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Hanno Gottschalk

Hanno Gottschalk appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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Computer VisionMachine Learningmath.OCmath-phmath.MPmath.NAmath.PRArtificial Intelligencemath.STNumerical AnalysisApplicationsComputational Engineering, Finance, and Scienceeess.IVgr-qcHuman-Computer Interactionphysics.flu-dynStatistics Theory

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Published work

27 published item(s)

preprint2026arXiv

Generative Adversarial Learning from Deterministic Processes

Physical AI is being successfully applied to data which does not follow the traditional paradigm of independent and identically distributed (i.i.d.) samples. In fact, physical AI is often trained on data which is not random at all, and is instead derived from chaotic dynamical systems like turbulence. We aim to explain the empirical success of these methods using the example of generative adversarial networks (GANs), whose statistical learning theory under the i.i.d. assumption is generally well understood. We prove that it is possible, using an infinite-dimensional model of generative adversarial learning (GAL), to learn the invariant distribution of a sufficiently chaotic dynamical system from a single deterministically evolving time series of its states or measurements thereof, and give explicit rates for the convergence to the solution in terms of the Jensen-Shannon divergence.

0 citations0 reviewsNo code linkedOpen access
preprint2023arXiv

A Convenient Infinite Dimensional Framework for Generative Adversarial Learning

In recent years, generative adversarial networks (GANs) have demonstrated impressive experimental results while there are only a few works that foster statistical learning theory for GANs. In this work, we propose an infinite dimensional theoretical framework for generative adversarial learning. We assume that the probability density functions of the underlying measure are uniformly bounded, $k$-times $α$-Hölder differentiable ($C^{k,α}$) and uniformly bounded away from zero. Under these assumptions, we show that the Rosenblatt transformation induces an optimal generator, which is realizable in the hypothesis space of $C^{k,α}$-generators. With a consistent definition of the hypothesis space of discriminators, we further show that the Jensen-Shannon divergence between the distribution induced by the generator from the adversarial learning procedure and the data generating distribution converges to zero. Under certain regularity assumptions on the density of the data generating process, we also provide rates of convergence based on chaining and concentration.

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preprint2023arXiv

Equivariant and Steerable Neural Networks: A review with special emphasis on the symmetric group

Convolutional neural networks revolutionized computer vision and natrual language processing. Their efficiency, as compared to fully connected neural networks, has its origin in the architecture, where convolutions reflect the translation invariance in space and time in pattern or speech recognition tasks. Recently, Cohen and Welling have put this in the broader perspective of invariance under symmetry groups, which leads to the concept of group equivaiant neural networks and more generally steerable neural networks. In this article, we review the architecture of such networks including equivariant layers and filter banks, activation with capsules and group pooling. We apply this formalism to the symmetric group, for which we work out a number of details on representations and capsules that are not found in the literature.

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preprint2022arXiv

Detecting and Learning the Unknown in Semantic Segmentation

Semantic segmentation is a crucial component for perception in automated driving. Deep neural networks (DNNs) are commonly used for this task and they are usually trained on a closed set of object classes appearing in a closed operational domain. However, this is in contrast to the open world assumption in automated driving that DNNs are deployed to. Therefore, DNNs necessarily face data that they have never encountered previously, also known as anomalies, which are extremely safety-critical to properly cope with. In this work, we first give an overview about anomalies from an information-theoretic perspective. Next, we review research in detecting semantically unknown objects in semantic segmentation. We demonstrate that training for high entropy responses on anomalous objects outperforms other recent methods, which is in line with our theoretical findings. Moreover, we examine a method to assess the occurrence frequency of anomalies in order to select anomaly types to include into a model's set of semantic categories. We demonstrate that these anomalies can then be learned in an unsupervised fashion, which is particularly suitable in online applications based on deep learning.

