Aretha L. Teckentrup

A. L. Teckentrup, R. Scheichl, M. B. Giles et al.

We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried out recently by Charrier, Scheichl and Teckentrup, to more difficult problems, posed on non--smooth domains and with discontinuities in the coefficient. For this wider class of model problem, we prove convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non--linear functional of the solution. We further improve the performance of the multilevel estimator by introducing level dependent truncations of the Karhunen--Loève expansion of the random coefficient. Numerical results complete the paper.

R. Scheichl, A. M. Stuart, A. L. Teckentrup

We are interested in computing the expectation of a functional of a PDE solution under a Bayesian posterior distribution. Using Bayes' rule, we reduce the problem to estimating the ratio of two related prior expectations. For a model elliptic problem, we provide a full convergence and complexity analysis of the ratio estimator in the case where Monte Carlo, quasi-Monte Carlo or multilevel Monte Carlo methods are used as estimators for the two prior expectations. We show that the computational complexity of the ratio estimator to achieve a given accuracy is the same as the corresponding complexity of the individual estimators for the numerator and the denominator. We {also include numerical simulations, in the context of the model elliptic problem, which demonstrate the effectiveness of the approach.

T. J. Dodwell, C. Ketelsen, R. Scheichl et al.

In this paper we address the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large--scale applications with high dimensional parameter spaces, e.g. in uncertainty quantification in porous media flow. We propose a new multilevel Metropolis-Hastings algorithm, and give an abstract, problem dependent theorem on the cost of the new multilevel estimator based on a set of simple, verifiable assumptions. For a typical model problem in subsurface flow, we then provide a detailed analysis of these assumptions and show significant gains over the standard Metropolis-Hastings estimator. Numerical experiments confirm the analysis and demonstrate the effectiveness of the method with consistent reductions of more than an order of magnitude in the cost of the multilevel estimator over the standard Metropolis-Hastings algorithm for tolerances $\varepsilon < 10^{-2}$.

Max Gunzburger, Aretha L Teckentrup

This paper is concerned with Lagrange interpolation by total degree polynomials in moderate dimensions. In particular, we are interested in characterising the optimal choice of points for the interpolation problem, where we define the optimal interpolation points as those which minimise the Lebesgue constant. We give a novel algorithm for numerically computing the location of the optimal points, which is independent of the shape of the domain and does not require computations with Vandermonde matrices. We perform a numerical study of the growth of the minimal Lebesgue constant with respect to the degree of the polynomials and the dimension, and report the lowest values known as yet of the Lebesgue constant in the unit cube and the unit ball in up to 10 dimensions.

Tianming Bai, Aretha L. Teckentrup, Konstantinos C. Zygalakis

This work is concerned with the use of Gaussian surrogate models for Bayesian inverse problems associated with linear partial differential equations. A particular focus is on the regime where only a small amount of training data is available. In this regime the type of Gaussian prior used is of critical importance with respect to how well the surrogate model will perform in terms of Bayesian inversion. We extend the framework of Raissi et. al. (2017) to construct PDE-informed Gaussian priors that we then use to construct different approximate posteriors. A number of different numerical experiments illustrate the superiority of the PDE-informed Gaussian priors over more traditional priors.

A. L. Teckentrup

We discuss the application of multilevel Monte Carlo methods to elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification in subsurface flow modeling. We give a brief review of recent advances in the numerical analysis of the multilevel algorithm under minimal assumptions on the random coefficient, and extend the analysis to cover also tensor--valued coefficients, as well as point evaluations. Our analysis includes as an example log--normal random coefficients, which are frequently used in applications.

Elliot J. Addy, Aretha L. Teckentrup

Sparse grids are popular tools for high-dimensional function approximation. In this work, we study the use of sparse grids for interpolation using separable Matérn kernels $Φ_{\boldsymbolν,\boldsymbolλ}(\mathbf{x},\mathbf{x}')=\prod_{j=1}^dϕ_{ν_j,λ_j}(x_j,x_j')$, with a particular focus on the anisotropic setting where the regularity $ν_j$ and the lengthscale $λ_j$ vary with dimension $j$. We combine the construction of anisotropic sparse grids, which exploit anisotropic $ν_j$ to improve convergence rates in smooth dimensions, with the construction of lengthscale-informed sparse grids, which diminish the error contribution of less varying dimensions using anisotropic $λ_j$. We provide theory and numerical experiments to showcase the benefits on asymptotic and pre-asymptotic error behaviour of sparse grid kernel interpolation.

Minho Park, Aretha Teckentrup

We consider the application of multilevel Monte Carlo methods to steady state Darcy flow in a random porous medium, described mathematically by elliptic partial differential equations with random coefficients. The levels in the multilevel estimator are defined by finite volume discretisations of the governing equations with different mesh parameters. To simulate different layers in the subsurface, the permeability is modelled as a piecewise constant or piecewise spatially correlated random field, including the possibility of piecewise log-normal random fields. The location of the layers is assumed unknown, and modelled by a random process. We prove new convergence results of the spatial discretisation error required to quantify the mean square error of the multilevel estimator, and provide an optimal implementation of the method based on algebraic multigrid methods and a novel variance reduction technique termed Coarse Grid Variates.