This mini-symposium aims at bringing together people working on kernel and other sampling-based approximation methods for high-dimensional problems, in particular, but not restricted to, quasi-Monte Carlo methods, and sparse grids methods. Kernel methods and the related Gaussian Process surrogate models are a powerful class of numerical methods, and they are often employed in problems arising in uncertainty quantification. Nonetheless, there is much to be explored in their theoretical analysis for UQ applications, which are often formulated as high-dimensional approximation or integration problems.
On the other hand, the theory and applicability of QMC and sparse grid approximation/integration techniques in high or infinite dimensional problems have seen considerable advances in the last years, yet being far from addressing all problems of interest in UQ.
The objective of this mini-symposium is to showcase the late theoretical results and exchange ideas on sampling-based high dimensional integration and approximation methods targeting UQ applications.
08:30
Higher order quasi-Monte Carlo rules for uncertainty quantification using periodic random variables
Vesa Kaarnioja | University of New South Wales | Australia
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Authors:
Vesa Kaarnioja | University of New South Wales | Australia
Frances Kuo | University of New South Wales | Australia
Ian Sloan | University of New South Wales | Australia
Many studies in uncertainty quantification have been carried out under the assumption of an input random field in which a countable number of independent random variables are each uniformly distributed on an interval, with these random variables entering linearly in the input random field (the so-called “affine” model). In this talk, we consider an alternative model of the input random field, where the random variables enter the input field as periodic functions instead. The field can be constructed to have the same mean and covariance function as the affine random field. This setting allows us to construct simple lattice QMC rules that obtain higher order convergence rates, which we apply to elliptic PDEs with random coefficients.
09:00
Analysis of the dynamical low rank equations for random semi-linear parabolic problems
Yoshihito Kazashi | École polytechnique fédérale de Lausanne | Switzerland
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Authors:
Yoshihito Kazashi | École polytechnique fédérale de Lausanne | Switzerland
Fabio Nobile | École polytechnique fédérale de Lausanne | Switzerland
In this joint work with Fabio Nobile, we will discuss a reduced basis method called the Dynamically Low Rank (DLR) approximation, to solve numerically semilinear parabolic partial differential equations with random parameters. The idea of this method is to approximate the solution of the problem as a linear combination of products of dynamical deterministic and stochastic basis functions, both of which evolve over time. The DLR approximation is given as a solution of a semi-discrete, highly nonlinear system of equations. Our interest in this talk is in an existence result: we apply the DLR method to a class of semi-linear random parabolic evolutionary equations, and discuss the existence of the solution of the resulting semi-discrete equation. It turns out that finding a suitable equivalent formulation of the original problem is important. After introducing this formulation, the DLR equation is recast to an abstract Cauchy problem in a suitable linear space, for which existence and uniqueness of the solution are established.
09:30
MDFEM for elliptic PDEs with lognormal and uniform random diffusion coefficients delivering higher-order convergence
Dirk Nuyens | KU Leuven | Belgium
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Authors:
Dirk Nuyens | KU Leuven | Belgium
Dong Nguyen | KU Leuven | Belgium
We make use of the Multivariate Decomposition Method (MDM) to decompose the infinite-dimensional random diffusion coefficient into a number of low-dimensional diffusion coefficients, for which each of the associated PDEs, which make use of these low-dimensional diffusion coefficients, can be treated independently.
These low-dimensional random diffusion problems can then be treated by standard methods, e.g., quasi-Monte Carlo (QMC) sampling of the random field in combination with a finite element method (FEM).