# Hermann G. Matthies's research while affiliated with Technische Universität Dresden and other places

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## Publications (245)

Very often, in the course of uncertainty quantification tasks or data analysis, one has to deal with high‐dimensional random variables. Here the interest is mainly to compute characterizations like the entropy, the Kullback–Leibler divergence, more general f$$ f $$‐divergences, or other such characteristics based on the probability density. The den...

This paper presents the machine learning-based ensemble conditional mean filter (ML-EnCMF) -- a filtering method based on the conditional mean filter (CMF) previously introduced in the literature. The updated mean of the CMF matches that of the posterior, obtained by applying Bayes' rule on the filter's forecast distribution. Moreover, we show that...

The main goal of this review is to provide a thorough scientific understanding of the interplay between stochastics and mechanics, by classifying what can be achieved by representing mechanical system parameters in terms of deterministic values (homogenization) versus random variables or random fields (stochastic upscaling). The latter is of specia...

Motivation: How to compute entropy, Kullback-Leibler (KL) and other divergences if probability density function (pdf) is not available?
The task considered here was the numerical computation of characterising statistics of
high-dimensional pdfs, as well as their divergences and distances,
where the pdf in the numerical implementation was assume...

Within the realm of isogeometric analysis, isogeometric collocation has been driven by the attempt to minimize the cost of quadrature associated with higher-order discretizations, with the goal of achieving higher-order accuracy at low computational cost. While the first applications of isogeometric collocation have mainly concerned linear problems...

We share some ideas how to compute the Kullback–Leibler divergence, entropy and other f-divergences in high-dimensions in low-rank tensor format. More details are here https://arxiv.org/abs/2111.07164

Spatial symmetries and invariances play an important role in the description of materials. When modelling material properties, it is important to be able to respect such invariances. Here we discuss how to model and generate random ensembles of tensors where one wants to be able to prescribe certain classes of spatial symmetries
and invariances for...

This paper presents the machine learning-based ensemble conditional mean filter (ML-EnCMF) — a filtering method based on the conditional mean filter (CMF) previously introduced in the literature. The updated mean of the CMF matches that of the posterior, obtained by applying Bayes' rule on the filter's forecast distribution. Moreover, we show that...

Very often, in the course of uncertainty quantification tasks or data analysis, one has to deal with high-dimensional random variables (RVs). A high-dimensional RV can be described by its probability density (pdf) and/or by the corresponding probability characteristic functions (pcf), or by a polynomial chaos (PCE) or similar expansion. Here the in...

We show that for the simulation of crack propagation in quasi-brittle, two-dimensional solids, very good results can be obtained with an embedded strong discontinuity quadrilateral finite element that has incompatible modes. Even more importantly, we demonstrate that these results can be obtained without using a crack tracking algorithm. Therefore,...

In this work, the visco-plastic behaviour of solids is formulated based on the Hellinger-Reissner variational principle. The latter provides a choice to obtain an better computation of stress by using discretization of both displacement and stress fields, see [3] and [4]. The formulation is further developed for dynamics problem. For this class of...

Phase-field modeling of fracture has gained popularity within the last decade due to the flexibility of the related computational framework in simulating three-dimensional arbitrarily complicated fracture processes. However, the numerical predictions are greatly affected by the presence of uncertainties in the mechanical properties of the material...

We propose a fast and economic representation of stationary random fields in trigonometric polynomial, utilizing the prowess of fast Fourier transform (FFT) and low‐rank tensor approximation. With the method we are able to generate large random fields with discretization size up to 2²⁰ which are otherwise well beyond the capacity of PCs. We also il...

This work deals with parameter identification problems in which uncertainties are modelled using random sets (RS), i.e. set-valued random variables. Dempster's rule of combination is applied for replacing the role of Bayes' rule to infer the posterior, which is also a RS. The considered framework allows accounting for mixed epistemic-aleatory uncer...

This article presents a novel deep learning‐based ensemble conditional mean filter (DL‐EnCMF) for nonlinear data assimilation. The filter's key component is the approximation of the conditional expectation (CE) using deep neural networks (DNNs). We implement the DL‐EnCMF for tracking the states of the Lorenz‐63 system. Numerical results show that t...

