With its advantages in wirelength reduction and routing flexibility compared with conventional Manhattan routing, X architecture has been proposed and applied to modern IC design. As a critical part in high-performanc...With its advantages in wirelength reduction and routing flexibility compared with conventional Manhattan routing, X architecture has been proposed and applied to modern IC design. As a critical part in high-performance integrated circuits, clock network design meets great challenges due to feature size decrease and clock frequency increase. In order to eliminate the delay and attenuation of clock signal introduced by the vias, and to make it more tolerant to process variations, in this paper, we propose an algorithm of a single layer zero skew clock routing in X architecture （called Pianar-CRX）. Our Planar- CRX method integrates the extended deferred-merge embedding algorithm （DME-X, which extends the DME algorithm to X architecture） with modified Ohtsuki＇s line-search algorithm to minimize the total wirelength and the bends. Compared with planar clock routing in the Manhattan plane, our method achieves a reduction of 6.81% in total wirelength on average and gets the resultant clock tree with fewer bends. Experimental results also indicate that our solution can be comparable with previous non-planar zero skew clock routing algorithm.展开更多
We present a parallel Schwarz type domain decomposition preconditioned recycling Krylov subspace method for the numerical solution of stochastic indefinite elliptic equations with two random coefficients.Karhunen-Loev...We present a parallel Schwarz type domain decomposition preconditioned recycling Krylov subspace method for the numerical solution of stochastic indefinite elliptic equations with two random coefficients.Karhunen-Loeve expansions are used to represent the stochastic variables and the stochastic Galerkin method with double orthogonal polynomials is used to derive a sequence of uncoupled deterministic equations.We show numerically that the Schwarz preconditioned recycling GMRES method is an effective technique for solving the entire family of linear systems and,in particular,the use of recycled Krylov subspaces is the key element of this successful approach.展开更多
基金Supported in part by the National Natural Science Foundation of China (Grant No. 60876026), the Specialized Research Fund for the Doctoral Program of Higher Education (Crant No. 200800030026)
文摘With its advantages in wirelength reduction and routing flexibility compared with conventional Manhattan routing, X architecture has been proposed and applied to modern IC design. As a critical part in high-performance integrated circuits, clock network design meets great challenges due to feature size decrease and clock frequency increase. In order to eliminate the delay and attenuation of clock signal introduced by the vias, and to make it more tolerant to process variations, in this paper, we propose an algorithm of a single layer zero skew clock routing in X architecture （called Pianar-CRX）. Our Planar- CRX method integrates the extended deferred-merge embedding algorithm （DME-X, which extends the DME algorithm to X architecture） with modified Ohtsuki＇s line-search algorithm to minimize the total wirelength and the bends. Compared with planar clock routing in the Manhattan plane, our method achieves a reduction of 6.81% in total wirelength on average and gets the resultant clock tree with fewer bends. Experimental results also indicate that our solution can be comparable with previous non-planar zero skew clock routing algorithm.
基金The research was supported in part by DOE under DE-FC02-04ER25595,and in part by NSF under grants CCF-0634894,and CNS-0722023.
文摘We present a parallel Schwarz type domain decomposition preconditioned recycling Krylov subspace method for the numerical solution of stochastic indefinite elliptic equations with two random coefficients.Karhunen-Loeve expansions are used to represent the stochastic variables and the stochastic Galerkin method with double orthogonal polynomials is used to derive a sequence of uncoupled deterministic equations.We show numerically that the Schwarz preconditioned recycling GMRES method is an effective technique for solving the entire family of linear systems and,in particular,the use of recycled Krylov subspaces is the key element of this successful approach.
