Schedule of Lectures: EECS 290A, Spring 2014
Jan 2014
2014/01/22 (Wed): NO CLASS (instructor out of town).
2014/01/27 (Mon):
- Syllabus and logistics. Algorithmic macromodelling and model order reduction. LTI MOR by moment matching: moments of LTI transfer functions.
2014/01/29 (Wed):
- LTI MOR by moment matching (contd.): computing moments efficiently, Pade approximation to fit a low-order rational function.
Feb 2014
2014/02/03 (Monday):
- LTI MOR by moment matching (contd.): Expressing the Pade rational function as an ODE in companion matrix form. Putting it together: AWE. Numerical experiments with AWE: RC ladder with 5, 10, 20 segments. AC analysis comparisons with AWE-reduced models.
2014/02/05 (Wed):
- Numerical experiments with AWE (contd): the moment disparity issue; poor matrix conditioning. transient NR failures even on a linear system. scaling the companion form to improve DAE matrix conditioning; moment scaling. Instability of reduced models. Introduction to Krylov subspaces.
- Krylov subspaces: definition and basic properties. Finding orthonormal bases for Krylov subspaces. The ))Gram-Schmidtprocess.Gram-Schmidtas a means to QR-factorize a matrix. Numerical inaccuracy (loss of orthogonality) inGram-Schmidt((.
2014/02/12 (Wed):
- Numerical instability of ))Gram-Schmidt- demo. ModifiedGram-Schmidt(( and improvements therefrom. Double orthogonalization and further improvements. Continuing numerical issues for Krylov vector orthonormalization, and their causes. The Arnoldi process and reasons for its numerical superiority.
2014/02/19 (Wed):
- Numerical superiority of Arnoldi over ))Gram-Schmidt((. Matrix forms for the Arnoldi process: decomposition using orthogonal and Hessenberg matrices. Formulae for recovering moments explicitly from Arnoldi-generated matrices. Futility of explicit calculation of moments using Arnoldi quantities. Reducing a large LTI DAE system "directly" to a small one based on the form of moment formulae, without their explicit calculation: implicit moment matching.
- Arnoldi based reduced models: the p=n case. Computational properties of Arnoldi: linear in original system size, quadratic in reduced system size. Mapping initial conditions into Arnoldi-reduced ROMs: orthogonal projection on the Krylov subspace. Geometrical view of Krylov-subspace based model reduction. MOR as a restriction of differential equations on subspaces.
2014/02/26 (Wed):
- Demos of Arnoldi-based MOR: explicit moments obtained via Arnoldi (and how they don't help); Arnoldi-based MOR and its superior performance wrt AWE; numerical dynamic range of Arnoldi-reduced models vs the original sparse model; dependence of numerical dynamic range on sparsity; good eigenapproximation properties of Arnoldi; transient simulation of Arnoldi-reduced models and their approximation characteristics; mapping initial conditions to Arnoldi-reduced models; projection views of Arnoldi-reduced models (3D->2D reduction).
March 2014
2014/03/03 (Mon): Guest lecture by Frank Liu, IBM: Modelling and simulation of river networks.
2014/03/05 (Wed): (2 hour class)
- Oscillators. Introduction and interesting features. Ideal and damped simple harmonic motion: formulae, phase plane representations.
- Numerical simulation of oscillators: artificial damping and phase instability issues. Self-sustaining, amplitude-stable oscillators - intuition and numerical demos. Topology of negative resistance LC oscillators.
2014/03/12 (Wed): (2.5 hour class)
- Feedback analysis of negative resistance oscillators. Linear Barkhausen criterion and its limitations. Nonlinear feedback analysis: splitting the system into a linear filter + a memoryless nonlinearity. Cutting the loop at the system level. Fourier components of the nonlinearity's output. Closing the loop: amplitude and phase conditions. Graphical depiction of the amplitude condition. Numerical demos and validation.
2014/03/19 (Wed): NO CLASS (instructor out of town).
2014/03/24 (Mon): NO CLASS (spring break).
2014/03/26 (Wed): NO CLASS (spring break).
