Projects in Sparse Grids and High Dimensional Approximation

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Exascale-ready PDE solvers We will soon reach the era of exascale: 10^18 floating point operations per second on high-end supercomputers. This will allow scientists to explore new research fields, but the sheer complexity of these systems brings along many issues. Among the most difficult ones is fault tolerance. A supercomputer with hundreds of thousands of computing elements will inevitably suffer from faults of different types, and algorithm designers should take this into consideration. We have developed a PDE solver that can run on large HPC systems and respond to simulated hardware faults. The objective of this thesis will be to test the exascale emulator GREMLINS on a high-dimensional PDE solver. Several test scenarios will be studied to understand how the solver could perform in future exascale systems. These scenarios include not only fault emulation, but also power caps, thermal caps, or limited bandwidth, among others. Alfredo Parra
Coupling general purpose PDE solvers with a Combination Technique Framework The solution of high-dimensional problems using grid-based methods is a numerically demanding task. One reason is the curse of dimensionality. The sparse grid combination technique has been successfully applied to the solution of elliptic PDEs (such as the Poisson equation) and hyperbolic PDEs (such as the advection equation). Coupling it to general purpose PDE solvers will allow to test many different PDEs and to identify possible application areas.
The work will include a comprehensive literature study and a comparison of existing PDE solver frameworks, their coupling to the existing combination technique framework written in Python, and a study of the numerical errors introduced by the combination technique for each PDE problem.
Alfredo Parra
Applying the Optimized Sparse Grid Combination Technique on the Schroedinger Equation The sparse grid combination technique has proved to be a viable method for solving high-dimensional PDE problems such as the Schroedinger equation. With the combination technique one solves a large eigenvalue problem on various grids of varying resolution. The solutions on these various grids are then combined to approximate a high-resolution grid. The computational effort of computing the solution on the different grids is substantially smaller than obtaining a single, finely-resolved approximation. A new method for solving eigenvalue problems with the combination technique has been developed. Its performance for the Schroedinger equation will have to be compared with existing regular and combination technique approaches.
The work includes the implementation of a solver of the Schroedinger equation on varying non-equidistant meshes. After its validation it will be used with the various traditional and new combination techniques for eigenvalue problems. The existing results will be compared with this new method.
Alfredo Parra