High-order Gaussian Processes

High-order Gaussian Process Methods for Computational Fluid Dynamics

Research Group: A. Reyes, D. Lee (UCSC); C. Graziani, P. Tzeferacos (U. of Chicago)

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.

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Figure 1. A comparison of stencil configurations of the polynomial interpolation/reconstruction (the second-order piecewise linear method, PLM; the third-order piecewise parabolic method, PPM) vs. GP. (Left and Center).
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Figure 2. Convergence rates of GP with Squared Exponential covariance kernel using three different radii r = 1, 2 and 3. The GP convergence rate is shown to be a function of radius r and is of (2r + 1)-order on the grid resolutions.