2016/12/31 by Boris N. Khoromskij, Khoromskij, B., S. Repin +1
Computer Science · Engineering · Mathematics · #65F10 #65F30 #65F50 #65N35 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1701.00039
openalex publication_date 2016/12/31 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
We consider an iteration method for solving an elliptic type boundary value problem A u=f, where a positive definite operator A is generated by a quasi--periodic structure with rapidly changing coefficients (typical period is characterized by a small parameter ε) . The method is based on using a simpler operator A0 (inversion of A0 is much simpler than inversion of A), which can be viewed as a preconditioner for A. We prove contraction of the iteration method and establish explicit estimates of the contraction factor q. Certainly the value of q depends on the difference between A and A0. For typical quasi--periodic structures, we establish simple relations that suggest an optimal A0 (in a selected set of "simple" structures) and compute the corresponding contraction factor. Further, this allows us to deduce fully computable two--sided a posteriori estimates able to control numerical solutions on any iteration. The method is especially efficient if the coefficients of A admit low rank representations and algebraic operations are performed in tensor structured formats. Under moderate assumptions the storage and solution complexity of our approach depends only weakly (merely linear-logarithmically) on the frequency parameter 1/ε, providing the FEM approximation of the order of O(ε1+p), p>0.