Optimization and parallelization of tensor andODE/PDE computations on GPU

We propose a multi-level GPU-based parallelization algorithm to solve the multi-compartment Hodgkin Huxley (HH) model equation that requires solving the Hines matrix. We use a ‘parallel-in-time’ algorithm (like the Parareal strategy) for obtaining outer level parallelism, and an Exact Domain Decomposition (EDD) algorithm with fine-decomposition for inner-level parallelism. We show that our technique can also be applied to any differential equation like the heat equations which induce tridiagonal systems. Typically, a solution to the HH equation runs for hundreds to tens of thousands of time-steps while solving a Hines matrix at each time step. Previous solutions by Michael Mascagni et al. (1991) and Hines et al. (2008) to this problem have tackled only solving the Hines matrix in parallel. Our approach uses the dynamic parallelism of CUDA to achieve multi-level parallelism on GPUs. Our solution outperforms the sequential time method on standard neuron morphologies upto 2.5x. We also show that iterative part of parareal method converges in 5-7 iterations on average with an accuracy of 10−6. We also propose a GPU optimization for the Higher Order Tensor Renormalization Group problem, where the tensor contraction operations inside HOTRG is optimized by a multi- GPU implementation using cuBLAS xt API.