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preprint2022arXiv

Generative Modeling of Turbulence

We present a mathematically well founded approach for the synthetic modeling of turbulent flows using generative adversarial networks (GAN). Based on the analysis of chaotic, deterministic systems in terms of ergodicity, we outline a mathematical proof that GAN can actually learn to sample state snapshots form the invariant measure of the chaotic system. Based on this analysis, we study a hierarchy of chaotic systems starting with the Lorenz attractor and then carry on to the modeling of turbulent flows with GAN. As training data, we use fields of velocity fluctuations obtained from large eddy simulations (LES). Two architectures are investigated in detail: we use a deep, convolutional GAN (DCGAN) to synthesise the turbulent flow around a cylinder. We furthermore simulate the flow around a low pressure turbine stator using the pix2pixHD architecture for a conditional DCGAN being conditioned on the position of a rotating wake in front of the stator. The settings of adversarial training and the effects of using specific GAN architectures are explained. We thereby show that GAN are efficient in simulating turbulence in technically challenging flow problems on the basis of a moderate amount of training data. GAN training and inference times significantly fall short when compared with classical numerical methods, in particular LES, while still providing turbulent flows in high resolution. We furthermore analyse the statistical properties of the synthesized and LES flow fields, which agree excellently. We also show the ability of the conditional GAN to generalize over changes of geometry by generating turbulent flow fields for positions of the wake that are not included in the training data.

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preprint2022arXiv

Gradient-Based Quantification of Epistemic Uncertainty for Deep Object Detectors

The vast majority of uncertainty quantification methods for deep object detectors such as variational inference are based on the network output. Here, we study gradient-based epistemic uncertainty metrics for deep object detectors to obtain reliable confidence estimates. We show that they contain predictive information and that they capture information orthogonal to that of common, output-based uncertainty estimation methods like Monte-Carlo dropout and deep ensembles. To this end, we use meta classification and meta regression to produce confidence estimates using gradient metrics and other baselines for uncertainty quantification which are in principle applicable to any object detection architecture. Specifically, we employ false positive detection and prediction of localization quality to investigate uncertainty content of our metrics and compute the calibration errors of meta classifiers. Moreover, we use them as a post-processing filter mechanism to the object detection pipeline and compare object detection performance. Our results show that gradient-based uncertainty is itself on par with output-based methods across different detectors and datasets. More significantly, combined meta classifiers based on gradient and output-based metrics outperform the standalone models. Based on this result, we conclude that gradient uncertainty adds orthogonal information to output-based methods. This suggests that variational inference may be supplemented by gradient-based uncertainty to obtain improved confidence measures, contributing to down-stream applications of deep object detectors and improving their probabilistic reliability.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Semi-supervised domain adaptation with CycleGAN guided by a downstream task loss

Domain adaptation is of huge interest as labeling is an expensive and error-prone task, especially when labels are needed on pixel-level like in semantic segmentation. Therefore, one would like to be able to train neural networks on synthetic domains, where data is abundant and labels are precise. However, these models often perform poorly on out-of-domain images. To mitigate the shift in the input, image-to-image approaches can be used. Nevertheless, standard image-to-image approaches that bridge the domain of deployment with the synthetic training domain do not focus on the downstream task but only on the visual inspection level. We therefore propose a "task aware" version of a GAN in an image-to-image domain adaptation approach. With the help of a small amount of labeled ground truth data, we guide the image-to-image translation to a more suitable input image for a semantic segmentation network trained on synthetic data (synthetic-domain expert). The main contributions of this work are 1) a modular semi-supervised domain adaptation method for semantic segmentation by training a downstream task aware CycleGAN while refraining from adapting the synthetic semantic segmentation expert 2) the demonstration that the method is applicable to complex domain adaptation tasks and 3) a less biased domain gap analysis by using from scratch networks. We evaluate our method on a classification task as well as on semantic segmentation. Our experiments demonstrate that our method outperforms CycleGAN - a standard image-to-image approach - by 7 percent points in accuracy in a classification task using only 70 (10%) ground truth images. For semantic segmentation we can show an improvement of about 4 to 7 percent points in mean Intersection over union on the Cityscapes evaluation dataset with only 14 ground truth images during training.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Special cosmological models derived from the semiclassical Einstein equation on flat FLRW space-times

This article presents numerical work on a special case of the cosmological semiclassical Einstein equation (SCE). The SCE describes the interaction of relativistic quantum matter by the expected value of the renormalized stress-energy tensor of a quantum field with classical gravity. Here we consider a free massless scalar field with general (not necessarily conformal) coupling to curvature. In a cosmological scenario with flat spatial sections for special choices of the initial conditions, we observe a separation of the dynamics of the quantum degrees of freedom from the dynamics of the scale factor, which extends a classical result by Starobinsky to general coupling. For this new equation of fourth order for the dynamics of the scale factor, we study numerical solutions. Typical solutions show a radiation-like Big Bang for the early universe and de Sitter-like expansion for the late universe. We discuss a specific solution to the cosmological horizon problem that can be produced by tuning parameters in the given equation. Although the model proposed here only contains massless matter, we give a preliminary comparison of the obtained cosmology with the $Λ$CDM standard model of cosmology and investigate parameter ranges in which the new models to a certain extent assimilates standard cosmology.

