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

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

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

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

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 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

Numerical Solution of Optimal Control Problems with Explicit and Implicit Switches

This dissertation deals with the efficient numerical solution of switched optimal control problems whose dynamics may coincidentally be affected by both explicit and implicit switches. A framework is being developed for this purpose, in which both problem classes are uniformly converted into a mixed–integer optimal control problem with combinatorial constraints. Recent research results relate this problem class to a continuous optimal control problem with vanishing constraints, which in turn represents a considerable subclass of an optimal control problem with equilibrium constraints. In this thesis, this connection forms the foundation for a numerical treatment. We employ numerical algorithms that are based on a direct collocation approach and require, in particular, a highly accurate determination of the switching structure of the original problem. Due to the fact that the switching structure is a priori unknown in general, our approach aims to identify it successively. During this process, a sequence of nonlinear programs, which are derived by applying discretization schemes to optimal control problems, is solved approximatively. After each iteration, the discretization grid is updated according to the currently estimated switching structure. Besides a precise determination of the switching structure, it is of central importance to estimate the global error that occurs when optimal control problems are solved numerically. Again, we focus on certain direct collocation discretization schemes and analyze error contributions of individual discretization intervals. For this purpose, we exploit a relationship between discrete adjoints and the Lagrange multipliers associated with those nonlinear programs that arise from the collocation transcription process. This relationship can be derived with the help of a functional analytic framework and by interrelating collocation methods and Petrov–Galerkin finite element methods. In analogy to the dual-weighted residual methodology for Galerkin methods, which is well–known in the partial differential equation community, we then derive goal–oriented global error estimators. Based on those error estimators, we present mesh refinement strategies that allow for an equilibration and an efficient reduction of the global error. In doing so we note that the grid adaption processes with respect to both switching structure detection and global error reduction get along with each other. This allows us to distill an iterative solution framework. Usually, individual state and control components have the same polynomial degree if they originate from a collocation discretization scheme. Due to the special role which some control components have in the proposed solution framework it is desirable to allow varying polynomial degrees. This results in implementation problems, which can be solved by means of clever structure exploitation techniques and a suitable permutation of variables and equations. The resulting algorithm was developed in parallel to this work and implemented in a software package. The presented methods are implemented and evaluated on the basis of several benchmark problems. Furthermore, their applicability and efficiency is demonstrated. With regard to a future embedding of the described methods in an online optimal control context and the associated real-time requirements, an extension of the well–known multi–level iteration schemes is proposed. This approach is based on the trapezoidal rule and, compared to a full evaluation of the involved Jacobians, it significantly reduces the computational costs in case of sparse data matrices

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