Polynomial and rational approximation techniques for non-intrusive uncertainty quantification [electronic resource]

With the ever-increasing power of computers and the advent parallel computing in the recent decades, scientists and engineers are relying more and more on numerical simulations in their studies of complex physical systems. Given the central role that simulations play in engineering design and decision making, it is crucial to assign confidence in their outputs. One important factor that leads to uncertainty in the output quantity of interest in a physical system, is uncertainty in the inputs like properties of materials, manufacturing details, initial and boundary conditions, \emph{etc}. Our goal in uncertainty quantification is (by definition) to quantify the effects of these input uncertainties on the output quantity of interest. In other words, we are trying to describe the behavior of this output variable as a function of the input uncertainties. In \emph{intrusive} UQ strategies, we solve the governing equations of the physical system in terms of both physical and uncertain variables. Solving these equations is usually significantly more challenging than solving the original (deterministic) equations, and often require writing new code that is substantially different from the deterministic solver. In \emph{non-intrusive} UQ on the other hand, we run the deterministic code for various values of the input parameters, and use the outputs of these simulations to construct an approximation for the behavior of the output quantity of interest as a function of the input uncertainties. Although non-intrusive methods come in many flavors, they are all based on some variation of the following fundamental problem in approximation theory: given the values of a function at a set of points in its domain, how can we efficiently and accurately approximate that function? In this dissertation, we study several variants of this problem. Sometimes, we are free in choosing the points at which the function is evaluated (\emph{i.e.}, the values of the input parameters for which we run the deterministic simulation). In this scenario, we need to choose the points in a way that, given a certain number of function evaluations, we can get the best quality of approximation. As an instance of the problem in this setting, we will look at a non-intrusive \emph{polynomial chaos expansion} (PCE) technique, in which we use weighted least squares to construct a multivariate polynomial surrogate. We present a novel optimization based method for finding the best points for this type of approximation. We are not always free to choose the grid points, and sometimes have to find the best approximation that we can, using a fixed set of points that are just given to us. For these problems, in univariate settings, we present an efficient and accurate method based on the \emph{Floater-Hormann} rational interpolation. For multivariate settings, we present a generalization of the nearest neighbor interpolation based on $L_1$ minimization. This method has similar convergence properties to those of the \emph{moving least squares} method, but unlike moving least squares, it does not come with any tunable parameters. We will also look at a hybrid setting, where some of the points are fixed, and we are free in choosing the rest. Assume that we have found the polynomial interpolant a function at a set of \emph{Chebyshev} points, and decide that we need to use a higher order polynomial interpolant. We present a method for finding the best points to use for finding the higher order interpolant, under the constraint that the previous set of points have to be reused.

Advisor: Iaccarino Gianluca primary advisor. Thesis advisor
Alonso Juan José
Moin Parviz advisor. Thesis advisor