Aretha L. Teckentrup

Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients

We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo method, and demonstrate numerically its superiority. The asymptotic cost of solving the stochastic problem with the multilevel method is always significantly lower than that of the standard method and grows only proportionally to the cost of solving the deterministic problem in certain circumstances. Numerical calculations demonstrating the effectiveness of the method for one- and two-dimensional model problems arising in groundwater flow are presented.

Finite Element Error Analysis of Elliptic PDEs with Random Coefficients and Its Application to Multilevel Monte Carlo Methods

We consider a finite element approximation of elliptic partial differential equations with random coefficients. Such equations arise, for example, in uncertainty quantification in subsurface flow modeling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, have only very limited spatial regularity and lead to variational problems that lack uniform coercivity and boundedness with respect to the random parameter. In our analysis we overcome these challenges by a careful treatment of the model problem almost surely in the random parameter, which then enables us to prove uniform bounds on the finite element error in standard Bochner spaces. These new bounds can then be used to perform a rigorous analysis of the multilevel Monte Carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness. To conclude, we give some numerical results that confirm the new bounds.

Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

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 in Charrier et al. (SIAM J Numer Anal, 2013), 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.

A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow

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

A Multilevel Stochastic Collocation Method for Partial Differential Equations with Random Input Data

\varepsilon < 10^{-2}

Posterior consistency for Gaussian process approximations of Bayesian posterior distributions

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Multilevel Monte Carlo methods for highly heterogeneous media

Stochastic collocation methods for approximating the solution of partial differential equations with random input data (e.g., coefficients and forcing terms) suffer from the curse of dimensionality...

Quasi-Monte Carlo and Multilevel Monte Carlo Methods for Computing Posterior Expectations in Elliptic Inverse Problems

We study the use of Gaussian process emulators to approximate the parameter-to-observation map or the negative log-likelihood in Bayesian inverse problems. We prove error bounds on the Hellinger distance between the true posterior distribution and various approximations based on the Gaussian process emulator. Our analysis includes approximations based on the mean of the predictive process, as well as approximations based on the full Gaussian process emulator. Our results show that the Hellinger distance between the true posterior and its approximations can be bounded by moments of the error in the emulator. Numerical results confirm our theoretical findings.

A multilevel stochastic collocation method for SPDEs

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.

How Deep Are Deep Gaussian Processes

We are interested in computing the expectation of a functional of a PDE solution under a Bayesian posterior distribution. Using Bayess 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.