Schedule

Assignments

Deadline	Description
TBD

Class Meetings

Date	Topic	Brief Description
2025-08-26	Python review.	Fundamental objects and syntax. Common packages.
2025-08-28	Basic definitions in linear algebra: vectors, matrices, determinant, inner product	Core objects and operations in linear algebra; norms and orthogonality; geometric interpretations. Conditioning and why it matters in computation.
2025-09-02	Gaussian elimination & solving linear systems	Elimination, back‑substitution, and residual/error checks. Numerical stability, pivoting idea (preview), and interpreting solutions in Ax=b problems.
2025-09-04	LU (partial pivoting) & Cholesky	Stable factorization for general matrices (PA = LU) and efficient solvers for symmetric positive definite (SPD) matrices (A = LL^T). When to use each in practice; complexity and reuse of factors.
2025-09-09	Matrix spaces: column space, null space	Rank, independence, and the structure of solution sets. Interpreting tall systems and consistency via column space; null space as degrees of freedom.
2025-09-11	Row space, orthogonal complements, bases, Gram–Schmidt	Building orthonormal bases; geometry of orthogonal complements. Modified Gram–Schmidt and the link to QR for stable projections.
2025-09-16	Rank‑deficient problems: tall A in Ax=b	Least‑squares formulation; normal equations vs. QR/SVD solutions. Regularization preview and diagnostics for ill‑conditioning.
2025-09-18	Eigendecomposition, matrix functions & orthogonal matrices	Spectral theorem, diagonalization, and matrix functions (polynomials, exponential). Orthogonal matrices and stability.
2025-09-23	SVD & underdetermined problems: wide A in Ax=b	SVD for pseudoinverses, minimal‑norm solutions, and low‑rank structure. Connections to compression, noise filtering, and constrained degrees of freedom.
2025-09-25	Positive definite matrices & applications	Quadratic forms, PD/PSD tests, and energy/convexity interpretations. Why PD matters in optimization, estimation, and numerical stability.
2025-09-30	Probability & random vectors: expectation, covariance, PSD	Random vectors, moments, and covariance as a PSD operator. Linear transformations, sample vs. population quantities, and empirical estimation.
2025-10-02	In‑class exam #1 (Linear Algebra)	Cumulative in‑class assessment covering the linear algebra module.
2025-10-07	Common distributions; linear transformations	Gaussian and exponential family basics; multivariate normal geometry. Transformations of random vectors and propagation of mean/covariance.
2025-10-09	Change of variables, Jacobians, uncertainty propagation	Jacobians and volume scaling; practical change‑of‑variables examples. First‑order and Monte‑Carlo uncertainty propagation in models.
2025-10-14	Maximum entropy & chemical‑engineering applications	Entropy as uncertainty; deriving distributions from constraints (maxent → exponential family). Links to statistical thermodynamics and prior modeling.
2025-10-16	Estimation theory: moments, MLE, Fisher information	Principles of parameter estimation; identifiability and variance bounds (Cramér–Rao). Using information matrices to reason about parameter precision.
2025-10-21	No class	Fall Break.
2025-10-23	No class	Fall Break.
2025-10-28	Hypothesis testing: LRTs, confidence ellipsoids	Likelihood‑based tests; interpreting p‑values and power. Multivariate confidence regions (ellipsoids) and Hotelling’s T² perspective.
2025-10-30	Bayesian inference: conjugacy, Maximum a Posteriori (MAP), updating	Prior → posterior mechanics for common models; MAP as regularization. Predictive distributions and sequential updating.
2025-11-04	Statistics capstone (incl. experimental design)	End‑to‑end inference on a chemical‑engineering case: objectives, efficient designs (D/A/E‑optimality), data collection/analysis, and communicating uncertainty.
2025-11-06	In‑class exam #2 (Statistics)	Cumulative in‑class assessment covering the statistics module.
2025-11-11	Generalized linear models (GLMs) & model assessment; Akaike and Bayesian Information Criteria (AIC/BIC) & CV	Link functions, deviance, and when GLMs are appropriate. Model comparison and validation under realistic data conditions. Heteroscedastic/Homoscedastic.
2025-11-13	Classification overview; Receiver Operator Characteristic (ROC) & precision–recall	Linear Discriminant Analysis (LDA), logistic classifier, SVM concepts. Evaluation under imbalance with ROC/PR and calibration.
2025-11-18	Newton–Raphson & quasi‑Newton for nonlinear equations	Root‑finding for vector systems; Jacobians/Hessians, line search vs. trust‑region ideas. Convergence behavior and practical safeguards.
2025-11-20	Nonlinear least squares: Gauss–Newton & Levenberg–Marquardt	Parameter estimation for nonlinear models; weighting, scaling, and robust losses. Implementation details that affect convergence.
2025-11-25	Constrained optimization: Lagrange multipliers & KKT	Equality/inequality constraints; optimality conditions and sensitivities. Brief look at QPs and engineering design constraints. (May opt for global optimization: genetic algorithms/particle swarm optimization instead).
2025-11-27	Thanksgiving Break — no class	No meeting (university holiday).
2025-12-02	MAP estimation & Bayesian nonlinear models (MCMC overview)	Priors as regularization in nonlinear settings; Laplace approximation intuition. Overview of MCMC (Metropolis–Hastings/HMC).
2025-12-04	Gaussian process regression & surrogate modeling	Nonparametric regression with kernels; posterior mean/variance and hyperparameter learning. Emulation of expensive ODE/PDE models for design and UQ.
2025-12-09	Capstone integration: parameter estimation + UQ	Full pipeline: model specification → estimation (deterministic/Bayesian) → validation → uncertainty propagation/sensitivity → decision support. Emphasis on reproducible computation.