# Ralf ZimmermannUniversity of Southern Denmark | SDU · Department of Mathematics and Computer Science

Ralf Zimmermann

Dr. rer. nat.

## About

60

Publications

18,613

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1,255

Citations

Citations since 2017

Introduction

Additional affiliations

August 2016 - present

July 2015 - July 2016

September 2014 - June 2015

## Publications

Publications (60)

In this paper, we propose two methods for multivariate Hermite interpolation of manifold-valued functions. On the one hand, we approach the problem via computing suitable weighted Riemannian barycenters. To satisfy the conditions for Hermite interpolation, the sampled derivative information is converted into a condition on the derivatives of the as...

In this paper, a novel gradient-enhanced (GE) polynomial chaos expansion (PCE) model is proposed for approximating complex computational models. We start from the assumption that the PCE model and its partial derivatives with respect to each of the input parameters are Gaussian processes and assign a Gaussian prior to the unknown coefficients. Then...

Classical model reduction methods disregard the special symplectic structure associated with Hamiltonian systems. A key challenge in projection-based approaches is to construct a sym-plectic basis that captures the essential system information. This necessitates the computation of a so-called proper symplectic decomposition (PSD) of a given sample...

Gradient-enhanced Kriging (GE-Kriging) is a well-established surrogate modelling technique for approximating expensive computational models. However, it tends to get impractical for high-dimensional problems due to the size of the inherent correlation matrix and the associated high-dimensional hyper-parameter tuning problem. To address these issues...

One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus form so-called matrix manifolds. This chapter reviews the numerical treatment of the most impor...

The real symplectic Stiefel manifold is the manifold of symplectic bases of symplectic subspaces of a fixed dimension. It features in a large variety of applications in physics and engineering. In this work, we study this manifold with the goal of providing theory and matrix-based numerical tools fit for basic data processing. Geodesics are fundame...

Solving the so-called geodesic endpoint problem, i.e., finding a geodesic that connects two given points on a manifold, is at the basis of virtually all data processing operations, including averaging, clustering, interpolation and optimization. On the Stiefel manifold of orthonormal frames, this problem is computationally involved. A remedy is to...

Solving the so-called geodesic endpoint problem, i.e., finding a geodesic that connects two given points on a manifold, is at the basis of virtually all data processing operations, including averaging, clustering, interpolation and optimization. On the Stiefel manifold of orthonormal frames, this problem is computationally involved. A remedy is to...

We address the problem of computing Riemannian normal coordinates on the real, compact Stiefel manifold of orthogonal frames. The Riemannian normal coordinates are based on the so-called Riemannian exponential and the Riemannian logarithm maps and enable to transfer almost any computational procedure to the realm of the Stiefel manifold. To compute...

The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this work, we aim to provide a collection of the es...

For a matrix $X\in \mathbb{R}^{n\times p}$, we provide an analytic formula that keeps track of an orthonormal basis for the range of $X$ under rank-one modifications.
More precisely, we consider rank-one adaptations $X_{new} = X+ab^T$ of a given $X$
with known matrix factorization $X = UW$, where $U\in\mathbb{R}^{n\times p}$ is column-orthogonal
an...

The main contribution of this paper is twofold: On the one hand, a general framework for performing Hermite interpolation on Riemannian manifolds is presented. The method is applicable if algorithms for the associated Riemannian exponential and logarithm mappings are available. This includes many of the matrix manifolds that arise in practical Riem...

The main contribution of this paper is twofold: On the one hand, a general framework for performing Hermite interpolation on Riemannian manifolds is presented. The method is applicable, if algorithms for the associated Riemannian exponential and logarithm mappings are available. This includes many of the matrix manifolds that arise in practical Rie...

One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus form so-called matrix manifolds. This work will be featured as a chapter in the upcoming Handbo...

In projection-based model reduction (MOR), orthogonal coordinate systems of comparably low dimension are used to produce ansatz subspaces for the efficient emulation of large-scale numerical simulation models. Constructing such coordinate systems is costly as it requires sample solutions at specific operating conditions of the full system that is t...