We show that for the simulation of crack propagation in quasi-brittle, two-dimensional solids, very good results can be obtained with an embedded strong discontinuity quadrilateral finite element that has incompatible modes. Even more importantly, we demonstrate that these results can be obtained without using a crack tracking algorithm. Therefore,...

In this paper we deal with a probability-based scale bridging for concrete material when passing the detailed information at the meso-scale (the scale where the aggregate vs. cement microstructure is visible) with a Voronoi-cell based microstructure representation towards the chosen reduced model at the macro-scale (the scale where the concrete is...

In variational phase-field modeling of brittle fracture, the functional to be minimized is not convex, so that the necessary stationarity conditions of the functional may admit multiple solutions. The solution obtained in an actual computation is typically one out of several local minimizers. Evidence of multiple solutions induced by small perturba...

Multi-scale processes governed on each scale by separate principles for evolution or equilibrium are coupled by matching the stored energy and dissipation in line with the Hill-Mandel principle. We are interested in cementitious materials, and consider here the macro- and meso-scale behaviour of such a material. The accurate representations of stor...

Parametric entities appear in many contexts, be it in optimisation, control, modelling of random quantities, or uncertainty quantification. These are all fields where reduced order models (ROMs) have a place to alleviate the computational burden. Assuming that the parametric entity takes values in a linear space, we show how is is associated to a l...

As a typical convex model, the parallelepiped plays an important role in the non-probabilistic uncertainty quantification with simultaneous dependent and independent variables. To overcome the complexity of the conventional geometric design-based method, this paper proposes a more efficient sample-driven procedure to construct the explicit mathemat...

The state of materials and accordingly the properties of structures are changing over the period of use, which may influence the reliability and quality of the structure during its life-time. Therefore identification of the model parameters of the system is a topic which has attracted attention in the content of structural health monitoring. The pa...

The state of materials and accordingly the properties of structures are changing over the period of use, which may influence the reliability and quality of the structure during its life-time. Therefore, identification of the model parameters of the system is a topic which has attracted attention in the content of structural health monitoring. The p...

Stochastic models share many characteristics with generic parametric models. In some ways they can be regarded as a special case. But for stochastic models there is a notion of weak distribution or generalised random variable, and the same arguments can be used to analyse parametric models. Such models in vector spaces are connected to a linear map...

To evaluate the cyclic behavior under different loading conditions using the kinematic and isotropic hardening theory of steel, a Chaboche viscoplastic material model is employed. The parameters of a constitutive model are usually identified by minimization of the distance between model response and experimental data. However, measurement errors an...

The state of materials and accordingly the properties of structures are changing over the period of use, which may influence the reliability and quality of the structure during its life-time. Therefore, identification of the model parameters of the system is a topic which has attracted attention in the content of structural health monitoring. The p...

In many engineering practices with uncertainty, the non-probabilistic methods play an increasing important role. To overcome the limitation of traditional single-uncertainty modeling methods in handling coupled uncertain problems, this paper develops a more general hybrid uncertainty analysis framework. The non-probabilistic hybrid uncertainties ar...

In variational phase-field modeling of brittle fracture, the functional to be minimized is not convex, so that the necessary stationarity conditions of the functional may admit multiple solutions. The solution obtained in an actual computation is typically one out of several local minimizers. Evidence of multiple solutions induced by small perturba...

We study the Cauchy problem in the framework of static linear elasticity and its resolution via the Steklov-Poincaré approach. In the linear Gaussian framework, the straightforward application of Bayes theory leads to formulas allowing to deduce the uncertainty on the identified field from the noise level. We use a truncated Ritz decomposition of t...

Scientific computations or measurements may result in huge volumes of high-dimensional data, for instance 10²⁰ or 100³⁰⁰ elements. Often these can be thought of representing a real-valued function on a high-dimensional domain. In this and also in most other cases the data can be conceptually arranged in the format of a tensor of high degree, and st...

A wide variety of uncertainty propagation methods have been developed to deal with the single uncertainty; however, different kinds of uncertainties may exist simultaneously in many engineering practices. By using random variables and interval variables to quantify the probabilistic and non-probabilistic uncertainties respectively, this paper propo...

