Graphics & Geometry Group

Scientific Computing

Lecturer: Prof. Dr. Mario Botsch
Assistants: Astrid Bunge, Martin Komaritzan
Lecture: Tue, 14-16, Room T2-205
Exercise: Wed, 12-14, Room GZI
Wed, 14-16, Room GZI
eKVV: 392022
Credits: 5 points
Scientific Computing


Many interesting projects in natural sciences and engineering require the computation of numerical solutions to certain mathematical problems, such as solving systems of equations or minimizing some cost function. This course introduces the most frequently used numerical methods in a compact manner, based on intuitive and interesting examples from computer graphics and physics-based dynamic simulations.

We will not focus on the theoretical derivation of the presented techniques. Instead, our goal is to effciently and robustly solve numerical problems in practical applications, which requires these three steps:

  1. Given an engineering problem, formulate it as a mathematical problem, for instance as a system of equations or an optimization problem.
  2. Given a mathematical problem, analyze its properties to understand which numerical methods can be employed for its solution.
  3. Given a numerical method, know which open-source implementation can be used and/or how to implement it yourself as an efficient and robust algorithm.

The numerical methods to be discussed include solving dense and sparse linear systems, least squares approximations, and partial differential equations. We will also discuss efficient C++ programming and shared memory parallelization.

To facilitate a better understanding we will implement most of the techniques that we discuss in the lecture in the programming assignments. Our exercises therefore consist of several mini-projects, which you can work on alone or in groups. Our tutors have weekly consulting hours, where students can get help if they have trouble with the implementation. At the end of each mini-project, students will present their results in the exercise course.




Week Lecture (Wednesday) Exercise (Wednesday)
Introduction (HTML, PDF) no exercise
Linear Systems, LU FactorizationC++ Crash Course
Least Squares, Cholesky FactorizationCurve Interpolation
and Approximation
QR Factorization
SVD, Numerical Stability
Heat Equation, Time Integration
Laplace Equation, Gradient DescentDiffusion
Conjugate Gradients, Sparse Matrices
Efficient C++ (HTML, PDF) Laplace Equation
Parallel Computing, OpenMP (HTML, PDF)
Wave Equation, Band CholeskyParallelization
Sparse Cholesky Factorization
GPU Computing (HTML, PDF) Wave Equation
Automatic Differentiation (PDF)
Conclusion (HTML, PDF)