【作者】 刘新亮；

【导师】 张镭；

【作者基本信息】 上海交通大学 ， 计算数学， 2016， 硕士

【摘要】 本文基于贝叶斯数值均匀化[1],介绍了通过变分方法寻找积分微分方程“最优的”基函数的方法框架。在[2]中,多重调和样条基函数(rough polyharmonic splines,RPS)被用来求解带有粗糙系数的散度型椭圆偏微分方程。RPS基函数可视为这个框架在微分方程上的应用。本文具体讨论这个框架在第二类Fredholm积分方程上的的应用。这个框架的主要思想是通过定义合适的基函数限制条件和内积形式,把方程的解正交分解到残差空间V0和有限维插值基函数空间上,使得残差以内积诱导出的范数意义下尽可能小的落在残差空间V0。这样一组基函数的存在性可以归结到由上述内积定义的RKHS空间的可再生核函数是否具有的严格对称正定的性质。之后我们介绍如何这种基函数推广应用到求解第二类Fredholm积分方程上的,分别论证Galerkin方法和collocation方法的可行性,在假设积分算子是紧算子,并且核函数K(s,t)满足一致李普西茨条件的前提下,collcation方法通过一步迭代得到的迭代collocation方法具有至少H32阶的收敛速度,而且相比于传统的多项式插值基函数,此方法对于解的光滑性的要求更弱。我们还对基函数的衰减性质进行了数值上的分析和讨论。更多还原

【Abstract】 We introduce a variational framework based on Bayesian numerical homogenization^[1]to identify optimal basis functions for integro-differential equation.Rough polyharmonic splines^[2]are developed especially for equation with rough coefficients as a type of technique of numerical homogenization.In this paper we apply this framework to Fredholm integral equations of the second kind.After imposing appropriate constraints to the interpolation basis functions and defining inner product associated with the integral equation,the residual of the interpolation attains its minimum with respect to the induced norm.And the residual is located in the space V₀,which is an orthogonal complement space to the space spanned by the interpolation basis function.Then we show how to apply the method to Fredholm integral equations of the second kind.Assuming the integral operator is compact and the kernel K（s,t）satisfies uniform Lipschitz conditon,the iterated colloca-tion method has a convergence rate of H³2.Compared with the traditional polynomial basis,the method has weak requirement on the smoothness of the solution.We also check the decay property of the basis and its behavior with respect to different kind of kernels numerically.更多还原