A Parallel Algebraic Multigrid Method for Elliptic Problems with Highly Discontinuous Coefficients

The aim of this thesis is the development of a parallel algebraic multigrid method suitable for solving linear systems arising from the discretization of scalar and systems of partial differential equations. Among others it is suitable from conforming finite element methods, finite volume methods, and discontinuous Galerkin methods. The method is especially tailored for the solution of diffusion problems with highly oscillating and discon- tinuous diffusion coefficients. The presented approach uses a new strength of connection measure for guiding the construction of the coarse level matrices. It uses a heuristic greedy aggregation algorithm that allows for aggressive coarsening. It is able to detect weak connections in the matrix graph even for anisotropic diffusion with bi- and trilinear finite elements and thus leads to semi- coarsening even for these cases. At the same time it keeps the stencil size from the finer levels and thus the total operator complexity low even for three dimensional problems. This leads to a very low memory consump- tion of our solver compared with other methods. We develop extensions of the solver to systems of partial differential equation by using special blocking approaches of the unknowns. These blockings are emulated by the underlying matrix and vector data struc- tures. As the blocking is already available to the compiler, it can be exploited to produce automatically more efficient code. For the solution of the linear systems stemming from Discontinuous Galerkin discretizations, we employ the subspace of continuous linear basis function as the space associated with the first coarse level. The further coarsening is done by using the above algorithm. For the method of Baumann and Oden we need to use overlapping Schwarz methods as smoothers to get a convergent method. Their local subspaces are con- structed using our aggregation algorithm on the blocks consisting of all unknowns associated with each element. Finally we present a parallelisation approach for iterative solvers and use it to parallelise our algebraic multigrid method. In our approach the information about the data decomposition is kept apart from the linear al- gebra solvers and data structures. It is used to keep the data stored in the local memory of the process consistent. Using our proposed consistency model, the efficient sequential linear algebra solvers and data structures can be reused without the need to rewrite the actual solver algorithms

Pure single photons from a trapped atom source

Single atoms or atom-like emitters are the purest source of on-demand single photons, they are intrinsically incapable of multi-photon emission. To demonstrate this degree of purity we have realized a tunable, on-demand source of single photons using a single ion trapped at the common focus of high numerical aperture lenses. Our trapped-ion source produces single-photon pulses at a rate of 200 kHz with g$^2(0) = (1.9 \pm 0.2) \times 10^{-3}$, without any background subtraction. The corresponding residual background is accounted for exclusively by detector dark counts. We further characterize the performance of our source by measuring the violation of a non-Gaussian state witness and show that its output corresponds to ideal attenuated single photons. Combined with current efforts to enhance collection efficiency from single emitters, our results suggest that single trapped ions are not only ideal stationary qubits for quantum information processing, but promising sources of light for scalable optical quantum networks.Comment: 7 pages plus one page supplementary materia

Site-resolved imaging of a fermionic Mott insulator

The complexity of quantum many-body systems originates from the interplay of strong interactions, quantum statistics, and the large number of quantum-mechanical degrees of freedom. Probing these systems on a microscopic level with single-site resolution offers important insights. Here we report site-resolved imaging of two-component fermionic Mott insulators, metals, and band insulators using ultracold atoms in a square lattice. For strong repulsive interactions we observe two-dimensional Mott insulators containing over 400 atoms. For intermediate interactions, we observe a coexistence of phases. From comparison to theory we find trap-averaged entropies per particle of $1.0\,k_{\mathrm{B}}$. In the band-insulator we find local entropies as low as $0.5\,k_{\mathrm{B}}$. Access to local observables will aid the understanding of fermionic many-body systems in regimes inaccessible by modern theoretical methods.Comment: 6+7 page

The iterative solver template library

The numerical solution of partial differential equations frequently requires the solution of large and sparse linear systems. Using generic programming techniques like in C++ one can create solver libraries that allow efficient realization of "fine grained interfaces", i. e. with functions consisting only of a few lines, like access to individual matrix entries. This prevents code replication and allows programmers to work more efficiently. In this paper we present the "Iterative Solver Template Library" (ISTL) which is part of the "Distributed and Unified Numerics Environment" (DUNE). It applies generic programming in C++ to the domain of iterative solvers of linear systems stemming from finite element discretizations. Those discretizations exhibit a lot of structure. Our matrix and vector interface supports a block recursive structure. I. E. each sparse matrix entry can be a sparse or a small dense matrix itself. Based on this interface we present efficient solvers that use the recursive block structure via template metaprogramming

An Evaluation and Comparison of GPU Hardware and Solver Libraries for Accelerating the OPM Flow Reservoir Simulator

Realistic reservoir simulation is known to be prohibitively expensive in terms of computation time when increasing the accuracy of the simulation or by enlarging the model grid size. One method to address this issue is to parallelize the computation by dividing the model in several partitions and using multiple CPUs to compute the result using techniques such as MPI and multi-threading. Alternatively, GPUs are also a good candidate to accelerate the computation due to their massively parallel architecture that allows many floating point operations per second to be performed. The numerical iterative solver takes thus the most computational time and is challenging to solve efficiently due to the dependencies that exist in the model between cells. In this work, we evaluate the OPM Flow simulator and compare several state-of-the-art GPU solver libraries as well as custom developed solutions for a BiCGStab solver using an ILU0 preconditioner and benchmark their performance against the default DUNE library implementation running on multiple CPU processors using MPI. The evaluated GPU software libraries include a manual linear solver in OpenCL and the integration of several third party sparse linear algebra libraries, such as cuSparse, rocSparse, and amgcl. To perform our bench-marking, we use small, medium, and large use cases, starting with the public test case NORNE that includes approximately 50k active cells and ending with a large model that includes approximately 1 million active cells. We find that a GPU can accelerate a single dual-threaded MPI process up to 5.6 times, and that it can compare with around 8 dual-threaded MPI processes