学术报告报告题目1: Nonasymptotic convergence rate of the quasi-Monte Carlo method: Applications to linear elliptic partial differential equations with lognormal coefficients and importance samplings
报告人: 刘洋博士（阿卜杜拉国王科技大学）
报告时间: 2023年9月12日（星期二）下午3:00-4:00
报告摘要：This study analyzes the nonasymptotic convergence behavior of the quasi-Monte Carlo (QMC) method with applications to linear elliptic partial differential equations (PDEs) with lognormal coefficients. We derive a nonasymptotic convergence estimate depending on the specific integrands, the input dimensionality, and the finite number of samples used in the QMC quadrature. We discuss the effects of the variance and dimensionality of the input random variable. Then, we apply the QMC method with importance sampling (IS) to approximate deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient in bounded domains of $\mathbb{R}^d$, where the random coefficient is modeled as a stationary Gaussian random field parameterized by the trigonometric and wavelet-type basis.
报告题目2: On the convergence rate of projection based quasi-Monte Carlo method
报告人: 欧阳督博士（清华大学）
报告时间: 2023年9月12日（星期二）下午4:00-5:00
报告摘要：We consider the problem of computing an approximation to the expectation $E[h(W)]$, where $h$ is a smooth function on $R^d$ and $W$ is a standard normal distributed random variable. We use a projection based quasi-Monte Carlo (QMC) method. By compositing the projection operator and the inverse distribution function of $W$, the modified integrand is bounded on $[0,1]^d$ and has bounded variation in the sense of Hardy and Krause. We subdivide the boundary growth conditions into several different cases, and obtain better convergence rate for “QMC friendly” conditions. Furthermore, by applying importance sampling, we obtain a convergence rate $O(n^{-1+\epsilon})$ for the boundary growth conditions that are not even “QMC friendly”. More importantly, we achieve the convergence rate $O(n^{-3/2+\epsilon})$ for randomized QMC methods. Our framework theoretically demonstrates the improvement of using importance sampling in QMC methods.
报告地点: 37号楼3A02室
邀请人: 何志坚教授
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数学学院
2023年9月6日