Algorithms for Large-scale Whole Genome Association Analysis

In order to associate complex traits with genetic polymorphisms, genome-wide association studies process huge datasets involving tens of thousands of individuals genotyped for millions of polymorphisms. When handling these datasets, which exceed the main memory of contemporary computers, one faces two distinct challenges: 1) Millions of polymorphisms come at the cost of hundreds of Gigabytes of genotype data, which can only be kept in secondary storage; 2) the relatedness of the test population is represented by a covariance matrix, which, for large populations, can only fit in the combined main memory of a distributed architecture. In this paper, we present solutions for both challenges: The genotype data is streamed from and to secondary storage using a double buffering technique, while the covariance matrix is kept across the main memory of a distributed memory system. We show that these methods sustain high-performance and allow the analysis of enormous datase

High Performance Solutions for Big-data GWAS

In order to associate complex traits with genetic polymorphisms, genome-wide association studies process huge datasets involving tens of thousands of individuals genotyped for millions of polymorphisms. When handling these datasets, which exceed the main memory of contemporary computers, one faces two distinct challenges: 1) Millions of polymorphisms and thousands of phenotypes come at the cost of hundreds of gigabytes of data, which can only be kept in secondary storage; 2) the relatedness of the test population is represented by a relationship matrix, which, for large populations, can only fit in the combined main memory of a distributed architecture. In this paper, by using distributed resources such as Cloud or clusters, we address both challenges: The genotype and phenotype data is streamed from secondary storage using a double buffer- ing technique, while the relationship matrix is kept across the main memory of a distributed memory system. With the help of these solutions, we develop separate algorithms for studies involving only one or a multitude of traits. We show that these algorithms sustain high-performance and allow the analysis of enormous datasets.Comment: Submitted to Parallel Computing. arXiv admin note: substantial text overlap with arXiv:1304.227

Applying OOC Techniques in the Reduction to Condensed Form for Very Large Symmetric Eigenproblems on GPUs

In this paper we address the reduction of a dense matrix to tridiagonal form for the solution of symmetric eigenvalue problems on a graphics processor (GPU) when the data is too large to fit into the accelerator memory. We apply out-of-core techniques to a three-stage algorithm, carefully redesigning the first stage to reduce the number of data transfers between the CPU and GPU memory spaces, maintain the memory requirements on the GPU within limits, and ensure high performance by featuring a high ratio between computation and communication

Design of a Robotic Instrument Manipulator for Endoscopic Deployment

This thesis describes the initial design process for an application of continuum robotics to endoscopic surgical procedures, specifically dissection of the colon. We first introduce the long-term vision for a benchtop dual-instrument endoscopic system with intuitive haptic controllers and then narrow our focus to the design and testing of the instrument manipulator itself, which must be actuated through the long, winding channel of a standard colonoscope. Based on design requirements for a target procedure, we analyze simulations of two types of continuum robots using recently established kinematic and mechanic modeling approaches: the concentric-tube robot (CTR) and the concentric agonist-antagonist robot (CAAR). In addition, we investigate solutions to the primary engineering challenge to this system, which is accurately transmitting joint motion through exible, hollow shafts. Based on our study of the manipulator simulations and transmission shafts, we select instrument designs for prototyping and testing. We present approaches for controlling the position of the robotic instrument in real-time using an input device, and demonstrate the degree of control we can achieve in various configurations by performing time trial experiments with our prototype robotic instruments. Our observations of the manipulator during testing inform us of sources of error, and we conclude this report with suggestions for future work, including shaft design and alternative continuum manipulator approaches

Many-Body Invariants for Super Spin Chains with Antiunitary Symmetries

Proposals for many-body invariants for super-spin chains with anti-unitary symmetries are evaluated. Symmetry protected phases are modeled as homotopy classes of gapped ground states. The formalism of matrix product states is systematically extended to fermionic systems with anti-unitary symmetries. A basis-independent diagrammatic approach capable of handling anti-unitary symmetries is developed. Suggestions from the literature for observables of a twisted entanglement entropy type are calculated and proven to be topological invariants of fermionic matrix product states. The viability of classifications via these invariants is discussed as well as the connection to the cohomology classification of one-dimensional fermionic symmetry protected phases. Taking the limit of diverging bond dimension while controlling the correlation length, the homotopy invariance is proved to persist

Topological and non-equilibrium superconductivity in low-dimensional strongly correlated quantum systems

