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We make a convergence analysis of the standard, harmonic, refined and refined harmonic extraction versions of Jacobi-Davidson SVD (JDSVD) type methods for computing an interior singular triplet of a large matrix $A$. At every iteration of these methods, a correction equation, i.e., inner linear system, is solved approximately by using iterative methods, which leads to four inexact JDSVD type methods, as opposed to the exact methods where inner linear systems are solved exactly. Accuracy of inner iterations critically affects the convergence and overall efficiency of the inexact JDSVD methods. A central problem is how accurately the correction equations should be solved so as to ensure that each of the inexact JDSVD methods can mimic its exact counterpart well, that is, they use almost the same outer iterations to achieve the convergence. In this paper, we prove that each inexact JDSVD method behaves like its exact counterpart if all the inner linear systems are solved with $low\ or\ modest$ accuracy during outer iterations. Based on the theory, we propose practical stopping criteria for inner iterations. Numerical experiments confirm our theory.

On Inner Iterations of Jacobi--Davidson Type Methods for Large SVD Computations