2014/03/31 (Mon): (2.5 hour class)
- Ring oscillators and their operation. Idealizations: perfect inverter + perfect delay, perfect inverter + RC delay. Analytical solution of idealized models; the role of the Golden Ratio. Practical electronic ring oscillators using BSIM inverters. A (synthetic) biological ring oscillator: the Elowitz repressilator. Interpretation of the repressilator equations as inverter + RC delay. Numerical demo. Relaxation oscillators: ideal inverting hysteresis. Modelling hysteresis realistically with smooth nonlinearities and fast time constants. Simple tanh-based example; numerical demos.
April 2014
2014/04/02 (Wed): (2 hour class)
- Making a relaxation oscillator out of the smooth "physical" tanh-based hysteresis model. Demos. The ))Fitzhugh-Nagumo(( neuron model as a relaxation oscillator. Graphical insight from steady state plots. Discussion of stability issues. Demos. Basic properties of autonomous DAEs and ODEs.
- Introduction to stochastic differential equations. Non-stationary (transient) noise. Overview of our development of SDEs: why "white noise" presents difficulties for stochastic analysis, the Wiener process as the integral of white noise, interpreting differential equations using integral forms, SDE notation, stochastic integrals.
2014/04/9 (Wed): MIDTERM EXAM
2014/04/14 (Mon): (2 hour class)
- Summary of our flow for developing SDEs. Naiive analysis of simple systems with white noise inputs using deterministic notions: ACF of the integral of white noise, variance of the solution to the Langevin equation, geometric Brownian motion.
2014/04/16 (Wed): (2 hour class)
- Naiive analysis of geometric Brownian motion (contd.); how predictions from stochastic calculus differ. Defining the Wiener process W(t) concretely. ACF of W(t) from its definition. The concept of constructing a stochastic process concretely from primitives; concept of uniqueness of a stochastic process. Sketch of ))Levy-Ciesielskiconstruction of W(t). Why dW/dt is like white noise. Review of Riemann integration: partitions, Riemann sums, limit of Riemann sum (epsilon-delta definition). Examples of Riemann integrals; continuity is sufficient for Riemann integration. Fundamental Theorem of Calculus, integral forms of deterministic differential equations, product differentiation rule and integration by parts. Motivation for theRiemann-Stieltjes(( integral.
- ))Riemann-Stieltjesintegrals. Properties of theRiemann-Stieltjesintegral. The deterministic chain rule and itsRiemann-Stieltjesdifferential form. Use of the chain rule to solve a linear ODE. Interpretation of differential forms asRiemann-Stieltjes(( sums. Stochastic integration: different types of stochastic integrals. Riemann integral of Brownian motion. Deterministic functions integrated with respect to stochastic measure. The use of Riemann(-Stieltjes) sums for numerics: numerical generation of the Wiener process and its derivative. Code and demo.
2014/04/23 (Wed): (2h10m class + 30m follow-up on video to complete the var(\int_0^T W(t)dW(t)) proof)
- Numerical demos of stochastic integrands wrt dt, and of deterministic integrands wrt stochastic measure. Match of theory and numerics for these cases; independence of integral wrt choice of ti* or lambda. Stochastic integrands wrt stochastic measure - integral of W(t) dW(t). Formula from deterministic integration; its mean and variance. Numerical integration/demos of W(t) dW(t). Dependence of mean on lambda; independence of variance on lambda. Low-level derivation using ))Riemann-Stieltjes(( sums of the mean and variance of the integral of W(t) dW(t). Prediction of lambda-dependent mean and lambda-independent variance; match with numerical demos.
- Andreas Freund's project presentation. ))Riemann-Stieltjes2(T)/2 plus a lambda-dependent deterministic offset. Ito, Stratonovich and" class="wiki wikinew text-danger tips">sum for \int_0^T W(t)dW(t) equals W
- [https://draco.eecs.berkeley.edu/videos/2014-Spring-290a/2014-04-28
andreasfinish-SDEs.html|video of 1-3:15pm lecture]. - class scribbles (PDF).
2014/04/30 (Wed):