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preprint2022arXiv

Towards Unsupervised Open World Semantic Segmentation

For the semantic segmentation of images, state-of-the-art deep neural networks (DNNs) achieve high segmentation accuracy if that task is restricted to a closed set of classes. However, as of now DNNs have limited ability to operate in an open world, where they are tasked to identify pixels belonging to unknown objects and eventually to learn novel classes, incrementally. Humans have the capability to say: I don't know what that is, but I've already seen something like that. Therefore, it is desirable to perform such an incremental learning task in an unsupervised fashion. We introduce a method where unknown objects are clustered based on visual similarity. Those clusters are utilized to define new classes and serve as training data for unsupervised incremental learning. More precisely, the connected components of a predicted semantic segmentation are assessed by a segmentation quality estimate. connected components with a low estimated prediction quality are candidates for a subsequent clustering. Additionally, the component-wise quality assessment allows for obtaining predicted segmentation masks for the image regions potentially containing unknown objects. The respective pixels of such masks are pseudo-labeled and afterwards used for re-training the DNN, i.e., without the use of ground truth generated by humans. In our experiments we demonstrate that, without access to ground truth and even with few data, a DNN's class space can be extended by a novel class, achieving considerable segmentation accuracy.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Two Video Data Sets for Tracking and Retrieval of Out of Distribution Objects

In this work we present two video test data sets for the novel computer vision (CV) task of out of distribution tracking (OOD tracking). Here, OOD objects are understood as objects with a semantic class outside the semantic space of an underlying image segmentation algorithm, or an instance within the semantic space which however looks decisively different from the instances contained in the training data. OOD objects occurring on video sequences should be detected on single frames as early as possible and tracked over their time of appearance as long as possible. During the time of appearance, they should be segmented as precisely as possible. We present the SOS data set containing 20 video sequences of street scenes and more than 1000 labeled frames with up to two OOD objects. We furthermore publish the synthetic CARLA-WildLife data set that consists of 26 video sequences containing up to four OOD objects on a single frame. We propose metrics to measure the success of OOD tracking and develop a baseline algorithm that efficiently tracks the OOD objects. As an application that benefits from OOD tracking, we retrieve OOD sequences from unlabeled videos of street scenes containing OOD objects.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Uncertainty Quantification and Resource-Demanding Computer Vision Applications of Deep Learning

Bringing deep neural networks (DNNs) into safety critical applications such as automated driving, medical imaging and finance, requires a thorough treatment of the model's uncertainties. Training deep neural networks is already resource demanding and so is also their uncertainty quantification. In this overview article, we survey methods that we developed to teach DNNs to be uncertain when they encounter new object classes. Additionally, we present training methods to learn from only a few labels with help of uncertainty quantification. Note that this is typically paid with a massive overhead in computation of an order of magnitude and more compared to ordinary network training. Finally, we survey our work on neural architecture search which is also an order of magnitude more resource demanding then ordinary network training.

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preprint2022arXiv

What should AI see? Using the Public's Opinion to Determine the Perception of an AI

Deep neural networks (DNN) have made impressive progress in the interpretation of image data, so that it is conceivable and to some degree realistic to use them in safety critical applications like automated driving. From an ethical standpoint, the AI algorithm should take into account the vulnerability of objects or subjects on the street that ranges from "not at all", e.g. the road itself, to "high vulnerability" of pedestrians. One way to take this into account is to define the cost of confusion of one semantic category with another and use cost-based decision rules for the interpretation of probabilities, which are the output of DNNs. However, it is an open problem how to define the cost structure, who should be in charge to do that, and thereby define what AI-algorithms will actually "see". As one possible answer, we follow a participatory approach and set up an online survey to ask the public to define the cost structure. We present the survey design and the data acquired along with an evaluation that also distinguishes between perspective (car passenger vs. external traffic participant) and gender. Using simulation based $F$-tests, we find highly significant differences between the groups. These differences have consequences on the reliable detection of pedestrians in a safety critical distance to the self-driving car. We discuss the ethical problems that are related to this approach and also discuss the problems emerging from human-machine interaction through the survey from a psychological point of view. Finally, we include comments from industry leaders in the field of AI safety on the applicability of survey based elements in the design of AI functionalities in automated driving.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Improving Video Instance Segmentation by Light-weight Temporal Uncertainty Estimates

