Rank Structured Approximation Method for Quasi-Periodic Elliptic Problems

We consider an iteration method for solving an elliptic type boundary value problem {\mathcal{A}u=f}, where a positive definite operator {\mathcal{A}} is generated by a quasi-periodic structure with rapidly changing coefficients (a typical period is characterized by a small parameter ϵ). The method is based on using a simpler operator {\mathcal{A}_{0}} (inversion of {\mathcal{A}_{0}} is much simpler than inversion of {\mathcal{A}}), which can be viewed as a preconditioner for {\mathcal{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 {\mathcal{A}} and {\mathcal{A}_{0}}. For typical quasi-periodic structures, we establish simple relations that suggest an optimal {\mathcal{A}_{0}} (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 {\mathcal{A}} admit low-rank representations and if 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 \frac{1}{\epsilon}.