论文标题
加权Sobolev和BESOV课程的最佳随机正交和雅各比的重量
Optimal randomized quadrature for weighted Sobolev and Besov classes with the Jacobi weight on the ball
论文作者
论文摘要
我们考虑数值集成$$ {\ rm int} _d（f）= \ int _ {\ Mathbb {b}^{d}}^{d}} f（x）w_μ（x）w_μ（x）dx $$，用于加权的sobolev类$ bw^^{r} _ {r} _ { $bb_τ^r（l_ {p，μ}）在随机情况下设置中，其中$w_μ，\，μ\ ge0，$是球上的经典jacobi重量$ \ bbb b^d $，$ \ bbb b^d $，$ 1 \ le p \ le p \ le p \ le \ le \ le \ le \ le \ le \ le \ iffty $，$ r> r> r> r> r> r> r> r> r> r> r> r> r> r> r> r> r> r> f $，和$ 0 <uft uft uft uft ufty ufty ufty ufty。对于上述两个类，我们在随机情况设置中获得最佳正交错误的订单为$ n^{ - r/d-1/2+（1/p-1/2）_+} $。与确定性案例设置中最佳正常错误的订单相比，当$ n^{ - r/d} $在确定性案例设置中，当$ p> 1 $时，随机性可以有效地改善收敛顺序。
We consider the numerical integration $${\rm INT}_d(f)=\int_{\mathbb{B}^{d}}f(x)w_μ(x)dx $$ for the weighted Sobolev classes $BW^{r}_{p,μ}$ and the weighted Besov classes $BB_τ^r(L_{p,μ})$ in the randomized case setting, where $w_μ, \,μ\ge0,$ is the classical Jacobi weight on the ball $\Bbb B^d$, $1\le p\le \infty$, $r>(d+2μ)/p$, and $0<τ\le\infty$. For the above two classes, we obtain the orders of the optimal quadrature errors in the randomized case setting are $n^{-r/d-1/2+(1/p-1/2)_+}$. Compared to the orders $n^{-r/d}$ of the optimal quadrature errors in the deterministic case setting, randomness can effectively improve the order of convergence when $p>1$.
