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Convergence results are provided for inexact inverse subspace iteration applied to the problem of finding the invariant subspace associated with a small number of eigenvalues of a large sparse matrix. These results are illustrated by the use of block-GMRES as the iterative solver. The costs of the inexact solves are measured by the number of inner iterations needed by the iterative solver at each outer step of the algorithm. It is shown that for a decreasing tolerance the number of inner iterations should not increase as the outer iteration proceeds, but it may increase for preconditioned iterative solves. However, it is also shown that an appropriate small rank change to the preconditioner can produce significant savings in costs, and in particular, can produce a situation where there is no increase in the costs of the iterative solves even though the solve tolerances are reducing. Numerical examples are provided to illustrate the theory.

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The evaluation of the response of elastic structures subjected to distributed random excitations is usually performed in the modal space. Random excitations (like acoustic diffuse fields) are usually modeled as weakly stationary random processes and are assumed to be homogeneous. Their characterization basically relies on the power spectral density (PSD) function of the pressure at a particular reference position and a suitable spatial correlation function. In the modal space, the distributed random excitation is characterized by a modal PSD matrix made from the joint acceptance functions related to the mode pairs. The joint acceptance function is a double surface integral involving the product of the considered mode shapes and the spatial correlation function. The paper shows how to evaluate efficiently this quadruple integral for cylindrical and truncated conical structures excited by an acoustic diffuse field. Basically, the procedure relies on the derivation of alternative expressions for the spatial correlation function. The related expressions prove to be more convenient for these geometries and are leading to a reduction of the double surface integral to a combination of simple integrals. A very substantial breakdown of the computational cost can be achieved using the resulting expressions.