EU Regional School - Vömel Seminar
Dr. Vömel - Practical Solution of Eigenproblems in Science and Engineering
Zurich University of Applied Sciences
Eigenvalue/vector problems play an important role in science and engineering; they arise in various contexts such as canonical forms of operators (mathematics), the solutions of wave equations (physics), and modal analysis of vibrational excitation (engineering).
In practice, there are two common types of solution methods. Dense eigensolvers allow a numerically stable computation in many relevant cases and can be essentially used as a black box. However, their cost is typically quadratic in memory and cubic in operations with respect to the problem dimension. Furthermore, they rarely can take advantage of natural sparsity in systems from finite difference, volumes, or element discretizations.
Sparse eigensolvers offer a more memory-economical, matrix-free solution approach where typically, one only needs the discrete operator as a matrix-vector product. Their main disadvantages are typically the dependence on preconditioners for fast convergence as well as a the difficulty of choosing inner linear solver, start vector, basis size, and restart strategy.
This lecture series provides information on theoretical ideas and practical issues of modern dense and sparse eigensolvers. Participants will be able to make informed decisions on algorithmic strategies and trade-offs in their own applications.
The course will cover the following topics.
- Examples, basic theory, importance of tridiagonal and Hessenberg form.
- Dense problems (all or a subset of eigenpairs): QR, Divide & Conquer, Inverse Iteration, MRRR
- Sparse problems I: Arnoldi and Lanczos, Shift-Invert with a nearly singular matrix
- Sparse problems II: PCG and Davidson variants, Preconditioning