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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:  ``\emph{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 talk, 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 $\calo(K^4+NK^2)$ to at most $\calo(NK^2)$. Numerical examples are given to demonstrate how this efficient and ready-to-use approach can benefit the stability and applicability of meshless collocation methods.

Kansa method, kernel collocation, radial basis function, adaptive greedy algorithm, basis selection.