Cokriging is a variable-fidelity surrogate modeling technique which emulates a target process based on the spatial correlation of sampled data of different levels of fidelity. In this work, we address two theoretical questions associated with the so-called new Cokriging method for variable-fidelity modeling:
(1) A mandatory requirement for the wel...

Designing a vehicle exterior shape takes place at the intersection between styling and aerodynamics – two disciplines with often competing notions. This calls for an interactive shape design process that delivers aerodynamic responses to design modifications in real-time. Reduced Order Modeling (ROM) is a well-known concept to accelerate aerodynami...

In many scientific applications, including model reduction and image processing,
subspaces are used as ansatz spaces for the low-dimensional approximation and reconstruction of the state vectors of interest. We introduce a procedure for adapting an existing subspace based on information from the least-squares problem that underlies the approximatio...

We make use of the non-intrusive dimensionality reduction method Isomap in order to emulate nonlinear parametric flow problems that are governed by the Reynolds-averaged Navier-Stokes equations. Isomap is a manifold learning approach that provides a low-dimensional embedding space that is approximately isometric to the manifold that is assumed to b...

We derive a numerical algorithm for evaluating
the Riemannian logarithm on the Stiefel manifold
with respect to the canonical metric.
In contrast to the existing optimization-based approach,
we work from a purely matrix-algebraic perspective.
Moreover, we prove that the algorithm converges locally and
exhibits a linear rate of convergence.

Variable-fidelity modeling (VFM), sometimes also termed multi-fidelity modeling, refers to the utilization of two or more data layers of different accuracy in order to construct an inexpensive emulator of a given numerical high-fidelity model. In practical applications, this situation arises when simulators of different accuracy for the same physic...

The advent and development of large-scale high-fidelity computational fluid dynamics (CFD) in aircraft design is requiring, more and more, procedures and techniques aimed at reducing its computational cost in order to afford accurate but fast simulations of, e.g., the aerodynamic loads. The adoption of reduced order modeling techniques in CFD repre...

Model reduction via Galerkin projection fails to provide considerable computational savings if applied to general nonlinear systems. This is because the reduced representation of the state vector appears as an argument to the nonlinear function, whose evaluation remains as costly as for the full model. Masked projection approaches, such as the miss...

A method is proposed for constructing local parametrizations of orthogonal bases and of subspaces by computing trajectories in the Stiefel and the Grassmann manifold, respectively. The trajectories are obtained by exploiting sensitivity information on the singular value decomposition with respect to parametric changes and a Taylor-like local linear...

Kriging is often impaired in terms of costs and accuracy by ill-conditioned covariance matrices of large dimension N. We propose to tackle both of these problem by using a pivoted Cholesky decomposition (PCD) and a rank-k formulation of Kriging. The PCD solves a rank-deficient but consistent system. By reformulating the maximum likelihood training...

A method is proposed for constructing local parametrizations of orthogonal bases and of subspaces by computing trajectories in the Stiefel and the Grassmann manifold, respectively. The trajectories are obtained by exploiting sensitivity information on the singular value decomposition with respect to parametric changes and a Taylor-like local linear...

Preliminary version of a follow-up to the LAA paper 'On the condition number anomaly of Gaussian correlation matrices'

In this article, we propose a strategy for speeding-up the computation of the aerodynamics of industrial high-lift configurations using a residual-based reduced-order model (ROM). The ROM is based on the proper orthogonal decomposition (POD) of a set of solutions to the Navier-Stokes equations governing fluid flow at different parameter values, fro...

In this short note, it is proved that the derivatives of the parametrized univariate Gaussian correlation matrix R_g (Î¸) = (exp(âÎ¸(x_i â x_j )^2_{i,j} â R^{nÃn} are rank-deficient in the limit Î¸ = 0 up to any order m < (n â 1)/2. This result generalizes the rank deficiency theorem for Euclidean distance matrices, which appear as the fir...

In aerodynamic applications, many model reduction methods use proper orthogonal decomposition (POD). In this work, a POD-based method, called missing point estimation (MPE), is modified and applied to steady-state flows with variation of the angle of attack. The main idea of MPE is to select a subset of the computational grid points (control volume...

