Lu Li

Lu Li, Connor Thompson, Gregory Henselman-Petrusek et al.

Cycle representatives of persistent homology classes can be used to provide descriptions of topological features in data. However, the non-uniqueness of these representatives creates ambiguity and can lead to many different interpretations of the same set of classes. One approach to solving this problem is to optimize the choice of representative against some measure that is meaningful in the context of the data. In this work, we provide a study of the effectiveness and computational cost of several $\ell_1$-minimization optimization procedures for constructing homological cycle bases for persistent homology with rational coefficients in dimension one, including uniform-weighted and length-weighted edge-loss algorithms as well as uniform-weighted and area-weighted triangle-loss algorithms. We conduct these optimizations via standard linear programming methods, applying general-purpose solvers to optimize over column bases of simplicial boundary matrices. Our key findings are: (i) optimization is effective in reducing the size of cycle representatives, (ii) the computational cost of optimizing a basis of cycle representatives exceeds the cost of computing such a basis in most data sets we consider, (iii) the choice of linear solvers matters a lot to the computation time of optimizing cycles, (iv) the computation time of solving an integer program is not significantly longer than the computation time of solving a linear program for most of the cycle representatives, using the Gurobi linear solver, (v) strikingly, whether requiring integer solutions or not, we almost always obtain a solution with the same cost and almost all solutions found have entries in {-1, 0, 1} and therefore, are also solutions to a restricted $\ell_0$ optimization problem, and (vi) we obtain qualitatively different results for generators in Erdős-Rényi random clique complexes.

Suejb Memeti, Lu Li, Sabri Pllana et al.

Many modern parallel computing systems are heterogeneous at their node level. Such nodes may comprise general purpose CPUs and accelerators (such as, GPU, or Intel Xeon Phi) that provide high performance with suitable energy-consumption characteristics. However, exploiting the available performance of heterogeneous architectures may be challenging. There are various parallel programming frameworks (such as, OpenMP, OpenCL, OpenACC, CUDA) and selecting the one that is suitable for a target context is not straightforward. In this paper, we study empirically the characteristics of OpenMP, OpenACC, OpenCL, and CUDA with respect to programming productivity, performance, and energy. To evaluate the programming productivity we use our homegrown tool CodeStat, which enables us to determine the percentage of code lines that was required to parallelize the code using a specific framework. We use our tool x-MeterPU to evaluate the energy consumption and the performance. Experiments are conducted using the industry-standard SPEC benchmark suite and the Rodinia benchmark suite for accelerated computing on heterogeneous systems that combine Intel Xeon E5 Processors with a GPU accelerator or an Intel Xeon Phi co-processor.