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Numerical Simulation and Modeling: EECS 219A, Fall 2011


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Numerical simulation and computational modelling are technologies that pervade science and engineering, from electronics (e.g., analog/RF/mixed-signal circuits, high-speed digital circuits, interconnect, etc.) to optics, nanotechnology, chemistry, biology and mechanics. This course provides a detailed introduction to the fundamental principles of these technologies.

Starting with an introduction to equation-based system modelling, the course covers fundamental concepts and algorithms in numerical simulation, including nonlinear and linear algebraic system solution, numerical algorithms for ODEs and DAEs, frequency-domain solution of linear(ized) systems and algorithms for simulating the effects of noise and parametric variability. The impact of good modelling and equation setup practice on the effectiveness of simulation algorithms, i.e., the importance of a holistic approach to modelling and simulation, is an underlying theme in this course. The fundamentals of biochemical modelling and simulation, including deterministic (RRE) models and stochastic modelling/simulation, are also covered in this course. An introduction is provided to reduced-order modelling of linear systems.

The course emphasizes hands-on programming and application to examples as an important means to understand and benefit from the material. Pre-requisites are reviewed in class to make the course as self-contained as possible.

This course is relevant to the BIO, CIR, DES, INC, MEMS, PHY and SCI areas of EECS, and is a part of Berkeley's Designated Emphasis in Computational Science and Engineering (CSE).

Syllabus
  1. Introduction: Computational modelling/simulation for analog, RF and digital circuits, biology, mechanics, nanotechnology, optoelectronics and other domains.
  2. Electronic device models: Constitutive relationships of common circuit elements. Linear and nonlinear resistors, capacitors, inductors, memristors. Simple semiconductor device models: diodes, BJTs, MOSFETs. Continuity, differentiability, smoothing.
  3. System equation setup: Setting up circuit equations using KVL, KCL and branch constitutive relations. Sparse tableau, nodal, and modified nodal circuit equation formulations. Standard DAE and ODE formulations. Mechanical systems.
  4. Quiescent steady state simulation: Numerical solution of nonlinear algebraic equations. The ))Newton-Raphson(( method. DC analysis. NR initialization and limiting. Damped Newton.
  5. Sparse matrix solution concepts: Jacobian matrices. The importance of sparsity. Gaussian Elimination. LU factorization. Pivoting for sparsity. Error buildup and stability. Iterative linear methods.
  6. Deterministic models of biochemical reactions: Biochemical reactions. Law of mass action. Reaction rate equations of individual reactions. Conservation. Stoichiometry matrices and RREs of coupled systems of reactions. ))Michaelis-Menten(( enzyme-catalyzed reactions.
  7. Time-domain ODE/DAE simulation: Existence and uniqueness issues; initial conditions. Forward Euler, Backward Euler and Trapezoidal methods. Transient analysis. Stability and accuracy. Linear multi-step (LMS) methods. Stiff differential equations, stiffly stable methods.
  8. Linear(ized) systems, frequency-domain simulation: Linear ODEs and DAEs. Relevance in applications. Linearization around quiescent steady state. Periodic steady states. AC analysis.
  9. Basic noise simulation: Review of random variables and stochastic processes — probability distributions, autocorrelation functions, power spectral density. Types of noise sources. Noise in linear time-invariant systems. Stationary noise analysis for DAE/ODE systems. Connections with AC analysis.
  10. Basic variability and sensitivity analysis: Linearization of parameter dependence. Propagation of Gaussian statistics. Principal components. Direct and adjoint DC sensitivity analysis. Basics of ))Monte-Carlo(( methods.
  11. Stochastic modelling and simulation: Well-stirred systems. Master equations. Gillespie's stochastic simulation algorithm (SSA). Tau leaping. Langevin equations. Deterministic rate equations.
  12. Basic model order reduction (MOR): The interconnect delay problem. Delay models, Elmore delay. Moments and moment matching. Padé approximation. Companion form realizations. Asymptotic Waveform Evaluation (AWE). Introduction to Krylov-subspace based MOR methods.
Course format
  • The grade for the course will be based on assigned homeworks, a midterm exam, a final examination (or individual project as approved by instructor), and class attendance/participation.
Credits
  • 4 credits. Course control number: 25800
Class location and times
Textbook and Materials
  • Slides/notes will be made available to the students.
Instructor
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