Jun.-Prof. Dr.-Ing. Simon Adrian

English or German (in the case of German-speaking participants only)

Summer

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

They will be able to chose numerical methods depending on the nature of the problem at hand and to judge the computational complexity of the algorithms and include this judgment in their choices. Moreover, they will be aware of current limitations and open research questions.

A 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 Multiple 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 is determined by

• Pass/not passed: homework assignments
• 75 %: oral examination
• 25 %: programming projects

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