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A fast method is presented for adaptive moving mesh generation in multi-dimensions using a domain decomposition parabolic Monge–Ampere approach. The domain decomposition procedure employed here is non-iterative and involves splitting the computational domain into overlapping subdomains. An adaptive mesh on each subdomain is then computed as the image of the solution of the L 2 optimal mass transfer problem using a parabolic Monge–Ampere method. The domain decomposition approach allows straightforward implementation for the parallel computation of adaptive meshes which helps to reduce computational time significantly. Results are presented to show the numerical convergence of the domain decomposition solution to the single domain solution. Several numerical experiments are given to demonstrate the performance and efficiency of the proposed method. The numerical results indicate that the domain decomposition parabolic Monge-Ampere method is more efficient than the standard implementation of the parabolic Monge-Ampere method on the whole domain, in particular when computing adaptive meshes in three spatial dimensions.

In this paper, we present an efficient adaptive moving mesh finite difference method for the modeling of early stage cancer cell invasion of tissue or extracellular matrix (ECM). The cancer cell invasion model is a nonlinear system of reaction–diffusion-taxis partial differential equations (PDEs) describing the time evolution of the cancer cells density and concentrations of proteins of the ECM. The solution of the model exhibits very rapid variations at the boundary of the healthy and cancer cells. Thus, using a uniform grid method to solve the cancer cell invasion model requires a very large number of grid points in order to resolve the steep gradients of the solution precisely. As a result, the computation can become prohibitively expensive and inefficient. In this effort, we propose a positivity preserving scheme for the spatial discretization of the model on an adaptive mesh to accurately capture the sharp structures of the solution. The adaptive mesh is generated by a coordinate transformation obtained as the solution of the optimal mass transfer problem. Several numerical experiments are presented to demonstrate the performance of the adaptive mesh method for solving the cancer cell invasion model. The numerical results show that the proposed adaptive mesh method is effective in capturing and predicting the correct behavior of the cancer cells movement within the ECM.

We propose an unsupervised machine-learning checkpoint-restart (CR) lossy algorithm for particle-in-cell (PIC) algorithms using Gaussian mixtures (GM). The algorithm features a particle compression stage and a particle reconstruction stage, where a continuum particle distribution function is constructed and resampled, respectively. To guarantee fidelity of the CR process, we ensure the exact preservation of charge, momentum, and energy for both compression and reconstruction stages, everywhere on the mesh. We also ensure the preservation of Gauss' law after particle reconstruction. As a result, the GM CR algorithm is shown to provide a clean, conservative restart capability while potentially affording orders of magnitude savings in input/output requirements. We demonstrate the algorithm using a recently developed exactly energy- and charge-conserving PIC algorithm on physical problems of interest, with compression factors $\gtrsim75$ with no appreciable impact on the quality of the restarted dynamics., Comment: Extended abstract for Supercheck21. arXiv admin note: substantial text overlap with arXiv:2007.12273