Introduction to basic algorithms and techniques for numerical computing. Error analysis, interpolation (including splines), numerical differentiation and integration, numerical linear algebra (including methods for linear systems, eigenvalue problems, and the singular value decomposition), root finding for nonlinear equations and systems, numerical ordinary differential equations, and approximation methods (including least squares, orthogonal polynomials, and Fourier transforms).
CM 730Introduction to Symbolic Computation0.50 units
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.
Duality in linear optimization. Farkas' Lemma and the theorems of the alternative. Duality theorem for linear programming. Complementary Slackness theorem. Simplex Method. Polyhedra and elementary convex geometry.
Combinatorial Optimization (4 weeks):
Linear diophantine equations. Facets of polyhedra. Integrality of polyhedra. Shortest paths and optimal flows. Total Unimodularity. Konig's Theorem. Max. Flow-Min. Cut Theorem. (Hungarian) Bipartite matching algorithm.
Continuous Optimization (4 weeks):
Convex functions. Analytic characterizations of convexity. Existence and uniqueness of optimal solutions. Separating and supporting hyperplane theorems. Lagrangean Duality. Karush-Kuhn-Tucker Theorem. Conic Optimization problems. Ellipsoild Method.
CM 750Numerical Solution of Partial Differential Equations0.50 units
Discretization methods for partial differential equations, including finite difference, finite volume and finite element methods. Application to elliptic, hyperbolic and parabolic equations. Convergence and stability issues, properties of discrete equations, and treatment of non-linearities. Stiffness matrix assembly and use of sparse matric software. Students should have completed a course in numerical computation at the undergraduate level.
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.
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 763Statistical Learning - Classification0.50 units
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 764Statistical Learning - Function Estimation0.50 units
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.