Julio E. Castrillon-Candas

Julio E. Castrillon-Candas, Fabio Nobile, Raul F. Tempone

In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by $N$ random variables. The elliptic problem is remapped on to a corresponding PDE with a fixed deterministic domain. We show that the solution can be analytically extended to a well defined region in $\C^{N}$ with respect to the random variables. A sparse grid stochastic collocation method is then used to compute the mean and standard deviation of the QoI. Finally, convergence rates for the mean and variance of the QoI are derived and compared to those obtained in numerical experiments.

Julio E. Castrillon-Candas, Fabio Nobile, Raul F. Tempone

In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by $N$ random variables. The random domain is split into large and small variations contributions. The large variations are approximated by applying a sparse grid stochastic collocation method. The small variations are approximated with a stochastic collocation-perturbation method. Convergence rates for the variance of the QoI are derived and compared to those obtained in numerical experiments. Our approach significantly reduces the dimensionality of the stochastic problem. The computational cost of this method increases at most quadratically with respect to the number of dimensions of the small variations. Moreover, for the case that the small and large variations are independent the cost increases linearly.

Julio E. Castrillon-Candas, Jie Xu

This work considers the problem of numerically approximating statistical moments of a Quantity of Interest (QoI) that depends on the solution of a linear parabolic partial differential equation. The geometry is assumed to be random and is parameterized by $N$ random variables. The parabolic problem is remapped to a fixed deterministic domain with random coefficients and shown to admit an extension on a well defined region embedded in the complex hyperplane. A Stochastic collocation method with an isotropic Smolyak sparse grid is used to compute the statistical moments of the QoI. In addition, convergence rates for the stochastic moments are derived and compared to numerical experiments.

Julio E. Castrillon-Candas

With the advent of massive data sets much of the computational science and engineering community has moved toward data-intensive approaches in regression and classification. However, these present significant challenges due to increasing size, complexity and dimensionality of the problems. In particular, covariance matrices in many cases are numerically unstable and linear algebra shows that often such matrices cannot be inverted accurately on a finite precision computer. A common ad hoc approach to stabilizing a matrix is application of a so-called nugget. However, this can change the model and introduce error to the original solution. It is well known from numerical analysis that ill-conditioned matrices cannot be accurately inverted. In this paper we develop a multilevel computational method that scales well with the number of observations and dimensions. A multilevel basis is constructed adapted to a kD-tree partitioning of the observations. Numerically unstable covariance matrices with large condition numbers can be transformed into well conditioned multilevel ones without compromising accuracy. Moreover, it is shown that the multilevel prediction exactly solves the Best Linear Unbiased Predictor (BLUP) and Generalized Least Squares (GLS) model, but is numerically stable. The multilevel method is tested on numerically unstable problems of up to 25 dimensions. Numerical results show speedups of up to 42,050 times for solving the BLUP problem, but with the same accuracy as the traditional iterative approach. For very ill-conditioned cases the speedup is infinite. In addition, decay estimates of the multilevel covariance matrices are derived based on high dimensional interpolation techniques from the field of numerical analysis. This work lies at the intersection of statistics, uncertainty quantification, high performance computing and computational applied mathematics.