In a Jacobi--Davidson (JD) type method for singular value decomposition (SVD) problems, called JDSVD, a large symmetric and generally indefinite correction equation is solved iteratively at each outer iteration, which constitutes the inner iterations and dominates the overall efficiency of JDSVD. In this paper, a convergence analysis is made on the minimal residual (MINRES) method for the correction equation. Motivated by the results obtained, at each outer iteration a new correction equation is derived that extracts useful information from current subspaces to construct effective preconditioners for the correction equation and is proven to retain the same convergence of outer iterations of JDSVD.The resulting method is called inner preconditioned JDSVD (IPJDSVD) method; it is also a new JDSVD method, and any viable preconditioner for the correction equations in JDSVD is straightforwardly applicable to those in IPJDSVD. Convergence results show that MINRES for the new correction equation can converge much faster when there is a cluster of singular values closest to a given target. A new thick-restart IPJDSVD algorithm with deflation and purgation is proposed that simultaneously accelerates the outer and inner convergence of the standard thick-restart JDSVD and computes several singular triplets. Numerical experiments justify the theory and illustrate the considerable superiority of IPJDSVD to JDSVD, and demonstrate that a similar two-stage IPJDSVD algorithm substantially outperforms the most advanced PRIMME\_SVDS software nowadays for computing the smallest singular triplets.

翻译：在称为JDSVD的奇异值分解问题Jacobi-Davidson类方法中，每个外层迭代需要迭代求解一个大型对称且通常不定的校正方程，这构成了内层迭代并主导着JDSVD的整体效率。本文对校正方程的最小残差法进行了收敛性分析。基于所得结果，我们在每个外层迭代推导出新的校正方程，该方程从当前子空间提取有效信息以构建校正方程的高效预处理器，并被证明能保持JDSVD外层迭代的相同收敛性。所得方法称为内层预处理JDSVD方法；它也是一种新型JDSVD方法，且JDSVD中任何可行的校正方程预处理器均可直接应用于IPJDSVD。收敛结果表明，当存在最接近给定目标的奇异值簇时，新校正方程的MINRES求解器可显著加速收敛。本文提出了结合收缩与净化机制的新型厚重启IPJDSVD算法，该算法在加速标准厚重启JDSVD内外层收敛的同时，能计算多个奇异三元组。数值实验验证了理论结果，表明IPJDSVD相对JDSVD具有显著优越性，并证明类似的两阶段IPJDSVD算法在计算最小奇异三元组时大幅优于当前最先进的PRIMME_SVDS软件。