High-precision randomized iterative methods for the random feature method

This paper focuses on solving large-scale, ill-conditioned, and overdetermined sparse least squares problems that arise from numerical partial differential equations (PDEs), mainly from the random feature method. To address these difficulties, we introduce (1) a count sketch technique to sketch the original matrix to a smaller matrix; (2) a QR factorization or a singular value decomposition for the smaller matrix to obtain the preconditioner, which is multiplied to the original matrix from the right-hand side; (3) least squares iterative solvers to solve the preconditioned least squares system. Therefore, the methods we develop are termed CSQRP-LSQR and CSSVDP-LSQR. Under mild assumptions, we prove that the preconditioned problem holds a condition number whose upper bound is independent of the condition number of the original matrix, and provide error estimates for both methods. Ample numerical experiments, including least squares problems arising from two-dimensional and three-dimensional PDEs and the Florida Sparse Matrix Collection, are conducted. Both methods are comparable to or even better than direct methods in accuracy and are computationally more efficient for large-scale problems. This opens up the applicability of the random feature method for PDEs over complicated geometries with high-complexity solutions.