2015/04/16 by Somaiyeh Rashedi, Sebastian Birk, Rashedi, Somaiyeh +5 · 1 citation
Computer Science · Mathematics · #65F10 #65F30 #65F50 #65H10 #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1504.04395
openalex publication_date 2015/04/16 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28
Block and global Krylov subspace methods have been proposed as methods\nadapted to the situation where one iteratively solves systems with the same\nmatrix and several right hand sides. These methods are advantageous, since they\nallow to cast the major part of the arithmetic in terms of matrix-block vector\nproducts, and since, in the block case, they take their iterates from a\npotentially richer subspace. In this paper we consider the most established\nKrylov subspace methods which rely on short recurrencies, i.e. BiCG, QMR and\nBiCGStab. We propose modifications of their block variants which increase\nnumerical stability, thus at least partly curing a problem previously observed\nby several authors. Moreover, we develop modifications of the "global" variants\nwhich almost halve the number of matrix-vector multiplications. We present a\ndiscussion as well as numerical evidence which both indicate that the\nadditional work present in the block methods can be substantial, and that the\nnew "economic" versions of the "global" BiCG and QMR method can be considered\nas good alternatives to the BiCGStab variants.\n