The differential transformation method and Miller's recurrence

The differential transformation method (DTM) enables the easy construction of a power-series solution to a nonlinear differential equation. The exponentiation operation has not been specifically addressed in the DTM literature, and constructing it iteratively is suboptimal. The recurrence for exponentiating a power series by J.C.P. Miller provides a concise implementation of exponentiation by a positive integer for DTM. An equally-concise implementation of the exponential function is also provided.Comment: 5 page

Stochastic Evolution of Graphs using Local Moves

Inspired by theories such as Loop Quantum Gravity, a class of stochastic graph dynamics was studied in an attempt to gain a better understanding of discrete relational systems under the influence of local dynamics. Unlabeled graphs in a variety of initial configurations were evolved using local rules, similar to Pachner moves, until they reached a size of tens of thousands of vertices. The effect of using different combinations of local moves was studied and a clear relationship can be discerned between the proportions used and the properties of the evolved graphs. Interestingly, simulations suggest that a number of relevant properties possess asymptotic stability with respect to the size of the evolved graphs.Comment: 17 pages, 17 figures. Basis for talk given at the LOOPS'05 conference (Potsdam, Germany: 13 Oct. 2005

User-Directed Loop-Transformations in Clang

Directives for the compiler such as pragmas can help programmers to separate an algorithm's semantics from its optimization. This keeps the code understandable and easier to optimize for different platforms. Simple transformations such as loop unrolling are already implemented in most mainstream compilers. We recently submitted a proposal to add generalized loop transformations to the OpenMP standard. We are also working on an implementation in LLVM/Clang/Polly to show its feasibility and usefulness. The current prototype allows applying patterns common to matrix-matrix multiplication optimizations.Comment: LLVM-HPC Workshop 2018 preprin

Loop Optimization Framework

The LLVM compiler framework supports a selection of loop transformations such as vectorization, distribution and unrolling. Each transformation is carried-out by specialized passes that have been developed independently. In this paper we propose an integrated approach to loop optimizations: A single dedicated pass that mutates a Loop Structure DAG. Each transformation can make use of a common infrastructure such as dependency analysis, transformation preconditions, etc.Comment: LCPC'18 preprin

ClangJIT: Enhancing C++ with Just-in-Time Compilation

The C++ programming language is not only a keystone of the high-performance-computing ecosystem but has proven to be a successful base for portable parallel-programming frameworks. As is well known, C++ programmers use templates to specialize algorithms, thus allowing the compiler to generate highly-efficient code for specific parameters, data structures, and so on. This capability has been limited to those specializations that can be identified when the application is compiled, and in many critical cases, compiling all potentially-relevant specializations is not practical. ClangJIT provides a well-integrated C++ language extension allowing template-based specialization to occur during program execution. This capability has been implemented for use in large-scale applications, and we demonstrate that just-in-time-compilation-based dynamic specialization can be integrated into applications, often requiring minimal changes (or no changes) to the applications themselves, providing significant performance improvements, programmer-productivity improvements, and decreased compilation time

Simulations of the Pairwise Kinematic Sunyaev-Zel'dovich Signal

The pairwise kinematic Sunyaev-Zel'dovich (kSZ) signal from galaxy clusters is a probe of their line-of-sight momenta, and thus a potentially valuable source of cosmological information. In addition to the momenta, the amplitude of the measured signal depends on the properties of the intra-cluster gas and observational limitations such as errors in determining cluster centers and redshifts. In this work we simulate the pairwise kSZ signal of clusters at z<1, using the output from a cosmological N-body simulation and including the properties of the intra-cluster gas via a model that can be varied in post-processing. We find that modifications to the gas profile due to star formation and feedback reduce the pairwise kSZ amplitude of clusters by ~50%, relative to the naive 'gas traces mass' assumption. We demonstrate that mis-centering can reduce the overall amplitude of the pairwise kSZ signal by up to 10%, while redshift errors can lead to an almost complete suppression of the signal at small separations. We confirm that a high-significance detection is expected from the combination of data from current-generation, high-resolution CMB experiments, such as the South Pole Telescope, and cluster samples from optical photometric surveys, such as the Dark Energy Survey. Furthermore, we forecast that future experiments such as Advanced ACTPol in conjunction with data from the Dark Energy Spectroscopic Instrument will yield detection significances of at least 20{\sigma}, and up to 57{\sigma} in an optimistic scenario. Our simulated maps are publicly available at: http://www.hep.anl.gov/cosmology/ksz.htmlComment: Journal versio

