Publication: Stochastic algorithms with Hermite cubic spline interpolation for global estimation of solutions of boundary value problems
All || By Area || By Year| Title | Stochastic algorithms with Hermite cubic spline interpolation for global estimation of solutions of boundary value problems | Authors/Editors* | Makarov, R.N. and Shkarupa, E.V. |
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| Where published* | SIAM Journal of Scientific Computing |
| How published* | Journal |
| Year* | 2008 |
| Volume | 30 |
| Number | 1 |
| Pages | 169-188 |
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| Keywords | |
| Link | |
| Abstract |
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 Greenâs 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. |
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