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Markus Hegland Jochen Garcke Vivien Challis 《Linear algebra and its applications》2007,420(2-3):249-275
The combination technique has repeatedly been shown to be an effective tool for the approximation with sparse grid spaces. Little is known about the reasons of this effectiveness and in some cases the combination technique can even break down. It is known, however, that the combination technique produces an exact result in the case of a projection into a sparse grid space if the involved partial projections commute.
The performance of the combination technique is analysed using a projection framework and the C/S decomposition. Error bounds are given in terms of angles between the spanning subspaces or the projections onto these subspaces. Based on this analysis modified combination coefficients are derived which are optimal in a certain sense and which can substantially extend the applicability and performance of the combination technique. 相似文献
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In recent years, Landweber iteration has been extended to solve linear inverse problems in Banach spaces by incorporating non-smooth convex penalty functionals to capture features of solutions. This method is known to be slowly convergent. However, because it is simple to implement, it still receives a lot of attention. By making use of the subspace optimization technique, we propose an accelerated version of Landweber iteration with non-smooth convex penalty which significantly speeds up the method. Numerical simulations are given to test the efficiency. 相似文献
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The direct numerical solution of the chemical master equation (CME) is usually impossible due to the high dimension of the
computational domain. The standard method for solution of the equation is to generate realizations of the chemical system
by the stochastic simulation algorithm (SSA) by Gillespie and then taking averages over the trajectories. Two alternatives
are described here using sparse grids and a hybrid method. Sparse grids, implemented as a combination of aggregated grids
are used to address the curse of dimensionality of the CME. The aggregated components are selected using an adaptive procedure.
In the hybrid method, some of the chemical species are represented macroscopically while the remaining species are simulated
with SSA. The convergence of variants of the method is investigated for a growing number of trajectories. Two signaling cascades
in molecular biology are simulated with the methods and compared to SSA results.
AMS subject classification (2000) 65C20, 60J25, 92C45 相似文献
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E. D. Davy J. Rosin W. H. Blome R. W. Morrison R. Eder W. Stucki L. van Itallie B. Kljatschkina J. M. A. Hegland A. Leulier B. Drevon L. Kofler A. Kofler A. I. Portnow E. N. Winogradowa L. Ekkert A. R. Bliss Jr. N. T. Chamberlin R. I. Grantham 《Analytical and bioanalytical chemistry》1935,102(9-10):376-384
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Markus Hegland 《Constructive Approximation》2007,26(2):205-224
A maximum a posteriori method has been developed for Gaussian priors over infinite-dimensional function spaces. In particular,
variational equations based on a generalisation of the representer theorem and an equivalent optimisation problem are presented.
This amounts to a generalisation of the ordinary Bayesian maximum a posteriori approach which is nontrivial as infinite-dimensional
domains do not admit any probability densities. Instead of the gradient of the density, the logarithmic gradient of the probability
distribution is used. Galerkin methods are proposed for the approximate solution of the variational equations. In summary,
a framework and some foundations are provided which are required for the application of numerical approximation to an important
class of machine learning problems. 相似文献
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