The most prominent approaches to model reduction share the general principle, that a given large-scale system is projected onto a suitable subspace spanned by a low-dimensional basis. The projection basis essentially determines the approximation quality of the resulting reduced order system. Nonlinear parametric dependencies may be taken into accou...

Spatial correlation matrices appear in a large variety of applications. For example, they are an essential component of spatial Gaussian processes, also known as spatial linear models or Kriging estimators, which are powerful and well-established tools for a multitude of engineering applications such as the design and analysis of computer experimen...

This paper presents a parametric reduced-order model (ROM) based on manifold learning (ML) for use in steady transonic aerodynamic applications. The main objective of this work is to derive an efficient ROM that exploits the low-dimensional nonlinear solution manifold to ensure an improved treatment of the nonlinearities involved in varying the inf...

A novel reduced-order modeling method based on proper orthogonal decomposition for predicting steady, turbulent flows subject to aerodynamic constraints is introduced. Model-order reduction is achieved by replacing the governing equations of computational fluid dynamics with a nonlinear weighted least-squares optimization problem, which aims at fin...

For transonic flows governed by the time-accurate Navier-Stokes equations, small, approximately periodic perturbations can be calculated accurately by transition to the frequency domain and truncating the Fourier expansion after the first harmonic. This is referred to as the linear frequency domain (LFD) method. In this paper, a parametric trajecto...

Variable-fidelity surrogate modeling offers an efficient way to generate aerodynamic data for aero-loads prediction based on a set of CFD methods with varying degree of fidelity and computational expense. In this paper, direct Gradient-Enhanced Kriging (GEK) and a newly developed Generalized Hybrid Bridge Function (GHBF) have been combined in order...

A new method for enhanced surrogate modeling of complex systems by exploiting gradient information is presented. The technique combines the proper orthogonal decomposition (POD) and interpolation methods capable of fitting both sampled input values and sampled derivative information like Kriging (aka spatial Gaussian processes). In contrast to exis...

This paper summarizes recent progress in developing meta-models for efficiently predicting the aerodynamic loads acting on industrial aircraft configurations. We introduce a physics-based approach to reduced-order modeling based on proper orthogonal decompostition of snapshots of the full-order CFD model, and a mathematical approach to variable-fid...

Spatial Gaussian processes, alias spatial linear models or Kriging estimators, are a powerful and well-established tool for the design and analysis of computer experiments in a multitude of engineering applications. A key challenge in constructing spatial Gaussian processes is the training of the predictor by numerically optimizing its associated m...

A reduced-order modelling (ROM) approach for predicting steady, turbulent aerodynamic flows based on computational fluid dynamics (CFD) and proper orthogonal decomposition (POD) is presented. Model-order reduction is achieved by parameter space sampling, solution space
representation via POD and restriction of a CFD solver to the POD subspace. Solv...

An alternative approach for the construction of the cokriging covariance matrix is developed and a more practical cokriging method in the context of surrogate-based analysis and optimization is proposed. The developed cokriging method is validated against an analytical problem and applied to construct global approximation models of the aerodynamic...

Efficiently updating an SVD-based data representation while keeping accurate track of the data mean when new observations are coming in is a common objective in many practical application scenarios. In this talk, two different SVD update algorithms capable of treating an arbitrary number of new observations are introduced following the symmetric EV...

In this paper we propose a methodology for the efficient and robust computation of the aerodynamics of high-lift configurations. A reduced-order modeling approach is considered, based on the proper orthogonal decomposition (POD) of a set of high-fidelity CFD solutions or snapshots from which a reduced basis is evaluated. From this reduced-order mod...

Efficiently updating an SVD-based data representation while keeping accurate track of the data mean when new observations are
coming in is a common objective in many practical application scenarios.
In this paper, two different SVD update algorithms
capable of treating an arbitrary number of new observations are introduced following the symmetric...

Via the proper orthogonal decomposition (POD) solving the full-order
governing equations of Computational Fluid Dynamics (CFD)
is replaced by determining a suitable linear combination
of POD basis vectors. As a consequence, only a low-order vector of POD
coefficients has to be determined in order to obtain an approximate flow solution.
There exi...

