Calculation of Generalized Polynomial-Chaos Basis Functions and Gauss Quadrature Rules in Hierarchical Uncertainty Quantification

Stochastic spectral methods are efficient techniques for uncertainty quantification. Recently they have shown excellent performance in the statistical analysis of integrated circuits. In stochastic spectral methods, one needs to determine a set of orthonormal polynomials and a proper numerical quadrature rule. The former are used as the basis functions in a generalized polynomial chaos expansion. The latter is used to compute the integrals involved in stochastic spectral methods. Obtaining such information requires knowing the density function of the random input {\it a-priori}. However, individual system components are often described by surrogate models rather than density functions. In order to apply stochastic spectral methods in hierarchical uncertainty quantification, we first propose to construct physically consistent closed-form density functions by two monotone interpolation schemes. Then, by exploiting the special forms of the obtained density functions, we determine the generalized polynomial-chaos basis functions and the Gauss quadrature rules that are required by a stochastic spectral simulator. The effectiveness of our proposed algorithm is verified by both synthetic and practical circuit examples.Comment: Published by IEEE Trans CAD in May 201

Spectral methods for circuit analysis

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1999.Includes bibliographical references (p. 119-124).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Harmonic balance (HB) methods are frequency-domain algorithms used for high accuracy computation of the periodic steady-state of circuits. Matrix-implicit Krylov-subspace techniques have made it possible for these methods to simulate large circuits more efficiently. However, the harmonic balance methods are not so efficient in computing steady-state solutions of strongly nonlinear circuits with rapid transitions. While the time-domain shooting-Newton methods can handle these problems, the low-order integration methods typically used with shooting-Newton methods are inefficient when high solution accuracy is required. We first examine possible enhancements to the standard state-of-the-art preconditioned matrix-implicit Krylovsubspace HB method. We formulate the BDF time-domain preconditioners and show that they can be quite effective for strongly nonlinear circuits, speeding up the HB runtimes by several times compared to using the frequency-domain block-diagonal preconditioner. Also, an approximate Galerkin HB formulation is derived, yielding a small improvement in accuracy over the standard pseudospectral HB formulation, and about a factor of 1.5 runtime speedup in runs reaching identical solution error. Next, we introduce and develop the Time-Mapped Harmonic Balance method (TMHB) as a fast Krylov-subspace spectral method that overcomes the inefficiency of standard harmonic balance for circuits with rapid transitions. TMHB features a non-uniform grid and a time-map function to resolve the sharp features in the signals. At the core of the TMHB method is the notion of pseudo Fourier approximations. The rapid transitions in the solution waveforms are well approximated with pseudo Fourier interpolants, whose building blocks are complex exponential basis functions with smoothly varying frequencies. The TMHB features a matrix-implicit Krylov-subspace solution approach of same complexity as the standard harmonic balance method. As the TMHB solution is computed in a pseudo domain, we give a procedure for computing the real Fourier coefficients of the solution, and we also detail the construction of the time-map function. The convergence properties of TMHB are analyzed and demonstrated on analytic waveforms. The success of TMHB is critically dependent on the selection of a non-uniform grid. Two grid selection strategies, direct and iterative, are introduced and studied. Both strategies are a priori schemes, and are designed to obey accuracy and stability requirements. Practical issues associated with their use are also addressed. Results of applying the TMHB method on several circuit examples demonstrate that the TMHB method achieves up to five orders of magnitude improvement in accuracy compared to the standard harmonic balance method. The solution error in TMHB decays exponentially faster than the standard HB method when the size of the Fourier basis increases linearly. The TMHB method is also up to six times faster than the standard harmonic balance method in reaching identical solution accuracy, and uses up to five times less computer memory. The TMHB runtime speedup factor and storage savings favorably increase for stricter accuracy requirements, making TMHB well suited for high accuracy simulations of large strongly nonlinear circuits with rapid transitions.by Ognen J. Nastov.Ph.D

Numerical study of performance of porous fin heat sink of functionally graded material for improved thermal management of consumer electronics

YesThe ever-increasing demand for high performance electronic and computer systems has unequivocally called for increased microprocessor performance. However, increasing microprocessor performance requires increasing the power and on-chip power density of the microprocessor, both of which are associated with increased heat dissipation. In recent times, thermal management of electronic systems has gained intense research attention due to increased miniaturization trend in the electronics industry. In the paper, we present a numerical study on the performance of a convective-radiative porous heat sink with functionally graded material for improved cooling of various consumer electronics. For the theoretical investigation, the thermal property of the functionally graded material is assumed as a linear and power-law function. We solved the developed thermal models using the Chebyshev spectral collocation method. The effects of inhomogeneity index of FGM, convective and radiative parameters on the thermal behaviour of the porous heat sink are investigated. The present study shows that increase in the inhomogeneity index of FGM, convective and radiative parameter improves the thermal efficiency of the porous fin heat sink. Moreover, for all values of Nc and Rd, the temperature gradient along the fin of FGM is negligible compared to HM fin in both linear and power-law functions. For comparison, the thermal predictions made in the present study using Chebyshev spectral collocation method agrees excellently with the established results of Runge-Kutta with shooting and homotopy analytical method.Supported in part from PhD sponsorship of the first author by the Tertiary Education Trust Fund of the Federal Government of Nigeria

Differential-Algebraic Equations

Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed

Numerical Simulation

Nowadays mathematical modeling and numerical simulations play an important role in life and natural science. Numerous researchers are working in developing different methods and techniques to help understand the behavior of very complex systems, from the brain activity with real importance in medicine to the turbulent flows with important applications in physics and engineering. This book presents an overview of some models, methods, and numerical computations that are useful for the applied research scientists and mathematicians, fluid tech engineers, and postgraduate students

