Numerical Approximation for Stochastic Nonlinear Fractional Diffusion Equation Driven by Rough Noise

基于Malliavin变分法的新渐近展开公式及其在分数布朗运动驱动的粗糙微分方程中的应用

New asymptotic expansion formula via Malliavin calculus and its application to rough differential equation driven by fractional Brownian motion

本文提出了一种基于Malliavin变分法的多维Wiener泛函期望新型渐近展开公式，在较弱条件下证明了一致估计，并应用于分数布朗运动驱动的粗糙微分方程，数值实验验证了其优于正态近似的有效性。

Local discontinuous Galerkin method for the fractional diffusion equation with integral fractional Laplacian

In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian $(-Δ)^{s}$ with $s\in(0,1)$ in two dimensions. We theoretically prove and numerically verify the numerical stability and convergence of the scheme with the convergence rate no worse than $\mathcal{O}(h^{k+\frac{1}{2}})$.

Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for time fractional sub-diffusion and diffusion-wave equations

In this article, we have studied the convergence behavior of the Dirichlet-Neumann and Neumann- Neumann waveform relaxation algorithms for time-fractional sub-diffusion and diffusion-wave equations in 1D & 2D for regular domains, where the dimensionless diffusion coefficient takes different constant values in different subdomains. We first observe that different diffusion coefficients lead to different relaxation parameters for optimal convergence. Using these optimal relaxation parameters, our analysis estimates the slow superlinear convergence of the algorithms when the fractional order of t…

分数阶非线性扩散方程的保结构格式

Structure-preserving scheme for fractional nonlinear diffusion equations

本文提出并分析了一种求解分数阶非线性扩散方程 Cauchy-Dirichlet 问题的保结构数值格式，严格证明了其能保持代数衰减和熄灭现象等连续方程性质，并提出了一种精确计算熄灭时间的新方法。

一维分数阶扩散方程解的正则性

Regularity of the Solution to 1-D Fractional Order Diffusion Equations

本文研究了一维有界域上稳态分数阶扩散方程解的正则性，推翻传统分数阶微分算子假设，推导出分数阶扩散算子核的闭式表达式，并提出基于雅可比多项式的谱近似方法。

Numerical approximation for fractional diffusion equation forced by a tempered fractional Gaussian noise

This paper discusses the fractional diffusion equation forced by a tempered fractional Gaussian noise. The fractional diffusion equation governs the probability density function of the subordinated killed Brownian motion. The tempered fractional Gaussian noise plays the role of fluctuating external source with the property of localization. We first establish the regularity of the infinite dimensional stochastic integration of the tempered fractional Brownian motion and then build the regularity of the mild solution of the fractional stochastic diffusion equation. The spectral Galerkin method i…

An efficient spectral-Galerkin method for fractional reaction-diffusion equations in unbounded domains

In this work, we apply a fast and accurate numerical method for solving fractional reaction-diffusion equations in unbounded domains. By using the Fourier-like spectral approach in space, this method can effectively handle the fractional Laplace operator, leading to a fully diagonal representation of the fractional Laplacian. To fully discretize the underlying nonlinear reaction-diffusion systems, we propose to use an accurate time marching scheme based on ETDRK4. Numerical examples are presented to illustrate the effectiveness of the proposed method.

分数扩散方程的边界条件

Boundary Conditions for Fractional Diffusion

本文推导了分数扩散方程的物理意义边界条件，通过质量平衡方法分析，并讨论了吸收和反射边界条件的理论性质与数值解。

Semi-implicit Taylor schemes for stiff rough differential equations

We study a class of semi-implicit Taylor-type numerical methods that are easy to implement and designed to solve multidimensional stochastic differential equations driven by a general rough noise, e.g. a fractional Brownian motion. In the multiplicative noise case, the equation is understood as a rough differential equation in the sense of T.~Lyons. We focus on equations for which the drift coefficient may be unbounded and satisfies a one-sided Lipschitz condition only. We prove well-posedness of the methods, provide a full analysis, and deduce their convergence rate. Numerical experiments sho…

Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport

We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders s in [0.5,1). We also present an application of the advection-diffu…

Weak error rates for option pricing under linear rough volatility

In quantitative finance, modeling the volatility structure of underlying assets is vital to pricing options. Rough stochastic volatility models, such as the rough Bergomi model [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887-904, 2016], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, $H < 1/2$, over reasonable timescales. Both time series of asset prices and option-derived price data indicate that $H$ often takes values close to $0.1$ or less, i.e., rougher than Brownian motio…