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首页> 外文期刊>SIAM Journal on Scientific Computing >Stochastic algorithms with Hermite cubic spline interpolation for global estimation of solutions of boundary value problems
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Stochastic algorithms with Hermite cubic spline interpolation for global estimation of solutions of boundary value problems

机译：带Hermite三次样条插值的随机算法用于边值问题解的整体估计

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Here we construct two new functional Monte Carlo algorithms for the numerical solution of three-dimensional Dirichlet boundary value problems for the linear and nonlinear Helmholtz equations. These algorithms are based on estimating the solution and, if necessary, its partial derivatives at grid nodes using first Monte Carlo methods followed by an appropriate interpolation scheme. This allows us to obtain an approximation of the solution in the entire domain, which is not commonly done with Monte Carlo. The Monte Carlo methods used in this paper include the random walk on spheres method and the walk in balls process (with possible branching in the nonlinear case) and the stochastic application of Green's formula for global approximation, cubic spline interpolation is used. One of the proposed approximation algorithms is based on Hermite cubic spline interpolation and utilizes estimates of the solution and its first partial derivatives. The other algorithm is based on Lagrange tricubic spline interpolation on a uniform grid and needs only estimates of the solution. An important problem is to find the optimal values of the interpolation algorithm parameters, such as the number of grid nodes and the sample volume for this we use a stochastic optimization approach; i.e., for both of the proposed approximation algorithms we construct upper bounds of the approximation errors and minimize computational cost functions constrained by a fixed error criterion with a stochastic technique. To study the effectiveness of these proposed methods, we make a comparison between three functional algorithms, which are based on the use of the Hermite cubic splines, on the Lagrange tricubic splines, and on more common multilinear interpolation.
机译：在这里，我们针对线性和非线性Helmholtz方程的三维Dirichlet边值问题的数值解，构造了两个新的功能性Monte Carlo算法。这些算法的基础是，首先使用蒙特卡洛方法，然后采用适当的插值方案，然后对网格节点处的解进行估计，并在必要时估计其偏导数。这使我们能够获得整个域中解的近似值，而蒙特卡洛通常不这样做。本文使用的蒙特卡罗方法包括球上随机游走法和球内游走过程（在非线性情况下可能会分支），以及格林公式在全局逼近中的随机应用，使用三次样条插值。所提出的一种近似算法基于Hermite三次样条插值，并利用了解的估计值及其一阶偏导数。另一种算法基于均匀网格上的Lagrange三三次样条插值，并且仅需要解的估计。一个重要的问题是要找到插值算法参数的最优值，例如网格节点的数量和样本量，为此我们采用了随机优化方法;即，对于两种提出的近似算法，我们都构造了近似误差的上限，并通过随机技术最小化了由固定误差准则约束的计算成本函数。为了研究这些建议方法的有效性，我们对三种功能算法进行了比较，这三种算法基于Hermite三次样条，Lagrange三三次样条以及更常见的多线性插值。

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