Numerical Solution of Linear Ordinary Differential Equations and Differential-Algebraic Equations by Spectral Methods

This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudo-spectral method which is more efficient than other spectral methods is suggested and tested on some examples. We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriate choice of Gauss-Chebyshev-Radau points, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours. Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities. In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously. In all chapters, wherever we talk about ODE or DAE we mean linear

Uncertainty analysis in a shipboard integrated power system using multi-element polynomial chaos

Thesis (Ph. D. in Ocean Engineering)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2007.Errata, dated Oct. 30, 2007, inserted between pages 3 and 4 of text.Includes bibliographical references (p. 301-307).The integrated power system has become increasingly important in electric ships due to the integrated capability of high-power equipment, for example, electromagnetic rail guns, advance radar system, etc. Several parameters of the shipboard power system are uncertain, caused by a measurement difficulty, a temperature dependency, and random fluctuation of its environment. To date, there has been little if any studies which account for these stochastic effects in the large and complex shipboard power system from either an analytical or a numerical perspective. Furthermore, all insensitive parameters must be identified so that the stochastic analysis with the reduced dimensional parameters can accelerate the process. Therefore, this thesis is focused on two main issues - stochastic and sensitivity analysis - on the shipboard power system. The stochastic analysis of the large and complex nonlinear systems with the non-Gaussian random variables or processes, in their initial states or parameters, are prohibited analytically and very time consuming using the brute force Monte Carlo method. As a result, numerical stochastic solutions of these systems can be efficiently solved by the generalized Polynomial Chaos (gPC) and Probabilistic Collocation Method (PCM).(cont.) In the case of the long-time integration and discontinuity in the stochastic solutions, the multi-element technique of PCM, which refines the solution in random space, can significantly improve the solutions' accuracy. Furthermore, the hybrid gPC+PCM is developed to extend the gPC ability to handle a system with nonlinear non-polynomial functions. Then, we systematically establish the convergence rate and compare the convergence performance among all numerical stochastic algorithms on various systems with both continuous and discontinuous solutions as a function of random dimension and the algorithms' accuracy governing parameters. To identify the most significant parameter in the large-scale complex systems, we propose new sensitivity analysis techniques - Monte Carlo Sampling, Collocation, Variance, and Inverse Variance methods - for static functions and show that they agree well with Morris method, which is one of the existing sensitivity analysis techniques for a function with large input dimensions. In addition, we extend the capability of the Sampling, Collocation, Variance, and the Morris methods to study both the parameters' sensitivity and the interaction of the ordinary differential equations.(cont.) In each approach, both strength and limitations of the sensitivity ranking accuracy and the convergence performance are emphasized. The convergence rate of the Collocation and Variance methods are more than an order of magnitude faster than that of Morris and Sampling methods for low and medium parameters' dimensions. At last, we successfully apply both stochastic and sensitivity analysis techniques to the integrated shipboard power system, with both open-and close-loop control of the propulsion system, to study a propagation of uncertainties and rank parameters in the order of their importance, respectively.by Pradya Prempraneerach.Ph.D.in Ocean Engineerin

Bivariate pseudospectral collocation algorithms for nonlinear partial differential equations.

Doctor of Philosophy in Applied Matheatics. University of KwaZulu-Natal, Pietermaritzburg 2016.Abstract available in PDF file

Transient simulation of complex electronic circuits and systems operating at ultra high frequencies

The electronics industry worldwide faces increasingly difficult challenges in a bid to produce ultra-fast, reliable and inexpensive electronic devices. Electronic manufacturers rely on the Electronic Design Automation (EDA) industry to produce consistent Computer A id e d Design (CAD) simulation tools that w ill enable the design of new high-performance integrated circuits (IC), the key component of a modem electronic device. However, the continuing trend towards increasing operational frequencies and shrinking device sizes raises the question of the capability of existing circuit simulators to accurately and efficiently estimate circuit behaviour. The principle objective of this thesis is to advance the state-of-art in the transient simulation of complex electronic circuits and systems operating at ultra high frequencies. Given a set of excitations and initial conditions, the research problem involves the determination of the transient response o f a high-frequency complex electronic system consisting of linear (interconnects) and non-linear (discrete elements) parts with greatly improved efficien cy compared to existing methods and with the potential for very high accuracy in a way that permits an effective trade-off between accuracy and computational complexity. High-frequency interconnect effects are a major cause of the signal degradation encountered b y a signal propagating through linear interconnect networks in the modem IC. Therefore, the development of an interconnect model that can accurately and efficiently take into account frequency-dependent parameters of modem non-uniform interconnect is of paramount importance for state-of-art circuit simulators. Analytical models and models based on a set of tabulated data are investigated in this thesis. Two novel, h igh ly accurate and efficient interconnect simulation techniques are developed. These techniques combine model order reduction methods with either an analytical resonant model or an interconnect model generated from frequency-dependent sparameters derived from measurements or rigorous full-wave simulation. The latter part o f the thesis is concerned with envelope simulation. The complex mixture of profoundly different analog/digital parts in a modern IC gives rise to multitime signals, where a fast changing signal arising from the digital section is modulated by a slower-changing envelope signal related to the analog part. A transient analysis of such a circuit is in general very time-consuming. Therefore, specialised methods that take into account the multi-time nature o f the signal are required. To address this issue, a novel envelope simulation technique is developed. This technique combines a wavelet-based collocation method with a multi-time approach to result in a novel simulation technique that enables the desired trade-off between the required accuracy and computational efficiency in a simple and intuitive way. Furthermore, this new technique has the potential to greatly reduce the overall design cycle