Multi-fidelity surrogate modeling refers to the enhanced prediction of the output of a
complex system by incorporating auxiliary fast-to-obtain data of lower fidelity; one such technique being Cokriging. In order to construct Cokriging predictors it is mandatory to estimate
certain co- and cross-variances based on sampled data.
In this paper, a si...

A reduced order modelling approach for predicting steady aerodynamic flows
based on Computational Fluid Dynamics (CFD) and global Proper Orthogonal
Decomposition (POD) using a suitable data transformation for obtaining
problem-adapted global basis modes is presented.
Model-order reduction is achieved by parameter space sampling, reduced solution...

A reduced order modelling approach for predicting steady aerodynamic flows and loads data based on Computational
Fluid Dynamics (CFD) and global Proper Orthogonal Decomposition (POD), that is, POD for multiple
different variables of interest simultaneously, is presented. A suitable data transformation for obtaining problem adapted
global basis mode...

Der Vortrag beginnt hierarchisch mit Übersichtsinformationen zum DLR, zum Standort Braunschweig, zum Institut für Aerodynamic und Strömungstechnik und schließlich zur Abteilung C^2A^2S^2E.
Anschließend wird die Arbeit der Gruppe AeroLoads-Prediction
im Allgemeinen und die dort entwickelte Methoden zur zur Strömungsvorhersage reduzierter Ordnung im...

Cokriging is a statistical interpolation method for the enhanced prediction of a less intensively sampled primary variable of interest with assistance of intensively sampled auxiliary variables. In the geostatistics community it is referred to as two- or multi-variable kriging. In this paper, a new cokriging method is proposed and used for variable...

The covariance structure of spatial Gaussian predictors (aka Kriging predictors) is generally
modeled by parameterized covariance functions; the associated hyperparameters in turn are
estimated via the method of maximum likelihood. In this work, the asymptotic behavior of the
maximum likelihood of spatial Gaussian predictor models as a function of...

Using a method by Traizet (J Differ Geom 60:103–153, 2002), which reduces the construction of minimal surfaces via the Weierstraß
Theorem and the implicit function theorem to solving algebraic equations in several complex variables, we will show the existence
of complete embedded minimal surfaces of finite total curvature with planar ends of least...

Variable-fidelity surrogate modeling offers an efficient way to generate aerodynamic data for aero-loads prediction based on a set of CFD methods with varying degree of fidelity and computational expense. In this paper, new algorithms, such as a Gradient-Enhanced Kriging method (direct GEK) and a generalized hybrid bridge function, have been develo...

## Projects

Projects (2)

Geometrically adapted projection bases for parameterized model reduction

As a rule, the practical execution of any data processing method on a curved manifold M necessitates to work in local coordinates.
This holds among others for basic tasks like averaging, clustering, interpolation and optimization.
Of special importance are the Riemannian normal coordinates, as they are radially isometric, which means that they preserve the Riemannian distance along geodesic rays..
The Riemannian normal coordinates rely on the Riemannian exponential map and the Riemannian logarithm map, which are local diffeomorphisms:
The exponential at a manifold location p sends a tangent vector v (i.e. the velocity vector of a manifold curve) to the endpoint q=c(1) of a geodesic curve c(t) that starts from p=c(0) with velocity v=c'(0).
The Riemannian logarithm at p maps a manifold location q to the starting velocity vector v of a geodesic c(t) that connects p=c(0) and q=c(1).
Hence, the Riemannian logarithm is associated with the
GEODESIC ENDPOINT PROBLEM:
"Given two manifold locations p,q, find a geodesic arc that connects p and q."
The Riemannian exponential and logarithm are important both from a theoretical perspective as well as in practical applications.
Examples range from data analysis and signal processing over computer vision to adaptive model reduction and subspace interpolation, solving differential equations on manifolds and optimization techniques on manifolds.
In this project, we tackle the geodesic endpoint problem on the Stiefel manifold of orthonormal frames.
Our main interest is in finding efficient algorithms that solve the geodesic endpoint problem locally.