GPU-Based Multi-Level Parallelization for Hines Solver and Some Related Problems

Gayatri, P K and Upadrasta, Ramakrishna (2019) GPU-Based Multi-Level Parallelization for Hines Solver and Some Related Problems. Masters thesis, Indian institute of technology Hyderabad.

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Abstract

We propose to optimize regular (affine) portions of CVODE-DENSE and CVODE-BAND solvers using the PLUTO polyhedral optimizer. We evaluate the effectiveness of polyhedral optimizations on two real-world ODEs; using both the varieties (BAND and DENSE) of solvers. We observed improvements of up to 2.4x on the DENSE solver and up to 1.5x for the BAND solver. This shows that SIMD vectorization is more suitable for BAND solver while loop tiling followed by OpenMP parallelization is more suitable for DENSE solver. Our contribution will make a viable case to apply on various loop transformations and polyhedral compilation techniques on ODE solvers. We also propose a GPU-based multi-level parallelization algorithm for solving the Hines matrix system generated by Hodgkin Huxley multi-compartment neuron model equations. The inner-level parallelism is achieved by employing fine-decomposition along with Exact Domain Decomposition method. The outer-level parallelism is obtained by applying parallel-in-time algorithm. CUDA’s dynamic parallelism technique is used by the proposed algorithm to achieve multi-level parallelism on GPUs. On well-known neuron morphologies, our approach beats the sequential time method up to 2.5x. Also, the iterative part of the parallel-in -time algorithm used, i.e. the Parareal method converges in five to eight iterations with an average precision of 10−6 . Finally, We suggest an approximation strategy for Renormalization Group of Higher dimensions problem. Proposed approximation strategy is applied to tensor contraction computations present in the problem. Tensors are stored in Coordinate format.

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