3D-VAR for parameterized partial differential equations: a certified reduced basis approach

Abstract

In this paper, we propose a reduced order approach for 3D variational data assimilation governed by parametrized partial differential equations. In contrast to the classical 3D-VAR formulation that penalizes the measurement error directly, we present a modified formulation that penalizes the experimentallyobservable misfit in the measurement space. Furthermore, we include a model correction term that allows to obtain an improved state estimate. We begin by discussing the influence of the measurement space on the amplification of noise and prove a necessary and sufficient condition for the identification of a "good" measurement space. We then propose a certified reduced basis (RB) method for the estimation of the model correction, the state prediction, the adjoint solution and the observable misfit with respect to the true state for real-time and many-query applications. A posteriori bounds are proposed for the error in each of these approximations. Finally, we introduce different approaches for the generation of the reduced basis spaces and the stability-based selection of measurement functionals. The 3D-VAR method and the associated certified reduced basis approximation are tested in a parameter and state estimation problem for a steady-state thermal conduction problem with unknown parameters and unknown Neumann boundary conditions.

An empirical interpolation method for two‐dimensional vector fields and vector measurements

“…The approximation error does not decrease necessary by increasing the size of the reduced space, and with it the number of interpolation points. The convergence of the method relies on the observability coefficient largely discussed in Reference 12 and directly employed in other sensors selection approaches 13 …”

“…The convergence of the method relies on the observability coefficient largely discussed in Reference 12 and directly employed in other sensors selection approaches. 13 In the following paragraphs, two options for the construction of a vector field interpolation operator are investigated.…”

A sequential sensor selection strategy for hyper-parameterized linear Bayesian inverse problems

“…In the following, we specify the forward model and the Bayesian inverse problem, before analysing it in Sec. 3.…”

“…In [3], we developed and utilized a numerical stability analysis for parameterized 3D-VAR data assimilation over a linear model correction term to find design criteria for stability-based sensor selection. In this contribution, we first re-interpret these results in the hyper-parameterized linear Bayesian inversion setting, and then show their relation to A-optimal experimental design.…”

How biased are our models? – a case study of the alpine region

“…The reduced basis method is able to provide us with this, in contrast to many other surrogate model techniques (Baş and Boyacı, 2007;Bezerra et al, 2008;Frangos et al, 2010;Khuri and Mukhopadhyay, 2010;Miao et al, 2019;Mo et al, 2019;Myers et al, 2016;Navarro et al, 2018). The reduced basis method is widely known in mathematical applications (i.e., Benner et al, 2015;Grepl, 2005;Hesthaven et al, 2016;Aretz-Nellesen et al, 2019;Kärcher et al, 2018;Prud'homme et al, 2002;Quarteroni et al, 2015;Rozza et al, 2007); however, only few geoscientific applications exist (Degen et al, 2020a). Nevertheless, some studies do use comparable approaches (Ghasemi and Gildin, 2016;Gosses et al, 2018;Rizzo et al, 2017;Rousset et al, 2014;Zlotnik et al, 2015).…”