论文标题
部分可观测时空混沌系统的无模型预测
A Modified Trapezoidal Rule for a Class of Weakly Singular Integrals in n Dimensions
论文作者
论文摘要
在本文中，我们提出和分析一类形式的一类弱的奇异积分，$ i = \ int _ {\ mathbb {r}^n} ϕ（x）s（x）s（x）s（x）dx $ in $ n $ dimensions，$ n $ n $ n $ n $ n（rmath $ n $ n $ n $ n $ n $ n（c） $ s $是弱奇异的内核。可允许的弱奇异核类需要$ s $满足扩张和对称属性，并且足够大，可以包含$ \ frac {p（x）} {| x |^r} $的功能，其中$ r> 0 $> r> 0 $ and $ p（x）$是任何单元素，就是任何单元素，使得$ \ \ \ \ \ \ \ deg} p {d p <r <r <r <r <\ r <\ p <\ r <\ r <\ p <\ p <\ p <\ t text + n $修改的梯形规则是通过校正术语添加的奇异性 - 调节性梯形规则，涉及奇异性周围网格点的校正权重。校正权重通过执行正交规则准确评估某些单一元素并求解相应的线性系统来确定校正权重。这些类型的方法的长期困难是建立了线性系统的非象征性，尽管有很强的数值证据。通过使用代数 - 组合参数，我们表明非单词始终保持并证明修改后的正交规则的一般融合顺序。我们提出数值实验以验证收敛顺序。
In this paper we propose and analyze a general arbitrarily high-order modified trapezoidal rule for a class of weakly singular integrals of the forms $I = \int_{\mathbb{R}^n}ϕ(x)s(x)dx$ in $n$ dimensions, where $ϕ\in C_c^N(\mathbb{R}^n)$ for some sufficiently large $N$ and $s$ is the weakly singular kernel. The admissible class of weakly singular kernel requires $s$ satisfies dilation and symmetry properties and is large enough to contain functions of the form $\frac{P(x)}{|x|^r}$ where $r > 0$ and $P(x)$ is any monomials such that $\text{deg} P < r < \text{deg} P + n$. The modified trapezoidal rule is the singularity-punctured trapezoidal rule added by correction terms involving the correction weights for grid points around singularity. Correction weights are determined by enforcing the quadrature rule exactly evaluates some monomials and solving corresponding linear systems. A long-standing difficulty of these type of methods is establishing the non-singularity of the linear system, despite strong numerical evidences. By using an algebraic-combinatorial argument, we show the non-singularity always holds and prove the general order of convergence of the modified quadrature rule. We present numerical experiments to validate the order of convergence.
