Balanced truncation is a well-established model order reduction method in system theory that has been applied to a variety of problems. Recently, a connection between linear Gaussian Bayesian inference problems and the system theoretic concept of balanced truncation was drawn for the first time. Although this connection is new, the application of balanced truncation to data assimilation is not a novel concept: It has already been used in four-dimensional variational data assimilation (4D-Var) in its discrete formulation. In this paper, the link between system theory  and data assimilation is further strengthened by discussing the application of balanced truncation to standard linear Gaussian Bayesian inference, and, in particular, the 4D-Var method.  similarities between both data assimilation problems allow a discussion of established methods as well as a generalisation of the state-of-the-art approach to arbitrary prior covariances as  reachability Gramians. Furthermore, we propose an enhanced approach using time-limited balanced truncation that allows to balance Bayesian inference for unstable systems and in addition mproves the numerical results for short observation periods.

Neural networks have emerged as powerful and versatile tools in the field of deep learning. As the complexity of the task increases, so do size and architectural complexity of the causing compression techniques to become a focus of current research. Parameter truncation can provide a significant reduction in memory and computational complexity. Originating from a model order reduction framework, the Discrete Empirical Interpolation Method is applied to the gradient descent training of neural networks and analyze for important parameters. The approach for various state-of-the-art neural networks is compared to established truncation methods. Further metrics like L₂ and Cross-Entropy Loss, as well as accuracy and compression rate are reported.

Four-dimensional variational data assimilation (4D-Var) is a data assimilation method often used in weather forecasting. Based on a numerical model and observations of a system, it predicts the system state beyond the last time of measurement. This requires the minimisation of a functional. At each step of the optimisation algorithm, a full nonlinear model evaluation and its adjoint is required. This quickly becomes very costly, especially in high dimensions. For this reason, a surrogate model is needed that approximates the full model well, but requires significantly less computational effort. In this paper, we propose time-limited balanced truncation to build such a reduced-order model. Our approach is able to deal with unstable system matrices. We demonstrate its performance in experiments and compare it with α-bounded balanced truncation, which is an another reduction approach for unstable systems.

We discuss discrete-time dynamical systems depending on a parameter μ. Assuming that the system matrix A(μ) is given, but the parameter μ is unknown, we infer the most-likely parameter μ_m≈μ from an observed trajectory x of the dynamical system. We use parametric eigenpairs (v_i(μ),lambda_i(μ) of the system matrix A(μ) computed with Newton's method based on a Chebyshev expansion. We then represent x in the eigenvector basis defined by the v_i(μ) and compare the decay of the components with predictions based on the lambda_i(μ). The resulting estimates for μ are combined using a kernel density estimator to find the most likely value for μ_m and a corresponding uncertainty quantification.

We present a Reduced Order Model (ROM) which exploits recent developments in Physics Informed Neural Networks (PINNs) for solving inverse problems for the Navier--Stokes equations (NSE). In the proposed approach, the presence of simulated data for the fluid dynamics fields is assumed. A POD-Galerkin ROM is then constructed by applying POD on the snapshots matrices of the fluid fields and performing a Galerkin projection of the NSE (or the modified equations in case of turbulence modeling) onto the POD reduced basis. A POD-Galerkin PINN ROM is then derived by introducing deep neural networks which approximate the reduced outputs with the input being time and/or parameters of the model. The neural networks incorporate the physical equations (the POD-Galerkin reduced equations) into their structure as part of the loss function. Using this approach, the reduced model is able to approximate unknown parameters such as physical constants or the boundary conditions. A demonstration of the applicability of the proposed ROM is illustrated by two cases which are the steady flow around a backward step and the unsteady turbulent flow around a surface mounted cubic obstacle.

We discuss two approaches to solving the parametric (or stochastic) eigenvalue problem A(μ)λ(μ)=A(μ)v(μ). One of them uses a Taylor expansion and the other a Chebyshev expansion. The parametric eigenvalue problem assumes that the matrix A depends on a parameter μ, where μ might be a random variable. Consequently, the eigenvalues and eigenvectors are also functions of μ. We compute a Taylor approximation of these functions about μ₀ by iteratively computing the Taylor coefficients. The complexity of this approach is O(n³) for all eigenpairs, if the derivatives of A(μ) at μ₀ are given.  The Chebyshev expansion works similarly. We first find an initial approximation iteratively which we then refine with Newton's method.  This second method is more expensive but provides a good approximation over the whole interval of the expansion instead around a single point.
 
We present numerical experiments confirming the complexity and demonstrating that the approaches are capable of tracking eigenvalues at intersection points.  Further experiments shed light on the limitations of the Taylor expansion approach with respect to the distance from the expansion point μ₀.

In this paper, we examine conditioning of the discretization of the Helmholtz problem. Although the discrete Helmholtz problem has been studied from different perspectives, to the best of our knowledge, there is no conditioning analysis for it. We aim to fill this gap in the literature. We propose a novel method in 1D to observe the near-zero eigenvalues of a symmetric indefinite matrix. Standard classification of ill-conditioning based on the matrix condition number is not true for the discrete Helmholtz problem. We relate the ill-conditioning of the discretization of the Helmholtz problem with the condition number of the matrix. We carry out analytical conditioning analysis in 1D and extend our observations to 2D with numerical observations. We examine several discretizations. We find different regions in which the condition number of the problem shows different characteristics. We also explain the general behavior of the solutions in these regions.

Der Sonderforschungsbereich SFB1294 „Datenassimilation: Die nahtlose Verschmelzung von Daten und Modellen“ wird seit 2017 von der Deutschen Forschungsgemeinschaft an der Universität of Potsdam gefördert und befindet sich gegenwärtig am Beginn der zweiten Förderphase. Gemeinsam mit den Projektpartnern der Humboldt Universität zu Berlin, der Technischen Universität Berlin, dem Weierstraß-Institut für Angewandte Analysis und Stochastik sowie dem Helmholtz-Zentrum Potsdam: Deutsches GeoForschungsZentrum GFZ, werden innerhalb des Verbundes sowohl die statistischen und mathematischen Grundlagen also auch mannigfaltige Anwendungen zeitabhängiger inverser Problemeuntersucht. In diesem Beitrag werden in kurzen Zügen sowohl die grundlegende mathematische Fragestellung also auch die aktuellen wissenschaftlichen Projekte des SFBs vorgestellt

MOFs and COFs are porous materials with a large variety of applications including gas storage and separation. Synthesised in a modular fashion from distinct building blocks, a near infinite number of structures can be constructed and the properties of the material can be tailored for a specific application. While this modularity is a very attractive feature it also poses a challenge. Attempting to identify the best performing material(s) for a given application is experimentally intractable. Current research efforts combine molecular simulations and machine learning techniques to evaluate the simulated performance of hundreds of thousands of materials to identify top performing MOFs and COFs for a given application. These approaches typically rely on moderated brute-force screening which is still resource-intensive as typically between 70‒100 % of the hundreds of thousands of materials must be simulated to create a training set for the machine learning models used, restricting screening to relatively simple molecules. In this work we demonstrate our novel Bayesian mining approach to materials screening which allows 62‒92 % of the top 100 porous materials for a range of applications to be readily identified from large materials databases after only assessing less than one percent of all materials. This is a stark contrast to the 0‒1 % achieved by conventional brute-force screening where porous materials are just chosen at random during a high throughput screening. Through this accelerated virtual screening process, the identification of high performing materials can be used to more rapidly inform experimental efforts and hence lead to an acceleration of the entire research and development pipeline of porous materials.