Instance segmentation with neural networks is an essential task in environment perception. In many works, it has been observed that neural networks can predict false positive instances with high confidence values and true positives with low ones. Thus, it is important to accurately model the uncertainties of neural networks in order to prevent safety issues and foster interpretability. In applications such as automated driving, the reliability of neural networks is of highest interest. In this paper, we present a time-dynamic approach to model uncertainties of instance segmentation networks and apply this to the detection of false positives as well as the estimation of prediction quality. The availability of image sequences in online applications allows for tracking instances over multiple frames. Based on an instances history of shape and uncertainty information, we construct temporal instance-wise aggregated metrics. The latter are used as input to post-processing models that estimate the prediction quality in terms of instance-wise intersection over union. The proposed method only requires a readily trained neural network (that may operate on single frames) and video sequence input. In our experiments, we further demonstrate the use of the proposed method by replacing the traditional score value from object detection and thereby improving the overall performance of the instance segmentation network.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Coarsening in Algebraic Multigrid using Gaussian Processes

Multigrid methods have proven to be an invaluable tool to efficiently solve large sparse linear systems arising in the discretization of partial differential equations (PDEs). Algebraic multigrid methods and in particular adaptive algebraic multigrid approaches have shown that multigrid efficiency can be obtained without having to resort to properties of the PDE. Yet the required setup of these methods poses a not negligible overhead cost. Methods from machine learning have attracted attention to streamline processes based on statistical models being trained on the available data. Interpreting algebraically smooth error as an instance of a Gaussian process, we develop a new, data driven approach to construct adaptive algebraic multigrid methods. Based on Gaussian a priori distributions, Kriging interpolation minimizes the mean squared error of the a posteriori distribution, given the data on the coarse grid. Going one step further, we exploit the quantification of uncertainty in the Gaussian process model in order to construct efficient variable splittings. Using a semivariogram fit of a suitable covariance model we demonstrate that our approach yields efficient methods using a single algebraically smooth vector.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

GivEn -- Shape Optimization for Gas Turbines in Volatile Energy Networks

This paper describes the project GivEn that develops a novel multicriteria optimization process for gas turbine blades and vanes using modern "adjoint" shape optimization algorithms. Given the many start and shut-down processes of gas power plants in volatile energy grids, besides optimizing gas turbine geometries for efficiency, the durability understood as minimization of the probability of failure is a design objective of increasing importance. We also describe the underlying coupling structure of the multiphysical simulations and use modern, gradient based multicriteria optimization procedures to enhance the exploration of Pareto-optimal solutions.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Time-Dynamic Estimates of the Reliability of Deep Semantic Segmentation Networks

In the semantic segmentation of street scenes with neural networks, the reliability of predictions is of highest interest. The assessment of neural networks by means of uncertainties is a common ansatz to prevent safety issues. As in applications like automated driving, video streams of images are available, we present a time-dynamic approach to investigating uncertainties and assessing the prediction quality of neural networks. We track segments over time and gather aggregated metrics per segment, thus obtaining time series of metrics from which we assess prediction quality. This is done by either classifying between intersection over union equal to 0 and greater than 0 or predicting the intersection over union directly. We study different models for these two tasks and analyze the influence of the time series length on the predictive power of our metrics.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Tracing locally Pareto optimal points by numerical integration

We suggest a novel approach for the efficient and reliable approximation of the Pareto front of sufficiently smooth unconstrained bi-criteria optimization problems. Optimality conditions formulated for weighted sum scalarizations of the problem yield a description of (parts of) the Pareto front as a parametric curve, parameterized by the scalarization parameter (i.e., the weight in the weighted sum scalarization). Its sensitivity w.r.t. parameter variations can be described by an ordinary differential equation (ODE). Starting from an arbitrary initial Pareto optimal solution, the Pareto front can then be traced by numerical integration. We provide an error analysis based on Lipschitz properties and suggest an explicit Runge-Kutta method for the numerical solution of the ODE. The method is validated on bi-criteria convex quadratic programming problems for which the exact solution is explicitly known, and numerically tested on complex bi-criteria shape optimization problems involving finite element discretizations of the state equation.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Optimal Reliability for Components under Thermomechanical Cyclic Loading

