## Computational Higher-Order Quasi-Monte Carlo for Random Partial Differential Equations

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Author

Gantner, Robert Nicholas

Date

2017-08Type

- Doctoral Thesis

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yes
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Abstract

We consider partial differential equation (PDE) models which depend on random inputs and can be used to model uncertainties present in various applications in engineering and the sciences. Our focus will be on distributed inputs, for example an uncertain, spatially varying coefficient function, or an uncertain domain on which the PDE is posed.
In the framework of uncertainty quantification (UQ), we are interested in computing expectations, usually the mean and variance, of functionals depending on the solution to such equations. This can be used to computationally predict properties that would otherwise be prohibitively expensive to measure or that may not be accessible at all. In practice, measurements of certain parts of a system may be available, motivating their use in the "calibration" of computations to allow this additional knowledge to improve the estimated quantities. This is considered here in a Bayesian framework, where we assume given some perturbed measurements of the solution, and are interested in computing expectations with respect to a posterior probability measure conditional on the given data.
A central aspect required for the Monte Carlo and quasi-Monte Carlo methods considered here is the parametrization of uncertainties, which yields a formulation as a parametric PDE that may formally depend on an infinite sequence of parameters. Truncation of such infinite-parametric equations yields an equation depending on a possibly large number of parameters, over which the expectation must be computed to solve the UQ problem. Thus, the central mathematical problem to be tackled to compute such expectations is the approximation of high-dimensional integrals resulting from truncation of a countably parametric operator equation.
The main contribution of this project is the implementation of a recently developed, higher-order quasi-Monte Carlo (HOQMC) method that allows efficient approximation of such high-dimensional integrals, under suitable assumptions on the integrand function. Software for the application of these methods has been published and is made available under a free, open source license, and allows evaluation of HOQMC rules on a variety of platforms, notably also state-of-the-art, massively parallel supercomputers.
With this technology at hand, various applications are considered. First of all, such HOQMC methods allow high-precision computations of the error committed by truncation to a finite number of dimensions. Such computations are conducted for various models and set the stage for a proof of a new, higher convergence rate that seems to be tight in the case of affine parametrizations. The main application of HOQMC methods considered here is the quantification of uncertainties for PDEs with random inputs. Various settings fulfilling the conditions required by the theory are considered, and a high degree of consistency is observed between the numerical results and theoretical predictions. The studies presented in the third part of this dissertation show in particular a massive reduction in the number of solutions (and thus the required computational work) of the PDE model required to achieve a given accuracy.
In addition to a higher-order method for the approximation of expectations, we also consider a multilevel approach to uncertainty quantification, which is based on the combination of solutions obtained on different resolutions in order to further reduce the required computational work. This method requires a careful choice of various parameters, which is detailed here along with the specification of these parameters for various relevant cases. Also in this case, a high degree of agreement with the predicted rates is observed, and these rates are in most cases significantly higher than a corresponding single-level approach Show more

Permanent link

https://doi.org/10.3929/ethz-b-000182695Publication status

publishedExternal links

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Contributors

Examiner: Schwab, ChristophExaminer: Arbenz, Peter

Examiner: Dick, Josef

Publisher

ETH ZurichSubject

Uncertainty Quantification; High-Performance Computing; Quasi-Monte CarloOrganisational unit

03435 - Schwab, Christoph / Schwab, Christoph
Funding

149819 - Numerical Analysis of Evolution Equations: Singularities, random inputs and inverse problems (SNF)

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