@article{GraessleHinzeLangetal.2019, author = {Gr{\"a}ßle, Carmen and Hinze, Michael and Lang, Jens and Ullmann, Sebastian}, title = {POD model order reduction with space-adapted snapshots for incompressible flows}, volume = {Advances in Computational Mathematics}, number = {45}, doi = {doi:10.1007/s10444-019-09716-7}, pages = {2401 -- 2428}, year = {2019}, abstract = {We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two approaches of deriving stable POD-Galerkin reduced-order models for this context. In the first approach, the pressure term and the continuity equation are eliminated by imposing a weak incompressibility constraint with respect to a pressure reference space. In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the velocity reduced space with supremizers computed on a velocity reference space. For problems with inhomogeneous Dirichlet conditions, we show how suitable lifting functions can be obtained from standard adaptive finite element computations. We provide a numerical comparison of the considered methods for a regularized lid-driven cavity problem.}, language = {en} } @article{UllmannRotkvicLang2016, author = {Ullmann, Sebastian and Rotkvic, Marko and Lang, Jens}, title = {POD-Galerkin reduced-order modeling with adaptive finite element snapshots}, doi = {10.1016/j.jcp.2016.08.018}, year = {2016}, abstract = {We consider model order reduction by proper orthogonal decomposition (POD) for parametrized partial differential equations, where the underlying snapshots are computed with adaptive finite elements. We address computational and theoretical issues arising from the fact that the snapshots are members of different finite element spaces. We propose a method to create a POD-Galerkin model without interpolating the snapshots onto their common finite element mesh. The error of the reduced-order solution is not necessarily Galerkin orthogonal to the reduced space created from space-adapted snapshot. We analyze how this influences the error assessment for POD-Galerkin models of linear elliptic boundary value problems. As a numerical example we consider a two-dimensional convection-diffusion equation with a parametrized convective direction. To illustrate the applicability of our techniques to non-linear time-dependent problems, we present a test case of a two-dimensional viscous Burgers equation with parametrized initial data.}, language = {en} } @article{UllmannMuellerLang2018, author = {Ullmann, Sebastian and M{\"u}ller, Christopher and Lang, Jens}, title = {Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection-Diffusion-Reaction Equations}, volume = {Fluids}, number = {6}, doi = {doi:10.3390/fluids6080263}, pages = {263}, year = {2018}, abstract = {We consider the estimation of parameter-dependent statistics of functional outputs of steady-state convection-diffusion-reaction equations with parametrized random and deterministic inputs in the framework of linear elliptic partial differential equations. For a given value of the deterministic parameter, a stochastic Galerkin finite element (SGFE) method can estimate the statistical moments of interest of a linear output at the cost of solving a single, large, block-structured linear system of equations. We propose a stochastic Galerkin reduced basis (SGRB) method as a means to lower the computational burden when statistical outputs are required for a large number of deterministic parameter queries. Our working assumption is that we have access to the computational resources necessary to set up such a reduced-order model for a spatial-stochastic weak formulation of the parameter-dependent model equations. In this scenario, the complexity of evaluating the SGRB model for a new value of the deterministic parameter only depends on the reduced dimension. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a reduced basis generated by a proper orthogonal decomposition (POD) of snapshots of SGFE solutions at representative values of the parameter. We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds.We test the SGRB method numerically for a convection-diffusion-reaction problem, choosing the convective velocity as a deterministic parameter and the parametrized reactivity or diffusivity field as a random input. Compared to a standard reduced basis model embedded in a Monte Carlo sampling procedure, the SGRB model requires a similar number of reduced basis functions to meet a given tolerance requirement. However, only a single run of the SGRB model suffices to estimate a statistical output for a new deterministic parameter value, while the standard reduced basis model must be solved for each Monte Carlo sample.}, language = {en} }