Prof. Dr.-Ing. Simon Adrian

Summer

• Tuesday, 15:15-16:45, BigBlueButton
• Thursday, 15:15-16:45, BigBlueButton
• Kick-off: 3rd November

The lecture is live and allows interactive participation. In addition, the lecture will be recorded and made available online in Stud.IP. Active participation is highly recommended!

After completing the module, students will have an understanding of fast iterative and direct methods for solving elliptic partial differential equations such as the Laplace, Poisson or Helmholtz equation.

They are able to select numerical methods depending on the nature of the problem and to assess the computational complexity of the algorithms and incorporate this assessment into their choice. Furthermore, they are aware of current limitations and open research questions.

Successful completion will enable them to work in computational research groups.

Modern algorithms for the fast solution of elliptic PDEs will be studied. Possible topics include (the class will be adapted to the background and interests of the students each term):

• Linear Algebra: Matrix factorizations and low-rank approximations; randomized methods for low-rank approximation; fast algorithms for rank-structured matrices
• Solution of multi-body problems: Ewald summation, Barnes-Hutt, fast multipole method
• Introduction to integral equations to solve acoustic and electromagnetic problems. Discretization of integral equations (Nyström, Galerkin).
• Introduction to iterative solvers and operator preconditioning for integral equations.
• Fast direct solvers for integral equations (hierarchical block separable matrices).
• Fast direct solvers for elliptic PDEs (Laplace or Helmholtz equation): fast direct sparse solvers, sweeping schemes.

The final grade consists of

• Passed/failed: Homework
• 75 %: Oral exam
• 25 %: Programming project

Martinsson, Per-Gunnar. Fast direct solvers for elliptic PDEs. Society for Industrial and Applied Mathematics, 2019.