We consider the existence of optimal shapes in the context of the thermomechanical system of partial differential equations (PDE) using the recent approach based on elliptic regularity theory. We give an extended and improved definition of the set of admissible shapes based on a class of sufficiently differentiable deformation maps applied to a baseline shape. The obtained set of admissible shapes again allows one to prove a uniform Schauder estimate for the elasticity PDE. In order to deal with thermal stress, a related uniform Schauder estimate is also given for the heat equation. Special emphasis is put on Robin boundary conditions, which are motivated from convective heat transfer. It is shown that these thermal Schauder estimates can serve as an input to the Schauder estimates for the elasticity equation. This is needed to prove the compactness of the (suitably extended) solutions of the entire PDE system in some state space that carries a c2-Hölder topology for the temperature field and a C3-Hölder topology for the displacement. From this one obtains he property of graph compactness, which is the essential tool in an proof of the existence of optimal shapes. Due to the topologies employed, the method works for objective functionals that depend on the displacement and its derivatives up to third order and on the temperature field and its derivatives up to second order. This general result in shape optimization is then applied to the problem of optimal reliability, i.e. the problem of finding shapes that have minimal failure probability under cyclic thermomechanical loading.

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preprint2016arXiv

Quantum fields obtained from convoluted generalized white noise never have positive metric

It is proven that the relativistic quantum fields obtained from analytic continuation of convoluted generalized (Lévy type) noise fields have positive metric, if and only if the noise is Gaussian. This follows as an easy observation from a criterion by K. Baumann, based on the Dell'Antonio-Robinson-Greenberg theorem, for a relativistic quantum field in positive metric to be a free field.

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preprint2015arXiv

Calibration of Lévy Processes using Optimal Control of Kolmogorov Equations with Periodic Boundary Conditions

We present an optimal control approach to the problem of model calibration for Lévy processes based on a non parametric estimation procedure. The calibration problem is of considerable interest in mathematical finance and beyond. Calibration of Lévy processes is particularly challenging as the jump distribution is given by an arbitrary Lévy measure, which form a infinite dimensional space. In this work, we follow an approach which is related to the maximum likelihood theory of sieves. The sampling of the Lévy process is modelled as independent observations of the stochastic process at some terminal time $T$. We use a generic spline discretization of the Lévy jump measure and select an adequate size of the spline basis using the Akaike Information Criterion (AIC). The numerical solution of the Lévy calibration problem requires efficient optimization of the log likelihood functional in high dimensional parameter spaces. We provide this by the optimal control of Kolmogorov's forward equation for the probability density function (Fokker-Planck equation). The first order optimality conditions are derived based on the Lagrange multiplier technique in a functional space. The resulting Partial Integral-Differential Equations (PIDE) are discretized, numerically solved and controlled using scheme a composed of Chang-Cooper, BDF2 and direct quadrature methods. For the numerical solver of the Kolmogorov's forward equation we prove conditions for non-negativity and stability in the $L^1$ norm of the discrete solution. To set boundary conditions, we argue that any Lévy process on the real line can be projected to a torus, where it again is a Lévy process. If the torus is sufficiently large, the loss of information is negligible.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Minimal failure probability for ceramic design via shape control

We consider the probability of failure for components made of brittle mate- rials under one time application of a load as introduced by Weibull and Batdorf - Crosse and more recently studied by NASA and the STAU cooperation as an objective functional in shape optimization and prove the existence of optimal shapes in the class of shapes with a uniform cone property. The corresponding integrand of the objective functional has convexity properties that allow to derive lower-semicontinuity according to Fujii (Opt. Th. Appl. 1988). These properties require less restrictive regularity assumptions for the boundaries and state functions compared to [arXiv:1210.4954]. Thereby, the weak formulation of linear elasticity can be kept for the abstract setting for shape optimization as presented in the book by Haslinger and Maekinen.

