Laura Scarabosio

Wouter Gerrit van Harten, Laura Scarabosio

In this work, we address the efficient computation of parameterized systems of linear equations, with possible nonlinear parameter dependence. When the matrix is highly sensitive to the parameters, mean-based preconditioning might not be enough. For this scenario, we explore an approach in which several preconditioners are placed in the parameter space during a precomputation step. To determine the optimal placement of a limited number of preconditioners, we estimate the expected number of iterations with respect to a given preconditioner a priori and use a location-allocation strategy to optimize the placement of the preconditioners. We elaborate on our methodology for the Helmholtz problem with exterior Dirichlet scattering at high frequencies, and we estimate the expected number of GMRES iterations via a gray-box Gaussian process regression approach. We illustrate our approach in two practical applications: scattering in a domain with a parametric refractive index and scattering from a scatterer with parameterized shape. Using these numerical examples, we show how our methods leads to runtime savings of about an order of magnitude. Moreover, we investigate the effect of the parameter dimension and the importance of dimension anisotropy on their efficacy.

Laura Scarabosio, Barbara Wohlmuth, J. Tinsley Oden et al.

Methods for generating sequences of surrogates approximating fine scale models of two-phase random heterogeneous media are presented that are designed to adaptively control the modeling error in key quantities of interest (QoIs). For specificity, the base models considered involve stochastic partial differential equations characterizing, for example, steady-state heat conduction in random heterogeneous materials and stochastic elastostatics problems in linear elasticity. The adaptive process involves generating a sequence of surrogate models defined on a partition of the solution domain into regular subdomains and then, based on estimates of the error in the QoIs, assigning homogenized effective material properties to some subdomains and full random fine scale properties to others, to control the error so as to meet a preset tolerance. New model-based Multilevel Monte Carlo (mbMLMC) methods are presented that exploit the adaptive sequencing and are designed to reduce variances and thereby accelerate convergence of Monte Carlo sampling. Estimates of cost and mean squared error of the method are presented. The results of several numerical experiments are discussed that confirm that substantial saving in computer costs can be realized through the use of controlled surrogate models and the associated mbMLMC algorithms.

Laura Scarabosio

We consider the point evaluation of the solution to interface problems with geometric uncertainties, where the uncertainty in the obstacle is described by a high-dimensional parameter $\boldsymbol{y}\in[-1,1]^d$, $d\in\mathbb{N}$. We focus in particular on an elliptic interface problem and a Helmholtz transmission problem. Point values of the solution in the physical domain depend in general non-smoothly on the high-dimensional parameter, posing a challenge when one is interested in building surrogates. Indeed, high-order methods show poor convergence rates, while methods which are able to track discontinuities usually suffer from the so-called curse of dimensionality. For this reason, in this work we propose to build surrogates for point evaluation using deep neural networks. We provide a theoretical justification for why we expect neural networks to provide good surrogates. Furthermore, we present extensive numerical experiments showing their good performance in practice. We observe in particular that neural networks do not suffer from the curse of dimensionality, and we study the dependence of the error on the number of point evaluations (that is, the number of discontinuities in the parameter space), as well as on several modeling parameters, such as the contrast between the two materials and, for the Helmholtz transmission problem, the wavenumber.

Ustim Khristenko, Laura Scarabosio, Piotr Swierczynski et al.

We consider the generation of samples of a mean-zero Gaussian random field with Matérn covariance function. Every sample requires the solution of a differential equation with Gaussian white noise forcing, formulated on a bounded computational domain. This introduces unwanted boundary effects since the stochastic partial differential equation is originally posed on the whole $\mathbb{R}^d$, without boundary conditions. We use a window technique, whereby one embeds the computational domain into a larger domain, and postulates convenient boundary conditions on the extended domain. To mitigate the pollution from the artificial boundary it has been suggested in numerical studies to choose a window size that is at least as large as the correlation length of the Matérn field. We provide a rigorous analysis for the error in the covariance introduced by the window technique, for homogeneous Dirichlet, homogeneous Neumann, and periodic boundary conditions. We show that the error decays exponentially in the window size, independently of the type of boundary condition. We conduct numerical experiments in 1D and 2D space, confirming our theoretical results.

Laura Scarabosio

In the framework of uncertainty quantification, we consider a quantity of interest which depends non-smoothly on the high-dimensional parameter representing the uncertainty. We show that, in this situation, the multilevel Monte Carlo algorithm is a valid option to compute moments of the quantity of interest (here we focus on the expectation), as it allows to bypass the precise location of discontinuities in the parameter space. We illustrate how such lack of smoothness occurs for the point evaluation of the solution to a (Helmholtz) transmission problem with uncertain interface, if the point can be crossed by the interface for some realizations. For this case, we provide a space regularity analysis for the solution, in order to state converge results in the L1-norm for the finite element discretization. The latter are then used to determine the optimal distribution of samples among the Monte Carlo levels. Particular emphasis is given on the robustness of our estimates with respect to the dimension of the parameter space.