Weakly nonlinear circuit analysis based on fast multidimensional inverse Laplace transform

There have been continuing thrusts in developing efficient modeling techniques for circuit simulation. However, most circuit simulation methods are time-domain solvers. In this paper we propose a frequency-domain simulation method based on Laguerre function expansion. The proposed method handles both linear and nonlinear circuits. The Laguerre method can invert multidimensional Laplace transform efficiently with a high accuracy, which is a key step of the proposed method. Besides, an adaptive mesh refinement (AMR) technique is developed and its parallel implementation is introduced to speed up the computation. Numerical examples show that our proposed method can accurately simulate large circuits while enjoying low computation complexity. © 2012 IEEE.published_or_final_versio

Laguerre polynomials and the inverse Laplace transform using discrete data

We consider the problem of finding a function defined on $(0,\infty)$ from a countable set of values of its Laplace transform. The problem is severely ill-posed. We shall use the expansion of the function in a series of Laguerre polynomials to convert the problem in an analytic interpolation problem. Then, using the coefficients of Lagrange polynomials we shall construct a stable approximation solution.Comment: 14 page

Transient Analysis and Applications of Markov Reward Processes

In this thesis, the problem of computing the cumulative distribution function (cdf) of the random time required for a system to first reach a specified reward threshold when the rate at which the reward accrues is controlled by a continuous time stochastic process is considered. This random time is a type of first passage time for the cumulative reward process. The major contribution of this work is a simplified, analytical expression for the Laplace-Stieltjes Transform of the cdf in one dimension rather than two. The result is obtained using two techniques: i) by converting an existing partial differential equation to an ordinary differential equation with a known solution, and ii) by inverting an existing two-dimensional result with respect to one of the dimensions. The results are applied to a variety of real-world operational problems using one-dimensional numerical Laplace inversion techniques and compared to solutions obtained from numerical inversion of a two-dimensional transform, as well as those from Monte-Carlo simulation. Inverting one-dimensional transforms is computationally more expedient than inverting two-dimensional transforms, particularly as the number of states in the governing Markov process increases. The numerical results demonstrate the accuracy with which the one-dimensional result approximates the first passage time probabilities in a comparatively negligible amount of the time

Analysis of Beam Propagation in 90-degree Holographic Recording and Readout Using Transfer Functions and Numerical 2-D-Laplace Inversion

Recently, 2-D-Laplace analysis of recording and readout of edge-holograms was reported. Numerical Laplace inversion was examined for simple test cases. Inversion algorithms are applied to examine beam shaping and distortion in photovoltaic and photorefractive materials

Polynomial approximations for bivariate aggregate claims amount probability distributions

A numerical method to compute bivariate probability distributions from their Laplace transforms is presented. The method consists in an orthogonal projection of the probability density function with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF). A particular link to Lancaster probabilities is highlighted. The procedure allows a quick and accurate calculation of probabilities of interest and does not require strong coding skills. Numerical illustrations and comparisons with other methods are provided. This work is motivated by actuarial applications. We aim at recovering the joint distribution of two aggregate claims amounts associated with two insurance policy portfolios that are closely related, and at computing survival functions for reinsurance losses in presence of two non-proportional reinsurance treaties

Recent progresses in the development of more reliable predictions of average molecular weights and chain-lenght distributions for complex irreversible non-linear polymerizations

Prediction of the structure of branched polymers is a challenging problem, for which only more or less clumsy mathematical solutions have been made available for a long time in the case of kinetically controlled polymerizations. The matter has, nevertheless, considerable economic significance. For instance, a controlled amount of long branching is known to have many benefits on the rheologic properties the widely used polyolefins (Nele et al., 2003). Development of processes and their optimization could benefit a lot with models with better predictive capacities. Progresses in applied mathematics could at last bring about a considerable improvement in this situation. This paper reviews recent methods (Costa and Dias, 2003; Dias and Costa, 2003, 2005) allowing the direct computation of moments (i. e. avoiding Hulburt-Katz closures) of polymer chain length distributions, even in the presence of gel, overcoming past difficulties in their computational implementation. Description of non-linear free radical polymerizations is now possible, thanks to the development methods for solving highly stiff two point boundary value problems (Cash et. al., 2001). Chain length distributions are obtained by adapting algorithms better known with Laplace transform inversion (Papoulis, 1956; Weeks, 1966; Durbin, 1974). Numerous past inconsistencies leading to unwanted errors are now avoided through the use of well-founded chemical and mathematical principles