后向蒙特卡洛方法的精度与收敛性

Uncertainty quantification for the BGK model of the Boltzmann equation using multilevel variance reduced Monte Carlo methods

We propose a control variate multilevel Monte Carlo method for the kinetic BGK model of the Boltzmann equation subject to random inputs. The method combines a multilevel Monte Carlo technique with the computation of the optimal control variate multipliers derived from local or global variance minimization problems. Consistency and convergence analysis for the method equipped with a second-order positivity-preserving and asymptotic-preserving scheme in space and time is also performed. Various numerical examples confirm that the optimized multilevel Monte Carlo method outperforms the classical …

多级蒙特卡洛方法加速多层材料随机缺陷计算

Multi-level Monte Carlo acceleration of computations on multi-layer materials with random defects

本文提出多级蒙特卡洛方法以加速随机缺陷材料的计算，通过紧束缚模型测试问题验证了该方法在计算效率上的优势。

A note on concatenation of quasi-Monte Carlo and plain Monte Carlo rules in high dimensions

In this note, we study a concatenation of quasi-Monte Carlo and plain Monte Carlo rules for high-dimensional numerical integration in weighted function spaces. In particular, we consider approximating the integral of periodic functions defined over the $s$-dimensional unit cube by using rank-1 lattice point sets only for the first $d\, (<s)$ coordinates and random points for the remaining $s-d$ coordinates. We prove that, by exploiting a decay of the weights of function spaces, almost the optimal order of the mean squared worst-case error is achieved by such a concatenated quadrature rule as l…

多级拟蒙特卡洛积分的快速正交变换

Fast orthogonal transforms for multi-level quasi-Monte Carlo integration

结合通用快速正交变换方法与多级蒙特卡洛方法，提升拟蒙特卡洛积分效率。

标量不变的蒙特卡洛和多级蒙特卡洛估计均值和方差：一个应用于线性弹性骨组织模拟的应用

Scale-invariant Monte Carlo and multilevel Monte Carlo estimation of mean and variance: An application to simulation of linear elastic bone tissue

本文提出了一种标量不变的误差估计方法，用于高效估计线性弹性骨组织模拟中的均值和方差，通过归一化误差估计优化多级蒙特卡洛计算成本。

美式蒙特卡洛算法的自动反向微分（条件期望）

Automatic Backward Differentiation for American Monte-Carlo Algorithms (Conditional Expectation)

本文推导了包含条件期望算子的算法的反向微分方法，应用于美式金融衍生品估值，提出利用条件期望的线性组合和期望微分性质简化实现。

Digital Nets and Sequences for Quasi-Monte Carlo Methods

Quasi-Monte Carlo methods are a way of improving the efficiency of Monte Carlo methods. Digital nets and sequences are one of the low discrepancy point sets used in quasi-Monte Carlo methods. This thesis presents the three new results pertaining to digital nets and sequences: implementing randomized digital nets, finding the distribution of the discrepancy of scrambled digital nets, and obtaining better quality of digital nets through evolutionary computation. Finally, applications of scrambled and non-scrambled digital nets are provided.

On the effective dimension and multilevel Monte Carlo

I consider the problem of integrating a function $f$ over the $d$-dimensional unit cube. I describe a multilevel Monte Carlo method that estimates the integral with variance at most $ε^{2}$ in $O(d+\ln(d)d_{t}ε^{-2})$ time, for $ε>0$, where $d_{t}$ is the truncation dimension of $f$. In contrast, the standard Monte Carlo method typically achieves such variance in $O(dε^{-2})$ time. A lower bound of order $d+d_{t}ε^{-2}$ is described for a class of multilevel Monte Carlo methods.

拟蒙特卡罗的误差估计

Error estimation for quasi-Monte Carlo

本文介绍了量化拟蒙特卡罗估计误差的新旧方法，指出获取精度与估计精度之间的冲突及不确定性量化过于保守的挑战，并发现RQMC估计的近对称分布有望构建无需中心极限定理或一致方差估计的置信区间。

随机椭圆偏微分方程的拟连续水平蒙特卡洛方法

Quasi continuous level Monte Carlo for random elliptic PDEs

本文提出拟连续水平蒙特卡洛方法，用拟蒙特卡洛序列生成指数分布随机变量，显著降低方差，在随机椭圆偏微分方程示例中明显优于多水平和连续水平蒙特卡洛。

在 Monte Carlo 和多层 Monte Carlo 方法中使用 h-statistics 进行协方差估计

Covariance estimation using h-statistics in Monte Carlo and multilevel Monte Carlo methods

本文提出利用 h-statistics 的 MC 与 MLMC 无偏协方差估计方法，以闭式形式构造无偏均方误差，并通过一维随机椭圆 PDE 验证数值效果。