Memory-Efficient Quantum Circuit Simulation by Using Lossy Data Compression

In order to evaluate, validate, and refine the design of new quantum algorithms or quantum computers, researchers and developers need methods to assess their correctness and fidelity. This requires the capabilities of quantum circuit simulations. However, the number of quantum state amplitudes increases exponentially with the number of qubits, leading to the exponential growth of the memory requirement for the simulations. In this work, we present our memory-efficient quantum circuit simulation by using lossy data compression. Our empirical data shows that we reduce the memory requirement to 16.5% and 2.24E-06 of the original requirement for QFT and Grover's search, respectively. This finding further suggests that we can simulate deep quantum circuits up to 63 qubits with 0.8 petabytes memory.Comment: 2 pages, 2 figures. The 3rd International Workshop on Post-Moore Era Supercomputing (PMES

Amplitude-Aware Lossy Compression for Quantum Circuit Simulation

Classical simulation of quantum circuits is crucial for evaluating and validating the design of new quantum algorithms. However, the number of quantum state amplitudes increases exponentially with the number of qubits, leading to the exponential growth of the memory requirement for the simulations. In this paper, we present a new data reduction technique to reduce the memory requirement of quantum circuit simulations. We apply our amplitude-aware lossy compression technique to the quantum state amplitude vector to trade the computation time and fidelity for memory space. The experimental results show that our simulator only needs 1/16 of the original memory requirement to simulate Quantum Fourier Transform circuits with 99.95% fidelity. The reduction amount of memory requirement suggests that we could increase 4 qubits in the quantum circuit simulation comparing to the simulation without our technique. Additionally, for some specific circuits, like Grover's search, we could increase the simulation size by 18 qubits.Comment: 6pages, 6 figures. The 4th International Workshop on Data Reduction for Big Scientific Data (DRBSD-4

Large-Scale Structure Formation with Massive Neutrinos and Dynamical Dark Energy

Over the next decade, cosmological measurements of the large-scale structure of the Universe will be sensitive to the combined effects of dynamical dark energy and massive neutrinos. The matter power spectrum is a key repository of this information. We extend higher-order perturbative methods for computing the power spectrum to investigate these effects over quasi-linear scales. Through comparison with N-body simulations we establish the regime of validity of a Time-Renormalization Group (Time-RG) perturbative treatment that includes dynamical dark energy and massive neutrinos. We also quantify the accuracy of Standard (SPT), Renormalized (RPT) and Lagrangian Resummation (LPT) perturbation theories without massive neutrinos. We find that an approximation that neglects neutrino clustering as a source for nonlinear matter clustering predicts the Baryon Acoustic Oscillation (BAO) peak position to 0.25% accuracy for redshifts 1 < z < 3, justifying the use of LPT for BAO reconstruction in upcoming surveys. We release a modified version of the public Copter code which includes the additional physics discussed in the paper.Comment: Code available at http://www.hep.anl.gov/cosmology/pert.html . 16 pages, 10 figures. Matches version accepted by PR

Redshift-space distortions in massive neutrino and evolving dark energy cosmologies

Large-scale structure surveys in the coming years will measure the redshift-space power spectrum to unprecedented accuracy, allowing for powerful new tests of the LambdaCDM picture as well as measurements of particle physics parameters such as the neutrino masses. We extend the Time-RG perturbative framework to redshift space, computing the power spectrum P_s(k,mu) in massive neutrino cosmologies with time-dependent dark energy equations of state w(z). Time-RG is uniquely capable of incorporating scale-dependent growth into the P_s(k,mu) computation, which is important for massive neutrinos as well as modified gravity models. Although changes to w(z) and the neutrino mass fraction both affect the late-time scale-dependence of the non-linear power spectrum, we find that the two effects depend differently on the line-of-sight angle mu. Finally, we use the HACC N-body code to quantify errors in the perturbative calculations. For a LambdaCDM model at redshift z=1, our procedure predicts the monopole~(quadrupole) to 1% accuracy up to a wave number 0.19h/Mpc (0.28h/Mpc), compared to 0.08h/Mpc (0.07h/Mpc) for the Kaiser approximation and 0.19h/Mpc (0.16h/Mpc) for the current state-of-the-art perturbation scheme. Our calculation agrees with the simulated redshift-space power spectrum even for neutrino masses above the current bound, and for rapidly-evolving dark energy equations of state, |dw/dz| ~ 1. Along with this article, we make our redshift-space Time-RG implementation publicly available as the code redTime.Comment: 18 pages, 17 figures, 4 tables. Matches version accepted by PRD. redTime code available at http://www.hep.anl.gov/cosmology/pert.htm