European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:30   MS29:Low-rank modelling in uncertainty quantification (Part 1)
Chair: Jan Heiland
25 mins
Energy estimates and model order reduction for bilinear systems with Lévy noise
Martin Redmann
Abstract: In this talk, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique to reduce the order of spatially-discretized systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. The reduced system is obtained by removing these states from the original system. In this talk, we present an L²-error bound for the proposed method applied to stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far.
25 mins
Certified Reduced Basis Methods for Variational Data Assimilation
Nicole Aretz, Martin Grepl, Karen Veroy-Grepl
Abstract: In order to approximate the state of a physical system, data from physical measurements can be incorporated into a mathematical model to improve the state prediction. Discrepancies between data and models arise, since on the one hand, measurements are subject to errors and, on the other hand, a model can only approximate the actual physical phenomenon. In this talk, we present a model order reduction method for (an interpretation of) the 3D- and 4D-VAR methods of variational data assimilation for parametrized partial differential equations. The classical 3D- and 4D-VAR methods make informed perturbations in order to find a state closer to the observations while main physical laws described by the model are maintained. For the 3D-VAR method, we take inspiration from recent developments in state and parameter estimation and analyse the influence of the measurement space on the amplification of noise. Here, we prove a necessary and sufficient condition for the identification of a “good” measurement space which can, in turn, be used for a stability-based selection of measurement functionals. For both 3D- and 4D-VAR we propose a certified reduced basis (RB) method for the estimation of the model correction, the state prediction, the adjoint solution, and the observable misfit. Finally, we introduce different approaches for the generation of the RB spaces suited for different applications, and present numerical results testing their performance.
25 mins
Analysis of the dynamical low rank equations for random semi-linear parabolic problems
Yoshihito Kazashi, Fabio Nobile
Abstract: 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. This work is motivated by [1--3].
25 mins
Low-rank tensor train methods for Isogeometric analysis
Alexandra Buenger, Martin Stoll
Abstract: Isogeometric analysis (IgA) is a popular method for the discretization of partial differential equations motivated by the use of NURBS (Non-uniform rational B-splines) for geometric representations in industry and science. In IgA the domain representation as well as the discrete solution of a PDE are described by the same global spline functions. However, the use of an exact geometric representation comes at a cost. Due to the global nature and large overlapping support of the basis functions, system matrix assembly becomes especially costly in IgA. To reduce the computing time and storage requirements low-rank tensor methods have become a promising tool. We successfully constructed a framework appying low rank tensor train calculations to IgA to efficiently solve PDE-constrained optimization problems on complex three dimensional domains without assembly of the actual system matrices. The method exploits the Kronecker product structure of the underlying spline space, reducing the three dimensional system matrices to a low-rank format as the sum of a small number of Kronecker products $ M = \sum_{i=1}^n M_i^{(1)} \otimes M_i^{(2)} \otimes M_i^{(3)}$, where $n$ is determined by the chosen size of the low rank approximation. For assembly of the smaller matrices $M_i^{(d)}$ only univariate integration in the corresponding geometric direction $d$ is performed, thus significantly reducing computation time and storage requirements. The developed method automatically detects the ranks for a given domain and conducts all necessary calculations in a memory efficient low rank tensor train format. We present the applicability of this framework to efficiently solve large scale PDE-constrained optimization problems as well as an extension to statistical inverse problems using the iterative AMEn block solve algorithm which preserves and exploits the low rank format of the system matrices.