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Statistical Timing for Parametric Yield Prediction of Digital Integrated Circuits
, 2003
"... Uncertainty in circuit performance due to manufacturing and environmental variations is increasing with each new generation of technology. It is therefore important to predict the performance of a chip as a probabilistic quantity. This paper proposes three novel algorithms for statistical timing ana ..."
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Cited by 45 (7 self)
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Uncertainty in circuit performance due to manufacturing and environmental variations is increasing with each new generation of technology. It is therefore important to predict the performance of a chip as a probabilistic quantity. This paper proposes three novel algorithms for statistical timing analysis and parametric yield prediction of digital integrated circuits. The methods have been implemented in the context of the 42660 static timing analyzer. Numerical results are presented to study the strengths and weaknesses of these complementary approaches. Acrossthechip variability continues to be accommodated by 39516 's "Linear Combination of Delay (LCD)" mode. Timing analysis results in the face of statistical temperature and V dd variations are presented on an industrial ASIC part on which a bounded timing methodology leads to surprisingly wrong results.
Orthogonal polynomials and cubature formulae on spheres and on balls
 Department of Mathematics, University of Oregon
"... Abstract. Orthogonal polynomials on the standard simplex Σ d in R d are shown to be related to the spherical orthogonal polynomials on the unit sphere S d in R d+1 that are invariant under the group Z2× · · ·×Z2. For a large class of measures on S d cubature formulae invariant under Z2 × · · · ..."
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Cited by 28 (20 self)
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Abstract. Orthogonal polynomials on the standard simplex Σ d in R d are shown to be related to the spherical orthogonal polynomials on the unit sphere S d in R d+1 that are invariant under the group Z2× · · ·×Z2. For a large class of measures on S d cubature formulae invariant under Z2 × · · · × Z2 are shown to be characterized by cubature formulae on Σ d. Moreover, it is also shown that there is a correspondence between orthogonal polynomials and cubature formulae on Σ d and those invariant on the unit ball B d in R d. The results provide a new approach to study orthogonal polynomials and cubature formulae on spheres and on simplices. 1.
Cubature Formulae and Orthogonal Polynomials
, 2000
"... The connection between orthogonal polynomials and cubature formulae for the approximation of multivariate integrals has been studied for about 100 years. The article J. Radon published about 50 years ago has been very inuential. In this text we describe some of the results that were obtained during ..."
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Cited by 6 (0 self)
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The connection between orthogonal polynomials and cubature formulae for the approximation of multivariate integrals has been studied for about 100 years. The article J. Radon published about 50 years ago has been very inuential. In this text we describe some of the results that were obtained during the search for answers to questions raised by his article.
Polynomial Interpolation In Several Variables, Cubature Formulae, And Ideals
, 2000
"... We discuss polynomial interpolation in several variables from a polynomial ideal point of view. One of the result states that if I is a real polynomial ideal with real variety and if its codimension is equal to the cardinality of its variety, then for each monomial order there is a unique polynom ..."
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Cited by 6 (5 self)
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We discuss polynomial interpolation in several variables from a polynomial ideal point of view. One of the result states that if I is a real polynomial ideal with real variety and if its codimension is equal to the cardinality of its variety, then for each monomial order there is a unique polynomial that interpolates on the points in the variety. The result is motivated by the problem of constructing cubature formulae, and it leads to a theorem on cubature formula which can be considered as an extension of Gaussian quadrature formulae to several variables.
Stochastic Integration Rules for Infinite Regions
 SIAM Journal on Scientific Computation
"... Stochastic integration rules are derived for infinite integration intervals, generalizing rules developed by Siegel and O'Brien (1985) for finite intervals. Then random orthogonal transformations of rules for integrals over the surface of the unit msphere are used to produce stochastic rules for th ..."
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Cited by 6 (4 self)
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Stochastic integration rules are derived for infinite integration intervals, generalizing rules developed by Siegel and O'Brien (1985) for finite intervals. Then random orthogonal transformations of rules for integrals over the surface of the unit msphere are used to produce stochastic rules for these integrals. The two types of rules are combined to produce stochastic rules for multidimensional integrals over infinite regions with Normal or Studentt weights. Example results are presented to illustrate the effectiveness of the new rules. Key Words: Monte Carlo, multiple integrals, numerical integration, statistical computation. AMS Subject Classifications: 65C05, 65C10, 65D30, 65D32. 1 Introduction A common problem in applied science and statistics is to numerically compute integrals in the form E(g) = Z 1 \Gamma1 Z 1 \Gamma1 ::: Z 1 \Gamma1 g(`)p(`)d`; with ` = (` 1 ; ` 2 ; :::; ` m ) t . For statistics applications the function p(`) may be an unnormalized unimodal posteri...