Superconductivity in its various manifestations has been stimulating both experimental and theoretical progress in condensed-matter physics for more than a hundred years. The remarkable property of electrons to pair up and form quasi-particles gives rise to a plethora of phenomena featuring important practical applications not only in science, but, for instance, also in medicine and metrology. Recently, new directions in investigating this fascinating subject emerged, such as superconductivity out-of equilibrium and topological superconductors. Providing experimental evidence for enhanced superconducting correlations in optically pumped copper oxides at temperatures far above the equilibrium transition temperature, the first issue caused considerable excitement. On the other hand, topological superconductors are believed to provide realizations of highly fault-tolerant qubits by means of hosting non-Abelian quasi-particles, which can be the building blocks of scalable quantum computers. Experimentally verifying the emergence of these Majorana edge modes, exotic quasi-particles in heterostructures consisting of a conventional superconductor and semiconductors or topological insulators, is one of the most urgent questions to be answered right now. Both subjects cannot be accounted for with analytically solvable approximations only, and also provide very challenging numerical problems. We implemented a matrix-product state (MPS) based toolkit exploiting $U(1)$ symmetries, providing a flexible and efficient platform to study these complex systems. In order to efficiently simulate out-of equilibrium setups we studied, compared, and developed time-evolution algorithms for MPS enabling us to choose the most suitable method for a given task. We also developed a new framework to represent operators in an enlarged Hilbert space so that benefits from conserving $U(1)$ symmetries can also be exploited in systems that originally break such symmetries (projected purification). Using this method we could efficiently model mesoscopic phenomena such as a charging energy controlled by a gate electrode without further approximations. Equipped with this techniques we studied out-of equilibrium spectral functions to explore how to identify superconducting correlations more reliably on ultra-short timescales. We found conclusive evidence that in particular two-particle spectral functions yield excellent probes for the formation of a (quasi-)condensate out-of equilibrium. Furthermore, we also investigated the question whether in a particular model system there is the possibility of true long-range order out-of equilibrium by studying correlation matrices and the scaling of their eigenvalues. Here, we observe a change in the algebraic decay of the correlations, even though the extrapolated order parameter is still zero within the error bounds. Furthermore, we also investigated the effects of coupling a superconductor-semiconductor heterostructure, which is subject to an in-plane magnetic field and a charging energy controlled by a gate voltage, to normal leads. In the context of experimentally verifying the existence of Majorana edge modes, such systems are believed to be the most promising and recent studies seem to underline this expectation. However, in order to consistently analyze the experimental data, the effects of quantum fluctuations caused by hybridization of the heterostructure with the leads have to be understood. Here, only perturbative limits are available so far, i.e., the weak and strong tunneling limit, while the experimentally relevant regime is expected to be somewhere inbetween. We aimed to fill this gap using the projected purification method to calculate the ground state phase diagram over a wide parameter regime. Our results indicate that the experimental situation is much more involved than what is predicted from perturbative analysis

Exponential Thermal Tensor Network Approach for Quantum Lattice Models

We speed up thermal simulations of quantum many-body systems in both one-(1D) and two-dimensional (2D) models in an exponential way by iteratively projecting the thermal density matrix (rho) over cap = e(-beta(H) over cap) onto itself. We refer to this scheme of doubling beta in each step of the imaginary time evolution as the exponential tensor renormalization group (XTRG). This approach is in stark contrast to conventional Trotter-Suzuki-type methods which evolve (rho) over cap. on a linear quasicontinuous grid in inverse temperature beta equivalent to 1/T. As an aside, the large steps in XTRG allow one to swiftly jump across finite-temperature phase transitions, i.e., without the need to resolve each singularly expensive phase-transition point right away, e.g., when interested in low-energy behavior. A fine temperature resolution can be obtained, nevertheless, by using interleaved temperature grids. In general, XTRG can reach low temperatures exponentially fast and, thus, not only saves computational time but also merits better accuracy due to significantly fewer truncation steps. For similar reasons, we also find that the series expansion thermal tensor network approach benefits in both efficiency and precision, from the logarithmic temperature scale setup. We work in an (effective) 1D setting exploiting matrix product operators (MPOs), which allows us to fully and uniquely implement non-Abelian and Abelian symmetries to greatly enhance numerical performance. We use our XTRG machinery to explore the thermal properties of Heisenberg models on 1D chains and 2D square and triangular lattices down to low temperatures approaching ground-state properties. The entanglement properties, as well as the renormalization-group flow of entanglement spectra in MPOs, are discussed, where logarithmic entropies (approximately ln beta) are shown in both spin chains and square-lattice models with gapless towers of states. We also reveal that XTRG can be employed to accurately simulate the Heisenberg XXZ model on the square lattice which undergoes a thermal phase transition. We determine its critical temperature based on thermal physical observables, as well as entanglement measures. Overall, we demonstrate that XTRG provides an elegant, versatile, and highly competitive approach to explore thermal properties, including finite-temperature thermal phase transitions as well as the different ordering tendencies at various temperature scales for frustrated systems