Automated Transformations for PDE Systems with Application to Multigrid Solvers

Terascale Optimal PDE Simulations (TOPS), An Enabling Technology Center Scientific Discovery Through Advanced Computing: Integrated Software Infrastructure Centers

iii 1 Background and Significance 1 2 Preliminary Studies 3 2.1 PDE Time Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 PDE Nonlinear Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 PDE-constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Linear Solvers . . . ...

Mixed-Precision GPU-Multigrid Solvers with Strong Smoothers

• Sparse iterative linear solvers are the most important building block in (implicit) schemes for PDE problems • In FD, FV and FE discretisations • Lots of research on GPUs so far for Krylov subspace methods, ADI approaches and multigrid • But: Limited to simple preconditioners and smoothing operators •Numerically strong smoothers exhibit inherently sequential data dependencies (impossible to p...

Fast and accurate finite-element multigrid solvers for PDE simulations on GPU clusters

Petascale elliptic solvers for anisotropic PDEs on GPU clusters

Memory bound applications such as solvers for large sparse systems of equations remain a challenge for GPUs. Fast solvers should be based on numerically efficient algorithms and implemented such that global memory access is minimised. To solve systems with up to one trillion (10) unknowns the code has to make efficient use of several million individual processor cores on large GPU clusters. We ...

Review of Methods Inspired by Algebraic-Multigrid for Data and Image Analysis Applications

Algebraic Multigrid (AMG) methods were developed originally for numerically solving Partial Differential Equations (PDE), not necessarily on structured grids. In the last two decades solvers inspired by the AMG approach, were developed for non PDE problems, including data and image analysis problems, such as clustering, segmentation, quantization and others. These solvers share a common princip...