GPU Accelerated Solutions for Curl-Curl Equation in Frequency Domain Electromagnetic Computations

The Curl-Curl equation is the fundament of time-harmonic electromagnetic (EM) problems in geophysics. The efficiency of its solvers is the key to the Magnetotelluric/Controlled Source EM simulations, which account for over 95% of the computation cost in practical inversion problems. However, most published EM inversion codes are still CPU-based and cannot utilize the recent computation techniques of GPGPU. Based on the previously proposed divergence-free algorithm, this study aims to develop a GPU accelerated method to solve the linear systems from the staggered-grid finite difference approximation of curl-curl problem. The large sparse linear systems arise from the discretization are manipulated as compressed matrices to enable efficient transfer and storage in the GPU memory. The matrices and vectors are further decomposed for GPU to perform the basic accelerated matrix-vector and vector-vector operations in parallel. The system is then solved in a hybrid mixed-precision setup with KSP solver, to exploit the high throughput for lower-precision computation in GPUs. The new algorithm is tested in magnetotelluric forward and adjoint calculations, with both the synthetic and real-world examples (10M DoFs). On comparisons with conventional ModEM CPU algorithms, test results show promising 3-4x improvement regarding the solution speed on a NVidia A100 GPU, over our 28-core CPU server with dual Intel Xeon E5 processor. This may further take the large-scale frequency domain EM inversions onto the next, practical stage on small modern GPU platforms.