Fast QMC Matrix-Vector Multiplication
Fast QMC Matrix-Vector Multiplication
复制标题
快速 QMC 矩阵向量乘法
DOI：
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发表时间：
2015
影响因子：
3.1
通讯作者：
C. Schwab
中科院分区：
文献类型：
--
作者：
J. Dick;F. Kuo;Q. Gia;C. Schwab
Quasi-Monte Carlo（QMC）规则$ 1/n sum_ {n = 0}^{n-1} f（oldsymbol {y lodsymbol {y} _n a）$可用于近似表格的积分$ int _ {[0,1 ]^s} f（oldsymbol {y} a），mathrm {d} oldsymbol {y} $，其中$ a $是一个矩阵和$ oldsymbol {y} $是行向量。例如，从具有一般协方差矩阵的正态分布的模拟中，出现了这种积分，这是由带有随机系数的PDE解决方案的期望值的近似值或统计信息的应用。在本文中，我们设计qmc正交点$ oldsymbol {y} _0，...，oldsymbol {y} _ {n-1} in [0,1]^s $，使得矩阵$ y =（ oldsymbol {y} _ {0}^ op，...，oldsymbol {y} _ {n-1}^ op $ op $的行是正交点，一个人可以使用快速的傅立叶变换来计算矩阵矢量产品$ y OldSymbol {a}^ op $，$ oldSymbol {a} in Mathbb {r}^ s $，in $ mathcal {o}（O}（O}） n log n）$ operations和最多$ s-1 $额外的添加。所提出的方法可以应用于晶格规则，多项式晶格规则和某种类型的Korobov $ p $ -set。
该方法通过三个数值实验在计算上进行了说明。第一个测试考虑了具有正态分布和一般协方差矩阵的点的生成，第二个测试将QMC应用于具有均匀分布随机系数的高维，仿射参数，椭圆形的部分微分方程，第三个测试解决了有限元元素 - 元素元素的分配。具有高维，对数正常随机输入数据的椭圆形部分微分方程。与常规实现相比，所有数值测试均显示了快速QMC矩阵方法的计算时间的显着加速，因为尺寸变得较大。
Quasi-Monte Carlo (QMC) rules $1/N sum_{n=0}^{N-1} f(oldsymbol{y}_n A)$ can be used to approximate integrals of the form $int_{[0,1]^s} f(oldsymbol{y} A) ,mathrm{d} oldsymbol{y}$, where $A$ is a matrix and $oldsymbol{y}$ is row vector. This type of integral arises for example from the simulation of a normal distribution with a general covariance matrix, from the approximation of the expectation value of solutions of PDEs with random coefficients, or from applications from statistics. In this paper we design QMC quadrature points $oldsymbol{y}_0, ..., oldsymbol{y}_{N-1} in [0,1]^s$ such that for the matrix $Y = (oldsymbol{y}_{0}^ op, ..., oldsymbol{y}_{N-1}^ op)^ op$ whose rows are the quadrature points, one can use the fast Fourier transform to compute the matrix-vector product $Y oldsymbol{a}^ op$, $oldsymbol{a} in mathbb{R}^s$, in $mathcal{O}(N log N)$ operations and at most $s-1$ extra additions. The proposed method can be applied to lattice rules, polynomial lattice rules and a certain type of Korobov $p$-set.
The approach is illustrated computationally by three numerical experiments. The first test considers the generation of points with normal distribution and general covariance matrix, the second test applies QMC to high-dimensional, affine-parametric, elliptic partial differential equations with uniformly distributed random coefficients, and the third test addresses Finite-Element discretizations of elliptic partial differential equations with high-dimensional, log-normal random input data. All numerical tests show a significant speed-up of the computation times of the fast QMC matrix method compared to a conventional implementation as the dimension becomes large.
影响因子：
4.1
作者：
Graham, I. G.;Kuo, F. Y.;Sloan, I. H.
通讯作者：
Sloan, I. H.
影响因子：
2.1
作者：
Graham, I. G.;Kuo, F. Y.;Sloan, I. H.
通讯作者：
Sloan, I. H.