Teaching
Numerical Linear Algebra (MAE 290A-PhD level)- F20, F21, F22
Fundamental concepts in numerical linear algebra, i.e., fundamental matrix decompositions: LU, Cholesky, QR, Eigen, Schur, Jordan, SVD; how to compute them. Solution of large linear systems via iterative methods (Lanczos, Arnoldi, CG, GMRES). Stability of algorithms and conditioning of mathematical problems. Finite precision arithmetic and roundoff issues. Additional topics may include randomized methods for computing SVD/QR and solution of Lyapunov/Riccati equations for high-dimensional systems. For more information, see the course syllabus. I previously taught this course as "Numerical Methods for Linear Algebra and ODE Simulation" in Fall 2020, yet starting from F21 we updated the course content so that it solely focuses on numerical linear algebra going forward.
Model Reduction (MAE 274-PhD level) - Sp20, Sp22, Wi25
High-dimensional systems abound in engineering and science, as they arise in modeling of biological processes, when discretizing partial differential equations (of fluids, solids, magnetic fields, etc.), and in mechanical systems with thousands of degrees of freedom. Reduced-order modeling provides the mathematical tools, theory and algorithms to reproduce and predict behaviors in complex systems without doing the direct (and expensive) simulation of the full-order model. As such, reduced-order models are important in long-time prediction, control, uncertainty quantification and computational design. This course covers a wide range of model reduction techniques and their underlying theory: System-theoretic approaches such as balanced truncation, transfer-function interpolation, and controller reduction techniques; Projection-based approaches such as proper orthogonal decomposition and reduced-basis methods; and data-driven approaches such as dynamic mode decomposition and Loewner-based system identification. For more information, see the course syllabus.
Uncertainty Quantification (MAE 279-PhD level) - Sp21, Sp23
Today’s models for complex systems are often in form of partial differential equations (PDEs) with a large state dimension, and many parameters (e.g., material constants, viscosities in flows, boundary conditions, etc.) that are only measurable and known up to some variance. The field of uncertainty quantification is concerned with understanding and recognizing this uncertainty in the models we use, and therefore takes a stochastic/statistical perspective. This course is an introduction to uncertainty quantification and will cover both basic and advanced methods to dealing with uncertainties in simulation. We will start with basics in statistical modeling and the central limit theorem, then cover Monte Carlo sampling techniques and Sensitivity analysis (Sobol indices), move towards Polynomial Chaos expansions, Stochastic Galerkin methods and then also cover dimensionality reduction through Principal Component Analysis. The last part of the course covers the thinking of Bayes’ Theorem and the related Bayesian sampling methods (e.g., Markov Chain Monte Carlo). For more information, see the course syllabus.
Signals and Systems (MAE 143A-third-year UG) - WI20, WI21, WI22, F22, WI24, WI25
This course covers: Dynamic modeling and vector differential equations. Concepts of state, input, output.Linearization around equilibria. Laplace transform, solutions to ODEs. Transfer functions and convolution representation of dynamic systems. Discrete signals, difference equations, z-transform. Continuous and discrete Fourier transform. UCSD students can access this course on Canvas. For more information, see the course syllabus.
Linear Control (MAE 143B-third-year UG) - Sp25
This course covers: Analysis and design of feedback systems in the frequency domain. Transfer functions. Time response specifications. PID controllers and Ziegler-Nichols tuning. Stability via Routh-Hurwitz test. Root locus method. Frequence response: Bode and Nyquist diagrams. Dynamic compensators, phase-lead and phase-lag. Actuator saturation and integrator wind-up.