Improving the performance of the adjoint inversion scheme for mantle convection with the particle-in-cell method

Mantle convection ultimately drives Earth’s tectonics. Therefore, it is key to truthfully reconstruct the past mantle flow and associated dynamic evolution. Mathematically, mantle convection could be approximated as an advection-diffusion problem with a high effective viscosity. When supplemented with appropriate initial and boundary conditions, the past mantle flow can be numerically reconstructed. However, constructing physically and geologically valid initial conditions for Earth remains challenging due to the lack of direct observations. The adjoint inversion scheme demonstrates great potential for constraining such valid initial conditions through assimilating present-day mantle structures revealed by seismic tomography and other geophysical and geological observations. In practice, this method iteratively solves the forward and adjoint conservation equations until the mismatch between the model prediction and observation becomes small enough. 

However, traditional grid-based energy equation solvers suffer from numerical oscillations due to instability of the advection term. Numerical errors will accumulate through the iterative process, resulting in a final mismatch locked to a local minimum rather than the global minimum. To address this problem, we introduce the particle-in-cell (PIC) method, which solves the advection problem using Lagrangian tracers, to improve the solutions to both the forward and adjoint energy equations. Here, we present a series of synthetic experiments to validate the new method while exploring its limitations. Our preliminary results show that incorporating the PIC method into the adjoint inversion scheme significantly reduces numerical oscillations, thus improving the reliability of the reconstructed initial conditions. Further experiments show that the updated adjoint inversion scheme converges much faster and produces a smaller mismatch between the prediction and observation, thereby reducing computational cost while improving the validity of numerical results. This updated adjoint scheme will be applied to real Earth problems and the result will improve our understanding of complex geodynamic problems.