110311 Numerical Analysis

The course introduces to the analysis of basic classes of numerical algorithms used in large-scale scientific computing applications. The core topics will be numerical linear algebra (with emphasis on sparse matrices), numerical integration and data approximation (including adaptivity), discretization techniques for ODE systems and selected PDE problems. It is suitable for Mathematics students, as well as for undergraduate and beginning graduate students of other majors resp. programs with exposure to computational methods and scientific computing applications in the natural sciences.

To comfort the possibly different needs of different participants, the course will have a "choose option": Students with strong interest and background in Mathematics choose the "Theory" option: Midterm (30%) and Theory Project (40%), which includes additional reading and a mock implementation on the mathematical analysis of numerical algorithms. Students with clear emphasis on applying numerical methods choose the "Methods" option: Midterm (30%) and Application Project (40%) which includes the derivation, implementation, and testing of a more complex numerical model. For both options, students have to work through a number of Homework and small Programming assignments (30%). Students are encouraged to propose their own projects (pending instructor approval). The differentiation between graduate and undergraduate level participants will mainly happen in the Project. The course will not have a Final Exam, all course work has to be finished by the end of the semester (Dec. 7).

The only prerequisite is an introductory course in Numerical Methods such as ESM4A or equivalent basic knowledge (instructor approval necessary). Basic material will be very briefly reviewed as introduction to the slightly more advanced and recent concepts also covered in this course. Since ESM4A has ESM1/2 as prerequisite, it should be clear that familiarity with calculus (including the notion of a differential equation, and multivariate Taylor expansion) and linear algebra is assumed.