Stochastic transport with Lévy noise fully discrete numerical approximation

IF 4.4 2区 数学 Q1 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Andreas Stein , Andrea Barth
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引用次数: 0

Abstract

Semilinear hyperbolic stochastic partial differential equations (SPDEs) find widespread applications in the natural and engineering sciences. However, the traditional Gaussian setting may prove too restrictive, as phenomena in mathematical finance, porous media, and pollution models often exhibit noise of a different nature. To capture temporal discontinuities and accommodate heavy-tailed distributions, Hilbert space-valued Lévy processes or Lévy fields are employed as driving noise terms. The numerical discretization of such SPDEs presents several challenges. The low regularity of the solution in space and time leads to slow convergence rates and instability in space/time discretization schemes. Furthermore, the Lévy process can take values in an infinite-dimensional Hilbert space, necessitating projections onto finite-dimensional subspaces at each discrete time point. Additionally, unbiased sampling from the resulting Lévy field may not be feasible. In this study, we introduce a novel fully discrete approximation scheme that tackles these difficulties. Our main contribution is a discontinuous Galerkin scheme for spatial approximation, derived naturally from the weak formulation of the SPDE. We establish optimal convergence properties for this approach and combine it with a suitable time stepping scheme to prevent numerical oscillations. Furthermore, we approximate the driving noise process using truncated Karhunen-Loève expansions. This approximation yields a sum of scaled and uncorrelated one-dimensional Lévy processes, which can be simulated with controlled bias using Fourier inversion techniques.

带有莱维噪声的随机传输全离散数值近似法
半线性双曲随机偏微分方程(SPDE)在自然科学和工程科学中有着广泛的应用。然而,由于数学金融、多孔介质和污染模型中的现象经常表现出不同性质的噪声,传统的高斯设置可能被证明过于局限。为了捕捉时间不连续性并适应重尾分布,我们采用了希尔伯特空间值的莱维过程或莱维场作为驱动噪声项。此类 SPDE 的数值离散化面临着一些挑战。解在空间和时间上的低规整性导致收敛速度缓慢以及空间/时间离散化方案的不稳定性。此外,Lévy 过程可以在无限维的希尔伯特空间取值,因此必须在每个离散时间点投影到有限维子空间。此外,从所得到的 Lévy 场进行无偏采样可能并不可行。在本研究中,我们引入了一种新颖的完全离散近似方案来解决这些难题。我们的主要贡献是一种用于空间近似的非连续 Galerkin 方案,它是从 SPDE 的弱表述中自然衍生出来的。我们为这种方法建立了最佳收敛特性,并将其与合适的时间步进方案相结合,以防止数值振荡。此外,我们还使用截断的卡尔胡宁-洛埃夫展开来近似驱动噪声过程。这种近似方法产生了一个缩放且不相关的一维莱维过程之和,可以利用傅立叶反演技术在控制偏差的情况下对其进行模拟。
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来源期刊
Mathematics and Computers in Simulation
Mathematics and Computers in Simulation 数学-计算机:跨学科应用
CiteScore
8.90
自引率
4.30%
发文量
335
审稿时长
54 days
期刊介绍: The aim of the journal is to provide an international forum for the dissemination of up-to-date information in the fields of the mathematics and computers, in particular (but not exclusively) as they apply to the dynamics of systems, their simulation and scientific computation in general. Published material ranges from short, concise research papers to more general tutorial articles. Mathematics and Computers in Simulation, published monthly, is the official organ of IMACS, the International Association for Mathematics and Computers in Simulation (Formerly AICA). This Association, founded in 1955 and legally incorporated in 1956 is a member of FIACC (the Five International Associations Coordinating Committee), together with IFIP, IFAV, IFORS and IMEKO. Topics covered by the journal include mathematical tools in: •The foundations of systems modelling •Numerical analysis and the development of algorithms for simulation They also include considerations about computer hardware for simulation and about special software and compilers. The journal also publishes articles concerned with specific applications of modelling and simulation in science and engineering, with relevant applied mathematics, the general philosophy of systems simulation, and their impact on disciplinary and interdisciplinary research. The journal includes a Book Review section -- and a "News on IMACS" section that contains a Calendar of future Conferences/Events and other information about the Association.
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