This letter investigates parallelism approaches for equation and Jacobian evaluation in power flow calculations. Two levels of parallelism are proposed and analyzed: inter-model parallelism, which evaluates models in parallel, and intra-model parallelism, which evaluates calculations within each model in parallel. Parallelism techniques such as multi-threading and single instruction multiple data (SIMD) vectorization are discussed, implemented, and benchmarked as six calculation workflows. Case studies on the 70,000-bus synthetic grid show that equation evaluations can be accelerated by ten times, and the overall Newton power flow outperforms MATPOWER by 20%.
In press, SIAM J. Matrix Anal. Appl.Abstract. In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.
This paper constructs tridiagonal random matrix models for general (β > 0) β-Hermite (Gaussian) and β-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for β = 1, 2, 4. Furthermore, in the cases of the β-Laguerre ensembles, we eliminate the exponent quantization present in the previously known models.We further discuss applications for the new matrix models, and present some open problems.
Abstract.We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t, ... , t") projected onto the surface of the unit sphere, divided by n . The probability density of the real zeros is proportional to how fast this curve is traced out.We then relax Kac's assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the Fubini-Study metric.
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