Local-metrics error-based Shepard interpolation as surrogate for highly non-linear material models in high dimensions

Abstract

Many problems in computational materials science and chemistry require the evaluation of expensive functions with locally rapid changes, such as the turn-over frequency of first principles kinetic Monte Carlo models for heterogeneous catalysis. Because of the high computational cost, it is often desirable to replace the original with a surrogate model, e.g., for use in coupled multiscale simulations. The construction of surrogates becomes particularly challenging in high-dimensions. Here, we present a novel version of the modified Shepard interpolation method which can overcome the curse of dimensionality for such functions to give faithful reconstructions even from very modest numbers of function evaluations. The introduction of local metrics allows us to take advantage of the fact that, on a local scale, rapid variation often occurs only across a small number of directions. Furthermore, we use local error estimates to weigh different local approximations, which helps avoid artificial oscillations. Finally, we test our approach on a number of challenging analytic functions as well as a realistic kinetic Monte Carlo model. Our method not only outperforms existing isotropic metric Shepard methods but also state-of-the-art Gaussian process regression.

Interpolation Methods for Molecular Potential Energy Surface Construction

“…Later, Zhou and co-workers used modified Shepard interpolation with 30 000 nodes to produce a globally accurate PES to study the reaction probabilities and integral cross sections of the same reaction . Lorenzi and co-workers use error-based local metrics to improve the accuracy of modified Shepard interpolation in regimes where rapid changes occur and apply their method to highly nonlinear material models . Since not all reaction coordinates are always influential on the process being studied, and since numerical instabilities arise near linear structures, Everhuis and Collins proposed a framework where the appropriate set of PES inputs is determined locally .…”

“…Several authors have studied gas–surface adsorption dynamics in up to 9 dimensions, requiring between 1000 and 4000 nodes. ,− However, it should be noted that the previous two applications of Shepard interpolation approximate the PES locally, in the vicinity of a particular reaction path; see Figure a,b. In a 7-dimensional application of Lorenzi et al’s local-metrics error-based modification, the PES converged well enough with 1024 nodes to inform a computational fluid dynamics (CFD) simulation . Morris and Jordan constructed a 3D dipole-moment surface for water with 2304 symmetry-unique points in a tensor-product framework …”

Computational Fluid Dynamics of Reacting Flows at Surfaces: Methodologies and Applications

“…To overcome such a limitation, a more complex method, e.g., Shepard interpolation, has been proposed to improve the prediction of the spline approach. It revealed to provide an excellent approximation of the complex reaction rates which characterize kMC simulations [88,96,97]. However, the accuracy and efficiency of these two approaches are strongly dependent on the size of the kinetic scheme, and thus, their applicability for complex kinetic schemes is hampered [89,93].…”

Multilevel Adaptive Sparse Grid Quadrature for Monte Carlo models

“…As a realistic example, we turn to a stochastic model for the CO oxidation on a RuO 2 (110) heterogeneous catalyst [38,39] and address its parametric uncertainty. It is a reduced version of the original quantum chemistry based model [40] and exploits that the chemistry is mainly controlled by the so-called cus adsorption sites [5,41,42].…”

“…are the rates, with which the respective process occurs. For further details on the model, we refer to the references [38,39].…”