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.
16:30
Stochastic collocation method for computing eigenspaces of parameter-dependent operators
Mikael Laaksonen | Lappeenranta-Lahti University of Technology | Finland
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Authors:
Mikael Laaksonen | Lappeenranta-Lahti University of Technology | Finland
Harri Hakula | Aalto University | Finland
Luka Grubišić | University of Zagreb | Croatia
Multiparametric eigenvalue problems, i.e., eigenvalue problems of operators that depend on a large number of parameters, arise in a variety of contexts. For instance, one may think of uncertainty quantification of mechanical vibration problems with data uncertainty or optimization of the spectrum of structures that depend on many design parameters. In recent years several numerical methods have been suggested for solving multiparametric eigenvalue problems. In particular, these include stochastic collocation algorithms and stochastic Galerkin based iterative algorithms. By nature, such methods rely on the assumption that the solution is smooth with respect to the input parameters. More precisely, they exhibit optimal rates of convergence only if the eigenpair of interest depends complex-analytically on the vector of parameters.
We consider the eigenvalue problem for an elliptic self-adjoint operator that depends on a countable number of parameters in an affine fashion. We restrict ourselves to so-called isolated eigenspaces, i.e., eigenspaces for which the corresponding set of eigenvalues is separated from the rest of the spectrum for all parameter values. The spectral projection operator associated to such an eigenspace can be shown to be complex-analytic with respect to the input parameters. We construct an analytically smooth basis for the eigenspace of interest and show optimal convergence rates when the basis vectors are approximated using sparse stochastic collocation.
17:00
On adaptation of sparse quadrature and sparse polynomial-based Knothe Rosenblatt maps for high-dimensional Bayesian inversion
Joshua Chen | Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin | United States
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Authors:
Joshua Chen | Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin | United States
Daniele Bigoni | Massachusetts Institute of Technology | United States
Peng Chen | Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin | United States
Youssef Marzouk | Massachusetts Institute of Technology | United States
Transport approaches to the approximation of posterior measures in Bayesian inversion enable the inexpensive construction of deterministic or random quadratures with respect to the posterior distribution. This is obtained at the cost of the solution of a variational optimization problem, which involves the discretization of the prior measure, sometimes leading to a prohibitive number of quadrature points. Random quadrature (Monte Carlo) has an undesirable O(N^-1/2) convergence rate and error proportional to the standard deviation of the cost functional. Sparse quadratures can decrease the number of required points dramatically by exploiting the regularity properties of the likelihood. In adaptive transport map approaches, the approximation space is greedily enlarged, while balancing map complexity, computational cost and the quadrature error. In this talk, we discuss a variational transport approach based on Knothe-Rosenblatt transports parameterized by sparse polynomials. We pursue the adaptation of sparse quadratures to discretize a KL divergence cost functional by utilizing information from the map approximation space, along with the usual greedy quadrature criterion utilized in sparse grid approaches. The method will be showcased on high-dimensional problems arising in spatial statistics and the inversion of PDEs.
17:30
On a fast hierarchical sparse grid quadrature and applications
Abdellah Chkifa | Mohammed VI Polytechnic University | Morocco
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Author:
Abdellah Chkifa | Mohammed VI Polytechnic University | Morocco
The motivation of this work is the computation of quadratures of multivariate functions which arise in parametrized physical and engineering models, the simulation of which depends on d input parameters $y_1,..., y_d$ varying in a parameter domain P. The main challenges is the curse of dimensionality arising for d >> 1. We present a quadrature based on the Smolyak approach using sequences with certain ''binary'' symmetry properties which yield simple and straight-forward computation of such quadratures. In particular, these quadratures are hierarchical with the cost of enriching as low as one additional collocation. We present numerical experiments which compare such quadratures with QMC methods and Chebyshev-Frolov lattice quadratures.