High-order Gaussian Process Methods for Computational Fluid Dynamics
We develop a whole new family of high-order data interpolation and reconstruction schemes for numerical solutions of PDEs using Gaussian Process (GP) modeling. The GP research group has been developing, studying and implementing a new set of high-order GP schemes for finite difference (FD) and finite volume (FV) methods to solve advection-diffusion multi-physics problems. The new methods are specifically aimed to embody a natural tradeoff between computation (i.e., solution accuracy) and memory.
GP is a stochastic process commonly used for high fidelity modeling. The proposed study can successfully serve as an alternate way to the conventional (and most popular) piecewise high-order polynomial approaches, and can deliver very high-order accurate numerical solutions. By design, the proposed GP high-order approach is also shown to overcome the typical shortcomings in the traditional polynomial methods (e.g., essentially 1-D based interpolations/reconstructions, complexities in multi-dimensions, etc.). Therefore, the GP algorithm will significantly help computer simulations advance from the current petascale level to the next exascale level, particularly by utilizing the ideas of Bayesian modeling in terms of providing new set of methods that are more accurate, faster convergent, more flexible and yet memory frugal, thereby pertinent to the next generation computer architectures.
The GP high-order interpolations/reconstructions center around data training by utilizing covarying relationships between data locations via covariance kernel functions. The novel idea lies in this very nonparametric GP prediction techniques where, with adjustable choices of the covariance kernel functions for different types of data (e.g., continuous vs. discontinuous fluid variables), high-order interpolations/reconstructions can be controlled and established using a truly multidimensional stencil configuration. This unique feature is another very different modeling advantage over the one-dimensional stencil configuration in most of the polynomial-based approaches.
GP for non-fluid dynamics problems
We develop a new interpolation approach using Gaussian Process Modeling. The approach has been extended to produce high-order accurate interpolation method for configuring computational grids adaptively called adaptive mesh refinements in massively parallel simulations.
Another area of application is to reconstruct high-resolution images based on convolutional neural network (CNN).