Sparse Automatic Differentiation for Large-Scale Computations Using Abstract Elementary Algebra

Most numerical solvers and libraries nowadays are implemented to use mathematical models created with language-specific built-in data types (e.g. real in Fortran or double in C) and their respective elementary algebra implementations. However, built-in elementary algebra typically has limited functionality and often restricts flexibility of mathematical models and analysis types that can be applied to those models. To overcome this limitation, a number of domain-specific languages with more feature-rich built-in data types have been proposed. In this paper, we argue that if numerical libraries and solvers are designed to use abstract elementary algebra rather than language-specific built-in algebra, modern mainstream languages can be as effective as any domain-specific language. We illustrate our ideas using the example of sparse Jacobian matrix computation. We implement an automatic differentiation method that takes advantage of sparse system structures and is straightforward to parallelize in MPI setting. Furthermore, we show that the computational cost scales linearly with the size of the system.Comment: Submitted to ACM Transactions on Mathematical Softwar

Effect of Gain-Dependent Phase Shift on Fiber Laser Synchronization

Recent experiments have demonstrated synchronization of fiber laser arrays at low and moderate pump levels. It has been suggested that a key dynamical process leading to synchronized behavior is the differential phase shift induced by the gain media. We explore theoretically the role of this effect in generating inphase dynamics. We find that its presence can substantially enhance the degree of inphase stability to an extent that could be practically important. At the same time, our analysis shows that a gain-dependent phase shift is not a necessary ingredient in the dynamical selection of the inphase state, thus, leading us to reconsider the essential mechanism behind inphase selection in fiber laser arrays

Linear solvers for power grid optimization problems: a review of GPU-accelerated linear solvers

The linear equations that arise in interior methods for constrained optimization are sparse symmetric indefinite and become extremely ill-conditioned as the interior method converges. These linear systems present a challenge for existing solver frameworks based on sparse LU or LDL^T decompositions. We benchmark five well known direct linear solver packages using matrices extracted from power grid optimization problems. The achieved solution accuracy varies greatly among the packages. None of the tested packages delivers significant GPU acceleration for our test cases

Towards Efficient Alternating Current Optimal Power Flow Analysis on Graphical Processing Units

We present a solution of sparse alternating current optimal power flow (ACOPF) analysis on graphical processing unit (GPU). In particular, we discuss the performance bottlenecks and detail our efforts to accelerate the linear solver, a core component of ACOPF that dominates the computational time. ACOPF analyses of two large-scale systems, synthetic Northeast (25,000 buses) and Eastern (70,000 buses) U.S. grids [1], on GPU show promising speed-up compared to analyses on central processing unit (CPU) using a state-of-the-art solver. To our knowledge, this is the first result demonstrating a significant acceleration of sparse ACOPF on GPUs