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A Theoretical and Experimental Study of the Symmetric Rank One Update ; CU-CS-489-90
A cooperative conjugate gradient method for linear systems permitting multithread implementation of low complexity
This paper proposes a generalization of the conjugate gradient (CG) method
used to solve the equation for a symmetric positive definite matrix
of large size . The generalization consists of permitting the scalar control
parameters (= stepsizes in gradient and conjugate gradient directions) to be
replaced by matrices, so that multiple descent and conjugate directions are
updated simultaneously. Implementation involves the use of multiple agents or
threads and is referred to as cooperative CG (cCG), in which the cooperation
between agents resides in the fact that the calculation of each entry of the
control parameter matrix now involves information that comes from the other
agents. For a sufficiently large dimension , the use of an optimal number of
cores gives the result that the multithread implementation has worst case
complexity in exact arithmetic. Numerical experiments, that
illustrate the interest of theoretical results, are carried out on a multicore
computer.Comment: Expanded version of manuscript submitted to the IEEE-CDC 2012
(Conference on Decision and Control
Extending the square root method to account for additive forecast noise in ensemble methods
A square root approach is considered for the problem of accounting for model noise in the forecast step of the ensemble Kalman filter (EnKF) and related algorithms. The primary aim is to replace the method of simulated, pseudo-random additive so as to eliminate the associated sampling errors. The core method is based on the analysis step of ensemble square root filters, and consists in the deterministic computation of a transform matrix. The theoretical advantages regarding dynamical consistency are surveyed, applying equally well to the square root method in the analysis step. A fundamental problem due to the limited size of the ensemble subspace is discussed, and novel solutions that complement the core method are suggested and studied. Benchmarks from twin experiments with simple, low-order dynamics indicate improved performance over standard approaches such as additive, simulated noise, and multiplicative inflation
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