A Fast Block-Greedy Algorithm for Quasi-optimal Meshless Trial Subspace Selection

Meshless collocation methods are often seen as a flexible alternative to overcome difficulties that may occur with other methods. As various meshless collocation methods gain popularity, finding appropriate settings becomes an important open question. Previously, we proposed a series of sequential-greedy algorithms for selecting quasi-optimal meshless trial subspaces that guarantee stable solutions from meshless methods, all of which were designed to solve a more general problem: “Let $A$ be an $M \times N$ matrix with full rank $M$; choose a large $M \times K$ submatrix formed by $K\leq M$ columns of $A$ such that it is numerically of full rank.” In this paper, we propose a block-greedy algorithm based on a primal/dual residual criterion. Similar to all algorithms in the series, the block-greedy algorithm can be implemented in a matrix-free fashion to reduce the storage requirement. Most significantly, the proposed algorithm reduces the computational cost from the previous $\mathcal{O}(K^4+NK^2)$ to at m...

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