An hp-proper orthogonal decomposition-moving least squares approach for molecular dynamics simulation

The application of model order reduction for molecular dynamic systems exhibits intrinsic complexities due to the highly nonlinear and nonlocal properties of such systems. The most costly computational part of such work is the calculation of potential energy and inter-atomic force fields. We introduce the so-called hp-proper orthogonal decomposition-moving least squares (hp-POD-MLS) method not only to tackle this force computation, but also to improve the accuracy of reduced displacement approximations in an adaptive and controllable manner. The approach combines the proper orthogonal decomposition (POD) method with the moving least squares (MLS) one to interpolate the inter-atomic force field at an arbitrary time instance; this approximated force field is then fed into the reduced molecular dynamic (MD) governing equation to compute the reduced order displacement field. In fact, this POD-MLS approach is already able to handle efficiently the force computation in the online computational stage (i.e., the online force computational cost is independent of the dimension of the full MD space); however its corresponding approximated displacement error will be very large if the considered total time interval is long. The proposed hp-POD-MLS algorithm will divide adaptively and automatically the global temporal interval into many smaller local subintervals (h-refinement), and then build the sets of local reduced basis functions over each of these subintervals (p-refinement). The goal is to reduce the corresponding displacement errors to a desired tolerance over these subintervals. The effectiveness of the algorithm is demonstrated by performing reduced-order simulations of several molecular dynamic systems.