proved with the order alpha - 1/2, where alpha is an element of (1/2, 1) is the index of Caputo fractional derivative. Finally, numerical experiments are given to verify the theoretical results and demonstrate that the computational efficiency of the fast EM scheme has overwhelming advantages over the original EM method.

关键词： Fractional stochastic differential equations Euler-Maruyama method sum-of-exponentials approximation Strong convergence Computational efficiency

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Efficient Numerical Computation of Time-Fractional Nonlinear Schrodinger Equations in Unbounded Domain

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Communications in Computational Physics 2019年第1期25卷 218-243页

作者： Jiwei Zhang Dongfang Li Xavier Antoine Beijing Computational Science Research Center Beijing 100193P.R.China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Institut Elie Cartan de Lorraine UMR CNRS 7502Universit́e de LorraineInria Nancy-Grand EstSPHINX TeamF-54506 Vandoeuvre-l`es-Nancy CedexFrance

The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrodinger equation set in an unbounded *** first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in[47,48]and a linearization ***,the initial boundary-value problem for the fractional system with ABCs is discretized,a stability analysis is developed and the error estimate O(h^(2)+τ)is *** accel-erate the L1-scheme in time,a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional *** resulting algorithm is highly efficient for long time ***,we end the paper by reporting some numerical simulations to validate the properties(accuracy and efficiency)of the derived scheme.

关键词： Time-fractional nonlinear Schrodinger equation absorbing boundary condition sta-bility analysis convergence analysis sum-of-exponentials approximation

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A fast second-order predictor-corrector method for a nonlinear time-fractional Benjamin-Bona-Mahony-Burgers equation

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NUMERICAL ALGORITHMS 2024年第2期95卷 693-720页

作者： Zhou, Yongtao Li, Cui Stynes, Martin Qingdao Univ Technol Sch Sci Qingdao 266520 Peoples R China Henan Univ Econ & Law Sch Math & Informat Sci Zhengzhou 450046 Peoples R China Beijing Computat Sci Res Ctr Appl & Computat Math Div Beijing 100193 Peoples R China

A second-order predictor-corrector method of Nguyen and Jang (Fract. Calc. Appl. Anal. 20(2), 447-476, 2017) is generalised to graded meshes to solve nonlinear fractional initial-value problems whose typical solutions have a weak singularity at the initial time. In comparison with existing predictor-corrector methods in the literature, this new method significantly improves the numerical accuracy while reducing the computational cost. Moreover, to increase its computational efficiency still further, a corresponding fast algorithm based on the sum-of-exponentials approximation to the kernel of the scheme is described. An error analysis is given for problems whose right-hand sides satisfy a Lipschitz condition. The method (and its fast variant) is then extended to solve the nonlinear time-fractional Benjamin-Bona-Mahony-Burgers (BBMB) initial-boundary value problem, combined with a standard discretisation of the spatial derivatives on a uniform mesh. Estimates are derived for the discrete H-1-norm errors in the computed solution for the BBMB problem;to enable this analysis, a new Gronwall inequality is proved. Finally, several numerical experiments show the sharpness of our theoretical error bounds for both problems.

关键词： Predictor-corrector method Graded mesh Fractional initial-value problem Time-fractional Benjamin-Bona-Mahony-Burgers equation sum-of-exponentials approximation Gronwall inequality

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