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Projects in Sparse Grids and High Dimensional Approximation

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Possible topics are listed below. Crossed topics indicate topics already assigned to students. It might however be possible to continue the work of past students or to extend the scope of a topic. Therefore, do not hesitate to ask about crossed out topics if you are interested in one of them.

Topic Description Contact
Towards 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, including fault emulation, power caps, thermal caps, and limited bandwidth, among others. This thesis involves C++ programming, parallel programming, performance analysis, and numerics. Michael Obersteiner; Master thesis
Shared Memory Parallelization for massively-parallel Combination scenarios 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 is one solution to delay the curse of dimensionality and has been successfully implemented for distributed systems. In this work a hybrid parallelization using MPI and OpenMP will be implemented for the distributed combination technique framework. The current MPI-only implementation will be adapted to support OpenMP parallelization. This includes large-scale tests to analyse and optimize the resulting performance. Michael Obersteiner; Bachelor or Master thesis
Time-adaptive combination schemes 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 is one solution to delay the curse of dimensionality and has been successfully implemented for distributed systems. One problem is that each of the component grids has an individual spacing which allows for different time-steps due to different CFL numbers. A common approach is to use the minimal time-steps of all component grids to guarantee global convergence. However, this introduces a severe increase in computational work. In this work students will implement different time-adaptive schemes and check the resulting accuracy and efficiency of the methods. An Example code is SWE, a shallow water equation code. In addition to an implementation a theoretical analysis is done to predict and measure the benefit of using a time-adaptive scheme. Michael Obersteiner; Bachelor thesis or student project
Implementation of the dimension-adaptive Combination Technique into a parallel 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 is one solution to delay the curse of dimensionality and has been successfully implemented for distributed systems. Currently only regular combination schemes and truncated schemes are implemented in our distributed framework. One major drawback is that the code can not adapt to the application and increase resolutions in dimension that need higher accuracy. In this work the well-known dimension adaptive algorithm for the combination technique will be implemented for our black-box framework. In order to have a successful adaptive scheme a suited error estimator will be implemented to detect dimension that would benefit from further refinement. Prerequisites are a good understanding of parallel programming and C++. Interest in sparse grids and the combination technique is advised. Michael Obersteiner; Master thesis or IDP
Towards a new spatially adaptive combination technique 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 is one solution to delay the curse of dimensionality and has been successfully implemented for distributed systems. Currently dimension adaptive combination technique is a widely used approach to tackle imbalances in the required resolution for each dimension. If the underlying distribution however only requires high resolution in a limited spatial area this is currently solved by increasing the resolution of the whole domain. This, however, introduces additional computational and memory overhead that would not be needed. In this work a new spatially adaptive combination technique is analysed and tested for very basic applications (such as quadrature). An interest in mathematics and especially numerics is advised. Michael Obersteiner; (Bachelor thesis), IDP, Master thesis
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 in Python (such as FiPy or SfePy) 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.
Michael Obersteiner; Bachelor thesis
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.
Michael Obersteiner; Master thesis