A Stochastic Algorithm for High Dimensional Integrals over Unbounded Regions with Gaussian Weight
 Journal of Computational Applied Mathematics
, 1997
"... Details are given for a Fortran implementation of an algorithm that uses stochastic sphericalradial rules for the numerical computation of multiple integrals over unbounded regions with Gaussian weight. The implemented rules are suitable for high dimensional problems. A high dimensional example fro ..."
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Cited by 6 (1 self)
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Details are given for a Fortran implementation of an algorithm that uses stochastic sphericalradial rules for the numerical computation of multiple integrals over unbounded regions with Gaussian weight. The implemented rules are suitable for high dimensional problems. A high dimensional example from a computational finance application is used to illustrate the use of the rules. 1 Introduction An important applications problem is to numerically compute integrals in the form I(f) = (2) \Gamman=2 Z 1 \Gamma1 Z 1 \Gamma1 \Delta \Delta \Delta Z 1 \Gamma1 e \Gammax t x=2 f(x)dx 1 dx 2 \Delta \Delta \Delta dx n : with x = (x 1 ; x 2 ; :::; x n ) t . There has been much recent interest (see for example, Paskov and Traub (1995), Papageorgiou and Traub (1996), and Joy, Boyle and Tan (1996)) in integrals in the I(f) form that come from computational finance applications where the dimension is high ( e.g. n ? 100). This type of integration problem has traditionally been handled u...
HIGHERDIMENSIONAL INTEGRATION WITH GAUSSIAN WEIGHT FOR APPLICATIONS IN PROBABILISTIC DESIGN ∗
"... Abstract. Higherdimensional Gaussian weighted integration is of interest in probabilistic simulations. Motivated by the need for variance calculations with functions being at least quadratic, the family of degree 5 formulae is considered. Using an existing formula for the integration over the surfa ..."
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Cited by 3 (0 self)
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Abstract. Higherdimensional Gaussian weighted integration is of interest in probabilistic simulations. Motivated by the need for variance calculations with functions being at least quadratic, the family of degree 5 formulae is considered. Using an existing formula for the integration over the surface of an nsphere, an efficient, new formula for Gaussian weighted integration is obtained. Several other formulae that have appeared in the numerical integration literature are also given. The number of function evaluations required by the formulae is compared to a minimal bound result. The degree 5 formulae are applied to simple test problems and the relative errors are compared.
Constructing Cubature Formulae By The Method Of Reproducing Kernel
 Department of Mathematics, University of Oregon
, 2000
"... We examine the method of reproducing kernel for constructing cubature formulae on the unit ball and on the triangle in light of the compact formulae of the reproducing kernels that are discovered recently. Several new cubature formulae are derived. ..."
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Cited by 2 (2 self)
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We examine the method of reproducing kernel for constructing cubature formulae on the unit ball and on the triangle in light of the compact formulae of the reproducing kernels that are discovered recently. Several new cubature formulae are derived.
Methods for Generating Random Orthogonal Matrices
"... Random orthogonal matrices are used to randomize integration methods for ndimensional integrals over spherically symmetric integration regions. Currently available methods for the generation of random orthogonal matrices are reviewed, and some methods for the generation of quasirandom orthogonal m ..."
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Cited by 2 (0 self)
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Random orthogonal matrices are used to randomize integration methods for ndimensional integrals over spherically symmetric integration regions. Currently available methods for the generation of random orthogonal matrices are reviewed, and some methods for the generation of quasirandom orthogonal matrices are proposed. These methods all have O(n³) time complexity. Some new methods to generate both random and quasirandom orthogonal matrices will be described and analyzed. The new methods use products of butterfly matrices, and have time complexity O(log(n)n²). The use of these methods will be illustrated with results from the numerical computation of highdimensional integrals from a computational finance application.
A Mixed Adaptive Algorithm for Multiple Integrals over Infinite Regions
, 1997
"... An algorithm is proposed for the numerical estimation of mdimensional integrals over the region (\Gamma1; 1) m that often arise in Bayesian statistical analysis. A change of variables is used to transform the original integration region to [0; 1) \Theta Um , where Um is surface of the unit msph ..."
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Cited by 1 (1 self)
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An algorithm is proposed for the numerical estimation of mdimensional integrals over the region (\Gamma1; 1) m that often arise in Bayesian statistical analysis. A change of variables is used to transform the original integration region to [0; 1) \Theta Um , where Um is surface of the unit msphere. The described algorithm mixes a globally adaptive one dimensional algorithm for the infinite radial interval [0; 1) with stochastic rules for Um . Different stochastic integration rules are used for the tail portion of the integral and the finite spherical shells that result from the radial subdivision. The mixed algorithm shows superior accuracy in comparisons and requires few assumptions about the tail behavior of the integrand. Key Words: numerical integration, Bayesian analysis, randomized integration rules, Monte Carlo, multiple integrals, statistical computation. AMS Subject Classifications: 65C05, 65C10, 65D30, 65D32. 1 Introduction A common problem in applied science and stat...