In this paper physical multi-scale processes governed by their own principles for evolution or equilibrium on each scale are coupled by matching the stored and dissipated energy, in line with the Hill-Mandel principle. In our view the correct representations of stored and dissipated energy is essential to the representation irreversible material be...

Parametric models in vector spaces are shown to possess an associated linear map, leading directly to reproducing kernel Hilbert spaces and affine/linear representations in terms of tensor products. From this map, analogues of correlation operators can be formed such that the associated linear map factorises the correlation. Its spectral decomposit...

We study the Cauchy problem in the framework of static linear elasticity and
its resolution via the Steklov-Poincaré approach. In the linear Gaussian framework,
the straightforward application of Bayes theory leads to formulas allowing to deduce
the uncertainty on the identified field from the noise level. We use a truncated
Ritz decomposition of t...

Parametric entities appear in many contexts, be it in optimisation, control, modelling of random quantities, or uncertainty quantification. These are all fields where reduced order models (ROMs) have a place to alleviate the computational burden. Assuming that the parametric entity takes values in a linear space, we show how is is associated to a l...

We enhanced the efficiency of Fast Fourier transform (FFT) based Galerkin methods on numerical homogeni-sation problems by exploiting low-rank tensor approximations in canonical, Tucker, and tensor train formats. This leads to a significant reduction in computational complexity and memory requirement. The advantages of the approach are demonstrated...

As a typical method for hybrid uncertainty analysis, imprecise probability theory recently plays an increasing important role in many engineering systems. To extend its application, this paper proposes a new imprecise probability model and a more efficient numerical method for hybrid uncertainty propagation. Instead of crisp real values, fuzzy sets...

Due to the aggressive and changing environmental conditions, various time-varying uncertainties widely exist in many engineering heat transfer problems. This paper introduces a non-probabilistic interval process model to characterize the time-varying uncertainty with limited information, whose lower and upper bounds are quantified as time-dependent...

With the rapid development of reliability analysis theory, the safety assessment for engineering systems with hybrid epistemic uncertainties has received increasing attentions. To overcome the imperfection of traditional single-uncertainty modeling methods, this paper introduces evidence variables and fuzzy variables simultaneously to describe the...

The structural control of concrete gravity dams is of primary importance. In this context, numerical models play a fundamental role both to assess the vulnerability of gravity dams and to control their behaviour during normal operativity and after extreme events. In this regard, data monitoring represents an important source of information for nume...

The state of materials and accordingly the properties of structures are changing over the period of use, which may influence the reliability and quality of the structure during its lifetime. Therefore identification of the model parameters of the system is a topic which has attracted attention in the content of structural health monitoring. The par...

To evaluate the cyclic behavior under different loading conditions using the kinematic and isotropic hardening theory of steel, a Chaboche viscoplastic material model is employed. The parameters of a constitutive model are usually identified by minimization of the difference between model response and experimental data. However, measurement errors...

To evaluate the cyclic behavior under different loading conditions using the kinematic and isotropic hardening theory of steel, a Chaboche viscoplastic material model is employed. The parameters of a constitutive model are usually identified by minimization of the distance between model response and experimental data. However, measurement errors an...

Scientific computations or measurements may result in huge volumes of data. Often these can be thought of representing a real-valued function on a high-dimensional domain, and can be conceptually arranged in the format of a tensor of high degree in some truncated or lossy compressed format. We look at some common post-processing tasks which are not...

Scientific computations or measurements may result in huge volumes of data. Often these can be thought of representing a real-valued function on a high-dimensional domain, and can be conceptually arranged in the format of a tensor of high degree in some truncated or lossy compressed format. We look at some common post-processing tasks which are not...

The state of materials and accordingly the properties of structures are changing over the period of use, which may influence the reliability and quality of the structure during its lifetime. Therefore identification of the model parameters of the system is a topic which has attracted attention in the content of structural health monitoring. The par...

In this contribution, several case studies with data uncertainties are presented which have been performed in individual projects as part of the DFG (German Research Foundation) Priority Programme SPP 1886 “Polymorphic uncertainty modelling for the numerical design of structures.” In all case studies numerical models with uncertainties are derived...

