A Note on Adaptive Nonlinear Preconditioning Techniques

A Note on Adaptive Estimation

Adaptive Estimation An adaptive estimator is an efficient estimator for a model that is only partially specified. For example, consider estimating a parameter that describes a sample of observations drawn from a distribution F . One natural question is: Is it possible that an estimator of the parameter constructed without knowledge of F could be as efficient (asymptotically) as any well-behaved...

Adaptive Techniques for Improving the Performance of Incomplete Factorization Preconditioning

A Note on Preconditioning by Low-Stretch Spanning Trees

Boman and Hendrickson [BH01] observed that one can solve linear systems in Laplacian matrices in time O ( m ln(1/ǫ) ) by preconditioning with the Laplacian of a low-stretch spanning tree. By examining the distribution of eigenvalues of the preconditioned linear system, we prove that the preconditioned conjugate gradient will actually solve the linear system in time Õ ( m ln(1/ǫ) ) .

A Note on Preconditioning for Indefinite Linear Systems

Preconditioners are often conceived as approximate inverses. For nonsingular indeenite matrices of saddle-point (or KKT) form, we show how preconditioners incorporating an exact Schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues. Thus approximations of the Schur complement lead to preconditioners which can be very eeective even though they ar...

Note on a Nonlinear Volterra Equation

9. S. G. Krein, and O. I. Prozorovskaya, An analogue of Seidel's method for operator equations, Voronez. Gos. Univ., Trudy Sem. Functional. Anal. 5 (1957), 35-38. 10. W. V. Petryshyn, The generalized overrelaxation method for the approximate solution of operator equations in Hubert space, J. Soc. Indust. Appl. Math. 10 (1962), 675-690. 11. S. Schechter, Relaxation methods for linear equations, ...