Calculation of Distributed-Order Fractional Derivative on Tensor Cores-Enabled GPU

Due to an increased computational complexity of calculating the values of the distributed-order Caputo fractional derivative compared to the classical Caputo derivative there is a need to develop new techniques that accelerate it. In this paper for this purpose we propose to use a fast matrix "multiply and accumulate" operation available in GPU’s that contain the so-called tensor cores. We present and experimentally analyze the properties of GPU-algorithms that are based on the L1 finite-difference approximation of the derivative and incorporate them into the Crank-Nicholson scheme for the distributed-order time-fractional diffusion equation. The computation of derivative’s values on GPU was faster than the multi-threaded implementation on CPU only for a large number of time steps with growing performance gain when number of time steps increase. The usage of the single-precision data type increased the error up to \(2.7\%\) comparing with the usage of the double-precision data type. Half-precision computations in tensor cores increased the error up to \(29.5\%\). While solving a time-fractional diffusion equation, algorithms implemented for GPU with the usage of the single-precision data type were at least three times faster than the CPU-implementation for the number of time steps more than 1280. Data type precision had only slight influence on the solution error with significantly increased execution time when the double-precision data type was used for data storage and processing.