@misc{WelkBreussSridhar, author = {Welk, Martin and Breuß, Michael and Sridhar, Vivek}, title = {Matrix Morphology with Extremum Principle}, series = {Mathematical Morphology and Its Applications to Signal and Image Processing : 14th International Symposium, ISMM 2019, Saarbr{\"u}cken, Germany, July 8-10, 2019, Proceedings}, journal = {Mathematical Morphology and Its Applications to Signal and Image Processing : 14th International Symposium, ISMM 2019, Saarbr{\"u}cken, Germany, July 8-10, 2019, Proceedings}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-20866-0}, doi = {10.1007/978-3-030-20867-7_14}, pages = {177 -- 188}, abstract = {The fundamental operations of mathematical morphology are dilation and erosion. In previous works, these operations have been generalised in a discrete setting to work with fields of symmetric matrices, and also corresponding methods based on partial differential equations have been constructed. However, the existing methods for dilation and erosion in the matrix-valued setting are not overall satisfying. By construction they may violate a discrete extremum principle, which means that results may leave the convex hull of the matrices that participate in the computation. This may not be desirable from the theoretical point of view, as the corresponding property is fundamental for discrete and continuous-scale formulations of dilation and erosion in the scalar setting. Moreover, if such a principle could be established in the matrix-valued framework, this would help to make computed solutions more interpretable. In our paper we address this issue. We show how to construct a method for matrix-valued morphological dilation and erosion that satisfies a discrete extremum principle. We validate the construction by showing experimental results on synthetic data as well as colour images, as the latter can be cast as fields of symmetric matrices.}, language = {en} } @misc{WelkBreuss, author = {Welk, Martin and Breuß, Michael}, title = {The convex-hull-stripping median approximates affine curvature motion}, series = {Scale Space and Variational Methods in Computer Vision : 7th International Conference, SSVM 2019, Hofgeismar, Germany, June 30 - July 4, 2019, Proceedings}, journal = {Scale Space and Variational Methods in Computer Vision : 7th International Conference, SSVM 2019, Hofgeismar, Germany, June 30 - July 4, 2019, Proceedings}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-22367-0}, doi = {10.1007/978-3-030-22368-7_16}, pages = {199 -- 210}, abstract = {The median filter is one of the fundamental filters in image processing. Its standard realisation relies on a rank ordering of given data which is easy to perform if the given data are scalar values. However, the generalisation of the median filter to multivariate data is a delicate issue. One of the methods of potential interest for computing a multivariate median is the convex-hull-stripping median from the statistics literature. Its definition is of purely algorithmical nature, and it offers the advantageous property of affine equivariance. While it is a classic result that the standard median filter approximates mean curvature motion, no corresponding assertion has been established up to now for the convex-hull-stripping median. The aim of our paper is to close this gap in the literature. In order to provide a theoretical foundation for the convex-hull-stripping median of multivariate images, we investigate its continuous-scale limit. It turns out that the resulting evolution is described by the well-known partial differential equation of affine curvature motion. Thus we have established in this paper a relation between two important models from image processing and statistics. We also present some experiments that support our theoretical findings.}, language = {en} } @misc{EltaherBonhageRaabetal., author = {Eltaher, Mahmoud and Bonhage, Alexander and Raab, Thomas and Breuß, Michael}, title = {A modified Mask R-CNN approach for automated detection of archaeological sites on high resolution LiDAR-derived DEMs}, series = {Anthropogenetische Geomorphologie - Geomorophologie im Anthropoz{\"a}n : Virtuelle Jahrestagung des Arbeitskreises f{\"u}r Geomorphologie 2020, 28./29. September 2020, BTU Cottbus - Senftenberg}, journal = {Anthropogenetische Geomorphologie - Geomorophologie im Anthropoz{\"a}n : Virtuelle Jahrestagung des Arbeitskreises f{\"u}r Geomorphologie 2020, 28./29. September 2020, BTU Cottbus - Senftenberg}, publisher = {Cottbus}, issn = {2196-4122}, doi = {10.26127/BTUOpen-5363}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-53633}, pages = {50}, language = {en} } @misc{BreussHoeltgenRadow, author = {Breuß, Michael and Hoeltgen, Laurent and Radow, Georg}, title = {Towards PDE-Based Video Compression with Optimal Masks Prolongated by Optic Flow}, series = {Journal of Mathematical Imaging and Vision}, volume = {63}, journal = {Journal of Mathematical Imaging and Vision}, number = {2}, issn = {1573-7683}, doi = {10.1007/s10851-020-00973-6}, pages = {144 -- 156}, abstract = {Lossy image compression methods based on partial differential equations have received