Computing Tails of Compound Distributions Using Direct Numerical Integration

An efficient adaptive direct numerical integration (DNI) algorithm is developed for computing high quantiles and conditional Value at Risk (CVaR) of compound distributions using characteristic functions. A key innovation of the numerical scheme is an effective tail integration approximation that reduces the truncation errors significantly with little extra effort. High precision results of the 0.999 quantile and CVaR were obtained for compound losses with heavy tails and a very wide range of loss frequencies using the DNI, Fast Fourier Transform (FFT) and Monte Carlo (MC) methods. These results, particularly relevant to operational risk modelling, can serve as benchmarks for comparing different numerical methods. We found that the adaptive DNI can achieve high accuracy with relatively coarse grids. It is much faster than MC and competitive with FFT in computing high quantiles and CVaR of compound distributions in the case of moderate to high frequencies and heavy tails

Event Generators for WW Physics

The report summarizes the results of the activities of the Working Group on Event Generators for WW Physics at CERN during 1995.Comment: 99 Latex, including 30 figures, 24 tables. The report is part of: G.Altarelli,T.Sjostrand and F.Zwirner (eds), Physics at LEP2 CERN 96-0

Enabling High-Dimensional Hierarchical Uncertainty Quantification by ANOVA and Tensor-Train Decomposition

Hierarchical uncertainty quantification can reduce the computational cost of stochastic circuit simulation by employing spectral methods at different levels. This paper presents an efficient framework to simulate hierarchically some challenging stochastic circuits/systems that include high-dimensional subsystems. Due to the high parameter dimensionality, it is challenging to both extract surrogate models at the low level of the design hierarchy and to handle them in the high-level simulation. In this paper, we develop an efficient ANOVA-based stochastic circuit/MEMS simulator to extract efficiently the surrogate models at the low level. In order to avoid the curse of dimensionality, we employ tensor-train decomposition at the high level to construct the basis functions and Gauss quadrature points. As a demonstration, we verify our algorithm on a stochastic oscillator with four MEMS capacitors and 184 random parameters. This challenging example is simulated efficiently by our simulator at the cost of only 10 minutes in MATLAB on a regular personal computer.Comment: 14 pages (IEEE double column), 11 figure, accepted by IEEE Trans CAD of Integrated Circuits and System

Smolyak's algorithm: A powerful black box for the acceleration of scientific computations

We provide a general discussion of Smolyak's algorithm for the acceleration of scientific computations. The algorithm first appeared in Smolyak's work on multidimensional integration and interpolation. Since then, it has been generalized in multiple directions and has been associated with the keywords: sparse grids, hyperbolic cross approximation, combination technique, and multilevel methods. Variants of Smolyak's algorithm have been employed in the computation of high-dimensional integrals in finance, chemistry, and physics, in the numerical solution of partial and stochastic differential equations, and in uncertainty quantification. Motivated by this broad and ever-increasing range of applications, we describe a general framework that summarizes fundamental results and assumptions in a concise application-independent manner