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We establish a class of accelerated Hermitian and skew-Hermitian splitting (AHSS) iteration methods for large sparse saddle-point problems by making use of the Hermitian and skew-Hermitian splitting (HSS) iteration technique. These methods involve two iteration parameters whose special choices can recover the known preconditioned HSS iteration methods, as well as yield new ones. Theoretical analyses show that the new methods converge unconditionally to the unique solution of the saddle-point problem. Moreover, the optimal choices of the iteration parameters involved and the corresponding asymptotic convergence rates of the new methods are computed exactly. In addition, theoretical properties of the preconditioned Krylov subspace methods such as GMRES are investigated in detail when the AHSS iterations are employed as their preconditioners. Numerical experiments confirm the correctness of the theory and the effectiveness of the methods. [ABSTRACT FROM AUTHOR]

The optimal parameter of the Hermitian/skew-Hermitian splitting (HSS) iteration method for a real two-by-two linear system is obtained. The result is used to determine the optimal parameters for linear systems associated with certain two-by-two block matrices and to estimate the optimal parameters of the HSS iteration method for linear systems with n-by-n real coefficient matrices. Numerical examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

We describe an integrative data-driven mathematical framework that formulates any number of genome-scale molecular biological data sets in terms of one chosen set of data samples, or of profiles extracted mathematically from data samples, designated the "basis" set. By using pseudoinverse projection, the molecular biological profiles of the data samples are least-squares-approximated as superositions of the basis profiles. Reconstruction of the data in the basis simulates experimental observation of only the cellular states manifest in the data that correspond to those of the basis. Classification of the data samples according to their reconstruction in the basis, rather than their overall measured profiles, maps the cellular states of the data onto those of the basis and gives a global picture of the correlations and possibly also causal coordination of these two sets of states. We illustrate this framework with an integration of yeast genome-scale proteins' DNA-binding data with cell cycle mRNA expression time course data. Novel correlation between DNA replication initiation and RNA transcription during the yeast cell cycle, which might be due to a previously unknown mechanism of regulation, is predicted. [ABSTRACT FROM AUTHOR]

In this paper, we study an inexact inverse iteration with inner-outer iterations for solving the generalized eigenvalue problem Ax = λBx, and analyze how the accuracy in the inner iterations affects the convergence of the outer iterations. By considering a special stopping criterion depending on a threshold parameter, we show that the outer iteration converges linearly with the inner threshold parameter as the convergence rate. We also discuss the total amount of work and asymptotic equivalence between this stopping criterion and a more standard one. Numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We derive a stable technique, based upon matrix pencils, for the reconstruction of (or approximation by) polygonal shapes from moments. We point out that this problem can be considered the dual of 2 — D numerical quadrature over polygonal domains. An analysis of the sensitivity of the problem is presented along with some numerical examples illustrating the relevant points. Finally, an application to the problem of gravimetry is explored where the shape of a gravitationally anomalous region is to be recovered from measurements of its exterior gravitational field. [ABSTRACT FROM AUTHOR]

We analyze the preconditioned Chebyshev iteration in which at each step the linear system involving the preconditioner is solved inexactly by an inner iteration. We allow the tolerance used in the inner iteration to decrease from one outer iteration to the next. When the tolerance converges to zero, the asymptotic convergence rate is the same as for the exact method. Motivated by this result, we seek the sequence of tolerance values that yields the lowest cost to achieve a specified accuracy. We find that among all sequences of slowly varying tolerances, a constant one is optimal. Numerical calculations that verify our results are presented. Asymptotic methods, such as the W.K.B. method for linear recurrence equations, are used with an estimate of the accuracy of the asymptotic result. [ABSTRACT FROM AUTHOR]