much attention in recent years. They may yield high-quality results but rely on the computationally expensive task of finding an optimal selection of data. For the possible extension to video compression, this data selection is a crucial issue. In this context, one could either analyse the video sequence as a whole or perform a frame-by-frame optimisation strategy. Both approaches are prohibitive in terms of memory and run time. In this work, we propose to restrict the expensive computation of optimal data to a single frame and to approximate the optimal reconstruction data for the remaining frames by prolongating it by means of an optic flow field. In this way, we achieve a notable decrease in the computational complexity. As a proof-of-concept, we evaluate the proposed approach for multiple sequences with different characteristics. In doing this, we discuss in detail the influence of possible computational setups. We show that the approach preserves a reasonable quality in the reconstruction and is very robust against errors in the flow field.}, language = {en} } @misc{BreussKleefeld, author = {Breuß, Michael and Kleefeld, Andreas}, title = {Implicit Monotone Difference Methods for Scalar Conservation Laws with Source Terms}, series = {Acta Mathematica Vietnamica}, volume = {45}, journal = {Acta Mathematica Vietnamica}, number = {3}, issn = {2315-4144}, doi = {10.1007/s40306-019-00354-1}, pages = {709 -- 738}, abstract = {In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author (Breuß SIAM J. Numer. Anal. 43(3), 970-986 2005). Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests.}, language = {en} } @misc{BreussBuhlMansouriYarahmadietal., author = {Breuß, Michael and Buhl, Johannes and Mansouri Yarahmadi, Ashkan and Bambach, Markus and Peter, Pascal}, title = {A Simple Approach to Stiffness Enhancement of a Printable Shape by Hamilton-Jacobi Skeletonization}, series = {Procedia Manufacturing}, volume = {Vol. 47}, journal = {Procedia Manufacturing}, issn = {2351-9789}, doi = {10.1016/j.promfg.2020.04.147}, pages = {1190 -- 1196}, abstract = {The 3D-Printing technology is ready to produce parts with specific properties like individual stiffness. Based on a predefined outer shape, the inner structure of a printed part defines mainly the mechanical features. By Hamilton-Jacobi skeletonization, a stiffness enhancement of a printable shape can be achieved in a way, that a novel AM-corner includes linear axis function. Originating in the field of shape analysis in computer vision and graphics, the so-called medial axis transform (MAT) is designed for the computation of a structure that resembles the bone structure of biological shapes. The input for MAT computation is typically a shape's boundary. The arising topological skeletons have proven to provide a useful concept for many applications; however, their computation is generally intricate and also known to rely on many parameters, diminishing the accessibility of skeletonization methods. In this work, the classical Hamilton-Jacobi skeletonization approach is adopted to compute a stability enhancing shape structure. As the basic method has not been designed for the context of additive manufacturing, a set of suitable modifications are introduced to design an algorithm that suits our intended purpose. Unlike the traditional skeletonization schemes, the resulting method appears to be robust and in practice almost completely automated as we can identify useful generic parameter settings. By a finite element method (FEM) study, the elastic stress properties of the AM-corner with linear axis function is validated and printed with in metal (1.4404) with the 3D Selected Laser Melting (SLM) system AconityMIDI. The AM-knot with skeletonization guides approximately 20 times better than a standard knot. While the first obtained results are shown as 2.5 dimensional shapes, it is emphasized that the proposed algorithm offers many possibilities for extensions to three dimensions and variations in context of additive manufacturing.}, language = {en} } @misc{SchneidereitBreuss, author = {Schneidereit, Toni and Breuß, Michael}, title = {Solving Ordinary Differential Equations using Artificial Neural Networks - A study on the solution variance}, series = {Proceedings of the Conference Algoritmy 2020}, journal = {Proceedings of the Conference Algoritmy 2020}, publisher = {Open Journal Systems , Public Knowledge Project}, pages = {21 -- 30}, abstract = {Solving differential equations can be realised with simple artificial neural network architectures. Several methods make use of trial solutions with different construction approaches and can provide reliable results. However, many parameters, different optimisation methods and random weight initialisation result in a non constant variance to the exact solution. To our knowledge, this variance has not been studied yet. We investigate several parameters and constant versus random weight initialisation for two solution methods to determine their reliability with the use of backpropagation and ADAM optimisation.