CM 271 Introduction to Computational Mathematics
A rigorous introduction to the field of computational mathematics. The focus is on the interplay between continuous models and their solution via discrete processes. Topics include: pitfalls in computation, solution of linear systems, interpolation, discrete Fourier transforms and numerical integration. Applications are used as motivation.
CM 339 Algorithms
The study of efficient algorithms and effective algorithm design techniques. Program design with emphasis on pragmatic and mathematical aspects of program efficiency. Topics include divide and conquer algorithms, recurrences, greedy algorithms, dynamic programming, graph search and backtrack, problems without algorithms, NP-completeness and its implications.
CM 340 Computational Optimization
A first course in computational optimization. Linear optimization, the simplex method, implementation issues, duality theory. Introduction to computational discrete and continuous optimization.
CM 352 Computational Methods for Differential Equations
Modelling of systems which lead to differential equations (examples include vibrations, population dynamics, and mixing processes). Scalar first order differential equations, second-order differential equations, systems of differential equations. Stability and qualitative analysis. Implicit and explicit time-stepping. Comparison of different methods. Stiffness. Linearization and the role of the Jacobian.
CM 361 Computational Statistics and Data Analysis
Approximation and optimization of noisy functions. Simulation from univariate and multivariate distributions, multivariate normal distribution, mixture distributions and introduction to Markov Monte Carlo. Introduction to supervised statistical learning including discrimination methods.
CM 375 Computational Linear Algebra
Basic concepts and implementation of numerical linear algebra techniques and their use in solving application problems. Special methods for solving linear systems having special features. Direct methods: symmetric, positive definite, band, general sparse structures, ordering methods.
CM 432 Applied Cryptography
A broad introduction to cryptography, highlighting the major developments of the past twenty years. Symmetric ciphers, hash functions and data integrity, public-key encryption and digital signatures, key establishment, key management. Applications to Internet security, computer security, communications security, and electronic commerce.
CM 433 Introduction to Symbolic Computation
An introduction to the use of computers for symbolic mathematical computation, involving traditional mathematical computations such as solving linear equations (exactly), analytic differentiation and integration of functions, and analytic solution of differential equations.
CM 434 Techniques in Computational Number Theory
An introduction to: integer factorization, elliptic curves methods, primality testing, fast integer arithmetic, fast Fourier transforms and quantum computing. This course is taught with a philosophy that encourages experimentation.
CM 442 Nonlinear Optimization
A course on the fundamentals of nonlinear optimization, including both the mathematical and the computational aspects. Necessary and sufficient optimality conditions for unconstrained and constrained problems. Convexity and its applications. Computational techniques and their analysis.
CM 443 Deterministic OR Models
An applications-oriented course that illustrates how various mathematical models and methods of optimization can be used to solve problems arising in business, industry and science.
CM 452 Computational Methods for Partial Differential Equations
This course studies basic methods for the numerical solution of partial differential equations. Emphasis is placed on regarding the discretized equations as discrete models of the system being studied. Basic discretization methods on structured and unstructured grids. Boundary conditions. Implicit/explicit timestepping. Stability, consistency and convergence. Non-conservative versus conservative systems. Nonlinearities.
CM 461 Computational Inference
Introduction to and application of computational methods in statistical inference. Monte Carlo evaluation of statistical procedures, exploration of the likelihood function through graphical and optimization techniques including EM. Bootstrapping, Markov Chain Monte Carlo, and other computationally intensive methods.
CM 462 Data Visualization
Visualization of high dimensional data including interactive methods directed at exploration and assessment of structure and dependencies in data. Methods for finding groups in data including traditional and modern methods of cluster analysis. Dimension reduction methods including multi-dimensional scaling, nonlinear and other methods.
CM 463 Statistical Learning - Classification
Given known group membership, methods which learn from data how to classify objects into the groups are treated. Review of likelihood and posterior based discrimination. Main topics include logistic regression, neural networks, tree-based methods and nearest neighbour methods. Model assessment, training and tuning.
CM 464 Statistical Learning - Function Estimation
Methods for finding surfaces in high dimensions from incomplete or noisy functional information. Both data adaptive and methods based on fixed parametric structure will be treated. Model assessment, training and tuning.
CM 473 Medical Image Processing
An introduction to computational problems in medical imaging. Sources of medical images (MRI, CT, ultrasound, PET) as well as reconstruction methods for MRI and CT. Image manipulation and enhancement such as denoising and deblurring. Patient motion correction and optimal image alignment. Tissue classification and organ delineation using image topology.
CM 476 Numeric Computation for Financial Modeling
The interaction of financial models, numerical methods, and computing environments. Basic computational aspects of option pricing and hedging. Numerical methods for stochastic differential equations, strong and weak convergence. Generating correlated random numbers. Time-stepping methods. Finite difference methods for the Black-Scholes equation. Discretization, stability, convergence. Methods for portfolio optimization, effect of data errors on portfolio weights.