High Performance Linear Solvers and Eigenvalue Computations
Efficient algorithms for matrix functions and equations depend critically on the solutions of underlying linear systems and eigenvalue problems. This course aims at bringing to the students the state-of-the-art high performance methods for sparse linear systems and eigenvalue computations. Many practical aspects will be addressed to deliver high speed and robustness to the users of today's sophisticated high performance computers.
For the part of linear systems, we will discuss sparse factorization techniques for building sparse direct, and hybrid direct/iterative solvers, and efficient preconditioners. An outline of this part is
- Sparse matrix data structures;
- High performance computer architectures, parallel programming;
- Sparse direct solvers;
- Domain decomposition, hybrid direct/iterative solvers;
- Advanced topics, such as scalable memory- and communication-reducing algorithms;
- Applications and software issues.
For the part of eigenvalue computations, we will review existing work on Rayleigh quotient-based methods and present recent developments. An outline of this part is
- Basic theory,
- minimization principles;
- Rayleigh quotient-based optimization techniques;
- Convergence theory and asymptotic analysis;
- Preconditioning techniques;
- Applications and software issues;
- Other eigenvalue problems that admit certain minimization/maximization, including the singular value decomposition problems and the linear response eigenvalue problems.