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preprint2013arXiv

A Probabilistic Model for LCF

Fatigue life of components or test specimens often exhibit a significant scatter. Furthermore, size effects have a non-negligible influence on fatigue life of parts with different geometries. We present a new probabilistic model for low-cycle fatigue (LCF) in the context of polycrystalline metal. The model takes size effects and inhomogeneous strain fields into account by means of the Poisson point process (PPP). This approach is based on the assumption of independently occurring LCF cracks and the Coffin-Manson-Basquin (CMB) equation. Within the probabilistic model, we give a new and more physical interpretation of the CMB parameters which are in the original approach no material parameters in a strict sense, as they depend on the specimen geometry. Calibration and validation of the proposed model is performed using results of strain controlled LCF tests of specimens with different surface areas. The test specimens are made of the nickel base superalloy RENE 80.

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preprint2013arXiv

A triviality result in the AdS/CFT correspondence for Euclidean quantum fields with exponential interaction

We consider scalar quantum fields with exponential interaction on Euclidean hyperbolic space $\mathbb{H}^2$ in two dimensions. Using decoupling inequalities for Neumann boundary conditions on a tessellation of $\mathbb{H}^2$, we are able to show that the infra-red limit for the generating functional of the conformal boundary field becomes trivial.

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preprint2013arXiv

Optimal Reliability in Design for Fatigue Life

The failure of a component often is the result of a degradation process that originates with the formation of a crack. Fatigue describes the crack formation in the material under cyclic loading. Activation and deactivation operations of technical units are important examples in engineering where fatigue and especially low-cycle fatigue (LCF) play an essential role. A significant scatter in fatigue life for many materials results in the necessity of advanced probabilistic models for fatigue. Moreover, optimization of reliability is of vital interest in engineering, where with respect to fatigue the cost functionals are motivated by the predicted probability for the integrity of the component after a certain number of load cycles. The natural mathematical language to model failure, here understood as crack initiation, is the language of spatio-temporal point processes and their first failure times. This translates the problem of optimal reliability in the framework of shape optimization. The cost functionals derived in this way for realistic optimal reliability problems are too singular to be $H^1$-lower semi-continuous as many damage mechanisms, like LCF, lead to crack initiation as a function of the stress at the component's surface. Realistic crack formation models therefore impose a new challenge to the theory of shape optimization. In this work, we have to modify the existence proof of optimal shapes, for the case of sufficiently smooth shapes using elliptic regularity, uniform Schauder estimates and compactness of strong solutions via the Arzela-Ascoli theorem. This result applies to a variety of crack initiation models and in particular applies to a recent probabilistic model for LCF.

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preprint2013arXiv

Probabilistic Analysis of LCF Crack Initiation Life of a Turbine Blade under Thermomechanical Loading

An accurate assessment for fatigue damage as a function of activation and deactivation cycles is vital for the design of many engineering parts. In this paper we extend the probabilistic and local approach to this problem proposed in [1,2] and [3] to the case of non-constant temperature fields and thermomechanical loading. The method has been implemented as a finite element postprocessor and applied to an example case of a gas-turbine blade which is made of a conventionally cast nickel base superalloy.

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preprint2013arXiv

Risk estimation for LCF crack initiation

An accurate risk assessment for fatigue damage is of vital importance for the design and service of today's turbomachinery components. We present an approach for quantifying the probability of crack initiation due to surface driven low-cycle fatigue (LCF). This approach is based on the theory of failure-time processes and takes inhomogeneous stress fields and size effects into account. The method has been implemented as a finite-element postprocessor which uses quadrature formulae of higher order. Results of applying this new approach to an example case of a gas-turbine compressor disk are discussed.

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preprint2011arXiv

Dynamical Backreaction in Robertson-Walker Spacetime

The treatment of a quantized field in a curved spacetime requires the study of backreaction of the field on the spacetime via the semiclassical Einstein equation. We consider a free scalar field in spatially flat Robertson-Walker space time. We require the state of the field to allow for a renormalized semiclassical stress tensor. We calculate the sigularities of the stress tensor restricted to equal times in agreement with the usual renormalization prescription for Hadamard states to perform an explicit renormalization. The dynamical system for the Robertson Walker scale parameter $a(t)$ coupled to the scalar field is finally derived for the case of conformal and also general coupling.

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