Stochastic Surrogates for Measurements and Computer Models of Fluids

Abstract

Both measurements and computer simulations of fluids introduce a prediction problem. A Particle Image Velocimetry (PIV) measurement of a flow field results in a discrete grid of velocity vectors, from which we aim to predict the velocity field or related quantities. In Computational Fluid Dynamics (CFD), the output of the computer code depends on the input parameters, and we aim to propogate the uncertainty of the input parameters or aim to determine the choice of input parameters that optimizes the output. Presently, we discuss a general framework that uses a stochastic surrogate to address the prediction problem. In flow measurement, we predict the velocity field conditional on the discrete set of PIV data. For uncertainty propagation and optimization, we construct a surrogate of the CFD output, conditional on a discrete set of solves. As a surrogate we use Kriging, derived in a Bayesian framework. This approach has two main advantages. Firstly, when adjoint-based gradient information is available from the CFD solver, the gradient information can be included in the surrogate, reducing the computational cost of higher-dimensional uncertainty propagation and optimization problems. Secondly, measurement uncertainty of PIV data, CFD output, and CFD gradient output can be represented through the likelihood: notably, PIV data can have local variations of measurement uncertainty due to imperfect experimental conditions, while CFD gradient output is often considerably noisy|to the extent of detoriating results instead of improving them when the noise is not properly accounted for. Before applying the Kriging predictor, two well-known problems are addressed. Firstly, robustness of the Kriging predictor is found not to be related to conditioning| as is often suggested|but to numerical positive definitness of part of the gain matrix. We provide analytical estimates for conditions under which the Kriging predictor is robust. Secondly, estimating the hyperparameters becomes expensive for larger data sets, such as PIV measurements. We discuss two FFT-based methods that reduce the cost of hyperparameter estimation.