Fast Fourier transform (FFT) based methods have turned out to be an effective computational approach for numerical homogenisation. In particular, Fourier–Galerkin methods are computational methods for partial differential equations that are discretised with trigonometric polynomials. Their computational effectiveness benefits from efficient FFT bas...

Fast Fourier transform (FFT) based methods has turned out to be an effective computational approach for numerical homogenisation. Particularly, Fourier-Galerkin methods are computational methods for partial differential equations that are discretised with trigonometric polynomials. Its computational effectiveness benefits from efficient FFT based a...

In practical engineering, the changing structural temperature distribution has been considered as an important factor influencing the structural vibration property. By treating the uncertainties in material properties, boundary conditions and external loads as interval variables, this paper proposes a dual-stage uncertainty analysis framework to ev...

Uncertainty of random variables is commonly characterized from measurement data. In practice, data might be insufficient in order to obtain an accurate probability model. In this work, we assume that the type of distribution of the considered random variable is known a priori, and use a hierarchical parametric probability box (p‐box) – which is a s...

This paper considers Bayesian identification of macroscopic bone material characteristics given digital image correlation (DIC) data. As the evaluation of the full Bayesian posterior distribution is known to be computationally intense, here we consider the approximate estimation in a Newton‐like manner by using the theory of conditional expectation...

A random set is a generalisation of a random variable, i.e. a set-valued random variable. The random set theory allows a unification of other uncertainty descriptions such as interval variable, mass belief function in Dempster-Shafer theory of evidence, possibility theory, and set of probability distributions. The aim of this work is to develop a n...

Fiber reinforced concrete (FRC) stands for a concrete with an addition of randomly distributed short discrete fibers. By knowing the distribution of fibers in a specimen, its mechanical properties can be derived by representing the micro structure of the material using numerical tools like the finite element method. Elaborated sophisticated model u...

For many uncertainty-based engineering practices, the information or experimental data used to construct the uncertainty analysis model are often deficient, thus rendering traditional probabilistic methods ineffective. In this context, this paper proposes a novel model calibration method that combines non-probabilistic interval technology with Baye...

With the development of modern computing technology, the uncertainty propagation theory plays an increasing important role in the thermal engineering practice. Under this context, our paper proposes an efficient dual-level framework for hybrid uncertainty propagation analysis. Two kinds of epistemic uncertainties are considered simultaneously in in...

In numerical heat transfer, the model validation problem with respect to epistemic uncertainty, where only a small amount of experimental information is available, has been recognized as a challenging issue. To overcome the drawback of traditional probabilistic methods in dealing with limited data, this paper proposes a novel model validation appro...

Uncertainty of random variables is commonly characterized from measurement data.
In practice, data might be insufficient in order to obtain an accurate probability model.
In this work, we assume that the type of distribution of the considered random variable is known a priori, and use a hierarchical parametric probability box (p-box)
-- which is a...

The model validation with respect to epistemic uncertainty, where only a small amount of experimental data is available, is a challenging problem in practical engineering. Interval theory is a useful tool to deal with such epistemic uncertainty, and this paper aims to construct an interval theory-based analysis framework for model validation. Using...

Stochastic models share many characteristics with generic parametric models. In some ways they can be regarded as a special case. But for stochastic models there is a notion of weak distribution or generalised random variable, and the same arguments can be used to analyse parametric models. Such models in vector spaces are connected to a linear map...

With the development of reliability technique, the safety assessment for the problem with epistemic uncertainty has attracted widespread attention. Evidence theory is a useful tool to deal with such uncertainty, and this paper aims to develop an efficient approach for the evidence theory-based reliability analysis and optimization design. By using...

In many instances one has to deal with parametric models. Such models in vector spaces are connected to a linear map. The reproducing kernel Hilbert space and affine- / linear- representations in terms of tensor products are directly related to this linear operator. This linear map leads to a generalised correlation operator, in fact it provides a...

This paper focuses on inverse problems to identify parameters by incorporating information from measurements. These generally ill-posed problems are formulated here in a probabilistic setting based on Bayes' theorem because it leads to a unique solution of the updated distribution of parameters. Many approaches build on Bayesian updating in terms o...