Fall 2026
Benjamin Zhang
Course Description:
Introduces key mathematical and computational tools for modern data analysis, integrating linear algebraic techniques, statistical methods, and geometric and topological approaches. Topics include the singular value decomposition, PCS, and QR factorization; core statistical foundations; and unsupervised methods such as clustering , hierarchical clustering, and dimensionality-reduction techniques including SVD, PCA, and multidimensional scaling. Supervised learning is covered through regression basic classification, LDS, Fisher discriminant analysis, and kernal methods. The course also develops spectral methods—graph Laplacians, heat-kernel embeddings, and spectral clustering—and concludes with geometric and topological tools for data analysis, including metric embeddings, shape-comparison methods, persistent homology, and optimal transport.
Text:
TBA
Prerequisites:
Graduate level Analysis and Linear Algebra
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Spring 2018 - Michael Vogelius
Previous Course Title Name was: Numerical Analysis II (updated to above in Fall 2026)
Course Description:
We will study the numerical solution of linear systems of equations, the approximation of matrix eigenvalues and eigenvectors, the numerical solution of nonlinear systems of equations, numerical techniques for unconstrained function minimization, finite difference and finite element methods for two-point boundary value problems, and finite difference methods for some model problems in partial differential equations.
Despite the many solution techniques presented in elementary calculus and differential equations courses, mathematical models used in applications often do not have the simple forms required for using these methods. Hence, a quantitative understanding of the models requires the use of numerical approximation schemes. This course provides the mathematical background for understanding how such schemes are derived and when they are likely to work.
To illustrate the theory, in addition to the usual pencil and paper problems, some short computer programs will be assigned. To minimize the effort involved, however, the use of Matlab will be encouraged. This program has many built in features which make programming easy, even for those with very little prior programming experience.
This course is also part of the Mathematical Finance Master's Degree Program.
Text:
(Recommended) K. Atkinson “An Introduction to Numerical Analysis" 0-471-62489-6; A. Quarteroni, R. Sacco, and F. Saleri, /Numerical Mathematics, (second edition), Springer, 2004, 0-387-98959-5
Prerequisites:
Numerical Analysis I - 16:642:573 is desirable, but not required
Schedule of Sections
Previous Semesters:
- Spring 2018 Prof. Vogelius
- Spring 2017 Prof. Wujun Zhang
- Spring 2016 Prof. Richard Falk
- Spring 2015 Prof. Duk-soon Oh
- Spring 2014 Prof. Shadi Tahvildar-Zadeh
- Spring 2013 Prof. Shadi Tahvildar-Zadeh
- Spring 2012 Prof. Young-Ju Lee
- Spring 2011 Prof. Young-Ju Lee
- Numerical Analysis 642:574 -- Spring 2007