}, language = {en} } @misc{BaehrBuhlRadowetal., author = {B{\"a}hr, Martin and Buhl, Johannes and Radow, Georg and Schmidt, Johannes and Bambach, Markus and Breuß, Michael and F{\"u}genschuh, Armin}, title = {Stable honeycomb structures and temperature based trajectory optimization for wire-arc additive manufacturing}, series = {Optimization and Engineering}, volume = {22}, journal = {Optimization and Engineering}, number = {2}, issn = {1573-2924}, doi = {10.1007/s11081-020-09552-5}, pages = {913 -- 974}, abstract = {We consider two mathematical problems that are connected and occur in the layer-wise production process of a workpiece using wire-arc additive manufacturing. As the first task, we consider the automatic construction of a honeycomb structure, given the boundary of a shape of interest. In doing this, we employ Lloyd's algorithm in two different realizations. For computing the incorporated Voronoi tesselation we consider the use of a Delaunay triangulation or alternatively, the eikonal equation. We compare and modify these approaches with the aim of combining their respective advantages. Then in the second task, to find an optimal tool path guaranteeing minimal production time and high quality of the workpiece, a mixed-integer linear programming problem is derived. The model takes thermal conduction and radiation during the process into account and aims to minimize temperature gradients inside the material. Its solvability for standard mixed-integer solvers is demonstrated on several test-instances. The results are compared with manufactured workpieces.}, language = {en} } @incollection{BreussMansouriYarahmadi, author = {Breuß, Michael and Mansouri Yarahmadi, Ashkan}, title = {Perspective Shape from Shading : An Exposition on Recent Works with New Experiments}, series = {Advances in Photometric 3D-Reconstruction}, booktitle = {Advances in Photometric 3D-Reconstruction}, editor = {Durou, Jean-Denis and Falcone, Maurizio and Qu{\´e}au, Yvain and Tozza, Silvia}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-51865-3}, issn = {2191-6586}, doi = {https://doi.org/10.1007/978-3-030-51866-0_2}, pages = {31 -- 72}, abstract = {Shape from Shading (SFS) is a fundamental task in computer vision. By given information about the reflectance of an object's surface and the position of the light source, the SFS problem is to reconstruct the 3D depth of the object from a single grayscale 2D input image. A modern class of SFS models relies on the property that the camera performs a perspective projection. The corresponding perspective SFS methods have been the subject of many investigations within the last years. The goal of this chapter is to give an overview of these developments. In our discussion, we focus on important model aspects, and we investigate some prominent algorithms appearing in the literature in more detail than it was done in previous works.}, language = {en} } @misc{BonhageEltaherRaabetal., author = {Bonhage, Alexander and Eltaher, Mahmoud and Raab, Thomas and Breuß, Michael and Raab, Alexandra and Schneider, Anna}, title = {A modified Mask region-based convolutional neural network approach for the automated detection of archaeological sites on high-resolution light detection and ranging-derived digital elevation models in the North German Lowland}, series = {Archaeological Prospection}, volume = {28}, journal = {Archaeological Prospection}, number = {2}, issn = {1099-0763}, doi = {10.1002/arp.1806}, pages = {177 -- 186}, language = {en} } @misc{KoehlerBreuss, author = {K{\"o}hler, Alexander and Breuß, Michael}, title = {Towards Efficient Time Stepping for Numerical Shape Correspondence}, series = {Scale Space and Variational Methods in Computer Vision : 8th International Conference, SSVM 2021, Virtual Event, May 16-20, 2021, Proceedings}, journal = {Scale Space and Variational Methods in Computer Vision : 8th International Conference, SSVM 2021, Virtual Event, May 16-20, 2021, Proceedings}, editor = {Elmoataz, Abderrahim and Fadili, Jalal and Qu{\´e}au, Yvain and Rabin, Julien and Simon, Lo{\"i}c}, publisher = {Springer International Publishing}, isbn = {978-3-030-75548-5}, issn = {1611-3349}, doi = {10.1007/978-3-030-75549-2_14}, pages = {165 -- 176}, abstract = {The computation of correspondences between shapes is a principal task in shape analysis. To this end, methods based on partial differential equations (PDEs) have been established, encompassing e.g. the classic heat kernel signature as well as numerical solution schemes for geometric PDEs. In this work we focus on the latter approach. We consider here several time stepping schemes. The goal of this investigation is to assess, if one may identify a useful property of methods for time integration for the shape analysis context. Thereby we investigate the dependence on time step size, since the class of implicit schemes that are useful candidates in this context should ideally yield an invariant behaviour with respect to this parameter. To this end we study integration of heat and wave equation on a manifold. In order to facilitate this study, we propose an efficient, unified model order reduction framework for these models. We show that specific l0 stable schemes are favourable for numerical shape analysis. We give an experimental evaluation of the methods at hand of classical TOSCA data sets.