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Minicourse: Numerical Analysis. Prof. Daniel Appelo.

Numerical analysis is the mathematical study of algorithms that uses numerical approximations for problems of mathematical analysis. With the ever increasing power of computers such numerical approximations have become ubiquitous in solving and modeling problems in science and engineering. In this course we explore some specific topics in numerical analysis and illustrate how they can be used to study problems from various fields of science.  We will explore some topics in the field through engaging examples. For example, we will explore Bezier curves with applications to design (cars, teapots) and graphics (calligraphic letters and fonts). The topics considered are:
(1) Approximation with Bezier curves. We discuss the definitions of quadratic and cubic Bezier curves and their geometrical properties, their application in design and graphics. Exercises will involve designs of cars, teapots, calligraphic letters and fonts. We will apply numerical methods to ODES to
(2) Numerical solution of ODEs. The lectures will cover derivation of the classic Runge-Kutta methods. The exercises will cover ay-tracing for seismic exploration. and gradient flow to find the minima on a 2D energy landscape.   Related to this,
(3) Interpolation. The Interpolation problem and algorithms for its implementation in a computer will be considered. The exercises will e will show how gradient flows can be coupled with interpolation to construct a method to find the minimum energy pathway between minima. This is a warmup for one of the proposed REU projects. We will also briefly discuss how a computer works, and the effects of reordering of loops and cache blocking. Many matlab exercises will be incorporated to reinforce the material.
(4) Linear algebra and your computer. Brief discussion of how a computer works. The exercises will compare different implementations of basic linear algebra operations showing the impact of basic optimization e.g. reordering of loops and cache blocking.

Professor Appelo joined our department in Fall 2011. He is an excellent presenter, teacher and mentor, with a pleasant personality. This workshop gives him an opportunity to more closely connect with our students.\\