An economic cascadic tensor multigrid method for solving high dimensional elliptic linear partial differential problems
Provides a more efficient solver for high-dimensional elliptic PDEs, which are computationally expensive in scientific computing.
This paper proposes an economic cascadic tensor multigrid method for high-dimensional elliptic PDEs, reducing computational complexity from O(n^3) to O(n^2) and storage space. Numerical examples verify its effectiveness.
In this paper, based on the tensor form, we propose a class of economic cascadic tensor multigrid method(ECTMG) for solving high dimensional elliptic linear partial differential problems. Compared with traditional methods, the new method not only reduces the storage space but also lowers the computational complexity from $\mathcal{O}(n^{3})$ to $\mathcal{O}(n^{2})$. We analyze the convergence rate of the conjugate gradient method which is based on the tensor form($\mathrm{CG}\_\mathrm{BTF}$), and then provide the convergence analysis for the new method. Finally, the effectiveness of the new method is verified through numerical examples.