A Weighted POD Method for Elliptic PDEs with Random Inputs

Abstract

In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L 2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to asses the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and higher dimensional problems.

Stabilized Weighted Reduced Basis Methods for Parametrized Advection Dominated Problems with Random Inputs

“…We can see in figure 7 the comparison between the average errors and the average ∆ N between these algorithms for a test set of size 100, with the same distribution as the training set. The results show that both weighting and a correct sampling are essential to obtain the best convergence results [48,49]. Indeed, putting together these two aspects we get the best results, reaching an error that is one tenth of the error of the classical Greedy algorithm on uniform distribution.…”

Continuous conditional generative adversarial networks for data-driven solutions of poroelasticity with heterogeneous material properties

“…A reduced order model (ROM) that aims to produce a low-dimensional representation of FOM could be an alternative to handling field-scale inverse problems, optimization, or real-time reservoir management (Schilders et al, 2008;Amsallem et al, 2015;Choi et al, 2019;Choi, Boncoraglio, et al, 2020;McBane & Choi, 2021;Yoon, Oostrom, et al, 2009). The ROM methodology primarily relies on the parameterization of a problem (i.e., repeated evaluations of a problem depending on parameters), which could correspond to physical properties, geometric characteristics, or boundary conditions (Ballarin et al, 2019;Venturi et al, 2019;Hesthaven et al, 2016;Hoang et al, 2021;Copeland et al, 2021;Choi, Coombs, & Anderson, 2020;Carlberg et al, 2018). However, it is difficult to parameterize heterogeneous spatial fields of PDE coefficients such as heterogeneous material properties by a few parameters.…”

A weighted POD-reduction approach for parametrized PDE-constrained Optimal Control Problems with random inputs and applications to environmental sciences

“…To accommodate this issue, a Reduced Order Model (ROM) can be used to accelerate the approximation process, by providing a cheaply computable surrogate of the expensive truth approximation for any given parameter value, called a reduced basis approximation. The interested reader may refer to [Hesthaven et al, 2016, Prud'homme et al, 2001, Rozza et al, 2008 for a survey on ROM techniques and to [Bader et al, 2017, Dedè, 2010, Kärcher et al, 2018, Negri et al, 2015, Negri et al, 2013 for their application to parametrized Optimal Control Problems, to [Torlo et al, 2018, Venturi et al, 2019a, Venturi et al, 2019b for their application to UQ and to [Chen et al, 2017] for the application to OCPs in UQ. Given a parametric measurement µ ∈ R n for n ∈ N, in the general formulation of an Optimal Control Problem (OCP) parametrized by µ, one is to minimize a convex functional J(•, •; µ) : Y × U → R over all state-control pairs (y, u) ∈ Y × U that satisfy the governing PDE-state equation e(y, u; µ) = 0.…”

“…The low dimensional subspace should be constructed in some optimal way. In this work, we do this by means of the weighted Proper Orthogonal Decomposition algorithm, already proposed in [Venturi et al, 2019a]. This algorithm is a combination of a singular value decomposition and a quadrature rule.…”