Preconditioners
Block Tridiagonal and Banded Linear Equation Systems+
★Scaled partial pivoting at maximum bandwidth★ These special cases of sparse systems appear often in practice, yet solving them in parallel with pivoting is very challenging because of problems with data dependent execution flow. Here we develop algorithms which implement the data dependent decisions without any SIMD divergence leading to far superior performance.
Algebraic Operator Splittings+
The main motivation behind operator splitting methods is the simpler and faster implicit treatment of individual components in comparison to the entire operator which is typically difficult to invert. We develop an algebraic framework for operator splitting preconditioners for general invertible sparse matrices. In particular, it generalizes ADI and ILU methods to multiple factors and more general factor form.
Parallel AMG, Krylov, multi-colored GS and ILU solvers+
★Comprehensive solver library for large GPU-clusters★ Algebraic multigrid is one of the main tools in science and industry for the solution of large sparse linear equation systems. The AmgX library executes AMG and Krylov methods with different preconditioners on GPU-systems with a single GPU, multiple GPUs and on GPU-clusters. Innovations in numerical schemes, graph algorithms and work scheduling enable the massively parallel execution.
Balanced Geometric Multigrid+
Neither solvers with best numerical convergence nor solvers with best parallel efficiency are the best choice for the fast solution of PDE problems in practice. The fastest solvers require a delicate balance between their numerical and hardware characteristics. Balancing both aspects we can even parallelize completely sequential preconditioners with large parallel speedup and hardly any loss in numerical performance.