Randomized Gram-Schmidt process with application to GMRES
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A randomized Gram-Schmidt algorithm is developed for orthonormalization of high-dimensional vectors or QR factorization. The proposed process can be less computationally expensive than the classical Gram-Schmidt process while being at least as numerically stable as the modified Gram-Schmidt process. Our approach is based on random sketching, which is a dimension reduction technique consisting in estimation of inner products of high-dimensional vectors by inner products of their small efficiently-computable random projections, so-called sketches. This allows to perform the projection step in Gram-Schmidt process on sketches rather than high-dimensional vectors with a minor computational cost. This also provides an ability to efficiently certify the output. The proposed Gram-Schmidt algorithm can provide computational cost reduction in any architecture. The benefit of random sketching can be amplified by exploiting multi-precision arithmetic. We provide stability analysis for multi-precision model with coarse unit roundoff for standard high-dimensional operations. Numerical stability is proven for the unit roundoff independent of the (high) dimension of the problem. The proposed Gram-Schmidt process can be applied to Arnoldi iteration and result in new Krylov subspace methods for solving high-dimensional systems of equations or eigenvalue problems. Among them we chose randomized GMRES method as a practical application of the methodology.
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