}, language = {en} } @misc{SridharBreussKahra, author = {Sridhar, Vivek and Breuß, Michael and Kahra, Marvin}, title = {Fast Approximation of Color Morphology}, series = {Advances in Visual Computing : 16th International Symposium, ISVC 2021, Virtual Event, October 4-6, 2021, Proceedings, Part II}, journal = {Advances in Visual Computing : 16th International Symposium, ISVC 2021, Virtual Event, October 4-6, 2021, Proceedings, Part II}, editor = {Bebis, George}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-90435-7}, doi = {10.1007/978-3-030-90436-4_39}, pages = {488 -- 499}, abstract = {The basic filters in mathematical morphology are dilation and erosion. They are defined by a flat or non-flat structuring element that is usually shifted pixel-wise over an image and a comparison process that takes place within the corresponding mask. The algorithmic complexity of fast algorithms that realise dilation and erosion for color images usually depends on size and shape of the structuring element. In this paper we propose and investigate an easy and fast way to make use of the fast Fourier transform for an approximate computation of dilation and erosion for color images. Similarly in construction as many other fast algorithms, the method extends a recent scheme proposed for single-channel filtering. It is by design highly flexible, as it can be used with flat and non-flat structuring elements of any size and shape. Moreover, its complexity only depends on the number of pixels in the filtered images. We analyse here some important aspects of the approximation, and we show experimentally that we obtain results of very reasonable quality while the method has very attractive computational properties.}, language = {en} } @book{BaehrBuhlRadowetal., author = {B{\"a}hr, Martin and Buhl, Johannes and Radow, Georg and Schmidt, Johannes and Bambach, Markus and Breuß, Michael and F{\"u}genschuh, Armin}, title = {Stable honeycomb structures and temperature based trajectory optimization for wire-arc additive manufacturing}, address = {Cottbus}, doi = {10.26127/btuopen-5079}, pages = {49}, language = {en} } @misc{SchneidereitBreuss, author = {Schneidereit, Toni and Breuß, Michael}, title = {Polynomial Neural Forms Using Feedforward Neural Networks for Solving Differential Equations}, series = {Artificial Intelligence and Soft Computing}, journal = {Artificial Intelligence and Soft Computing}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-87985-3}, issn = {978-3-030-87986-0}, doi = {10.1007/978-3-030-87986-0_21}, pages = {236 -- 245}, abstract = {Several neural network approaches for solving differential equations employ trial solutions with a feedforward neural network. There are different means to incorporate the trial solution in the construction, for instance one may include them directly in the cost function. Used within the corresponding neural network, the trial solutions define the so-called neural form. Such neural forms represent general, flexible tools by which one may solve various differential equations. In this article we consider time-dependent initial value problems, which requires to set up the trial solution framework adequately. The neural forms presented up to now in the literature for such a setting can be considered as first order polynomials. In this work we propose to extend the polynomial order of the neural forms. The novel construction includes several feedforward neural networks, one for each order. The feedforward neural networks are optimised using a stochastic gradient descent method (ADAM). As a baseline model problem we consider a simple yet stiff ordinary differential equation. In experiments we illuminate some interesting properties of the proposed approach.}, language = {en} } @misc{SridharBreuss, author = {Sridhar, Vivek and Breuß, Michael}, title = {Sampling of Non-flat Morphology for Grey Value Images}, series = {Computer Analysis of Images and Patterns}, volume = {2021}, journal = {Computer Analysis of Images and Patterns}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-89130-5}, issn = {978-3-030-89131-2}, doi = {10.1007/978-3-030-89131-2_8}, pages = {88 -- 97}, abstract = {Sampling is a basic operation in image processing. In previous literature, a morphological sampling theorem has been established showing how sampling interacts with image reconstruction by morphological operations. However, while many aspects of morphological sampling have been investigated for binary images in classic works, only some of them have been extended to grey scale imagery. Especially, previous attempts to study the relation between sampling and grey scale morphology are restricted by construction to flat morphological filters. In order to establish a sampling theory for non-flat morphology, we establish an alternative definition for grey scale opening and closing relying on the umbra notion. Making use of this, we prove a sampling theorem about the interaction of sampling with fundamental morphological operations for non-flat morphology. This allows to make precise corresponding relations between sampling and image reconstruction, extending classic results for flat morphology of grey value images.}, language = {en} } @misc{MansouriYarahmadiBreuss, author = {Mansouri Yarahmadi, Ashkan and Breuß, Michael}, title = {Automatic Watermeter Reading in Presence of Highly Deformed Digits}, series = {Computer Analysis of Images and Patterns}, journal = {Computer Analysis of Images and Patterns}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-89130-5}, issn = {978-3-030-89131-2}, doi = {10.1007/978-3-030-89131-2_14}, pages = {153 -- 163}, abstract = {The task we face in this paper is to automate the reading of watermeters as can be found in large apartment houses. Typically water passes through such watermeters, so that one faces a wide range of challenges caused by water as the medium where the digits are positioned. One of the main obstacles is given by the frequently produced bubbles inside the watermeter that deform the digits. To overcome this problem, we propose the construction of a novel data set that resembles the watermeter digits with a focus on their deformations by bubbles. We report on promising experimental recognition results, based on a deep and recurrent network architecture performed on our data set.}, language = {en} } @misc{KahraSridharBreuss, author = {Kahra, Marvin and Sridhar, Vivek and Breuß, Michael}, title = {Fast Morphological Dilation and Erosion for Grey Scale Images Using the Fourier Transform}, series = {Scale Space and Variational Methods in Computer Vision}, journal = {Scale Space and Variational Methods in Computer Vision}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-75548-5}, issn = {978-3-030-75549-2}, doi = {10.1007/978-3-030-75549-2_6}, pages = {65 -- 77}, abstract = {The basic filters in mathematical morphology are dilation and erosion. They are defined by a flat or non-flat structuring element that is usually shifted pixel-wise over an image and a comparison process that takes place within the corresponding mask. Existing fast algorithms that realise dilation and erosion for grey value images are often limited with respect to size or shape of the structuring element. Usually their algorithmic complexity depends on these aspects. Many fast methods only address flat morphology. In this paper we propose a novel way to make use of the fast Fourier transform for the computation of dilation and erosion. Our method is by design highly flexible, as it can be used with flat and non-flat structuring elements of any size and shape. Moreover, its complexity does not depend on size or shape of the structuring element, but only on the number of pixels in the filtered images. We show experimentally that we obtain results of very reasonable quality with the proposed method.}, language = {en} } @misc{SchneidereitBreuss, author = {Schneidereit, Toni and Breuß, Michael}, title = {Collocation polynomial neural forms and domain fragmentation for solving initial value problems}, series = {Neural Computing and Applications}, volume = {34}, journal = {Neural Computing and Applications}, number = {9}, issn = {1433-3058}, doi = {10.1007/s00521-021-06860-4}, pages = {7141 -- 7156}, abstract = {Several neural network approaches for solving differential equations employ trial solutions with a feedforward neural network. There are different means to incorporate the trial solution in the construction, for instance, one may include them directly in the cost function. Used within the corresponding neural network, the trial solutions define the so-called neural form. Such neural forms represent general, flexible tools by which one may solve various differential equations. In this article, we consider time-dependent initial value problems, which require to set up the neural form framework adequately. The neural forms presented up to now in the literature for such a setting can be considered as first-order polynomials. In this work, we propose to extend the polynomial order of the neural forms. The novel collocation-type construction includes several feedforward neural networks, one for each order. Additionally, we propose the fragmentation of the computational domain into subdomains. The neural forms are solved on each subdomain, whereas the interfacing grid points overlap in order to provide initial values over the whole fragmentation. We illustrate in experiments that the combination of collocation neural forms of higher order and the domain fragmentation allows to solve initial value problems over large domains with high accuracy and reliability.}, language = {en} } @misc{KoehlerRigiBreuss, author = {K{\"o}hler, Alexander and Rigi, Ashkan and Breuß, Michael}, title = {Fast Shape Classification Using Kolmogorov-Smirnov Statistics}, series = {WSCG'2022 - 30. International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision'2022}, journal = {WSCG'2022 - 30. International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision'2022}, number = {CSRN 3201}, doi = {10.24132/CSRN.3201.22}, pages = {172 -- 180}, abstract = {The fast classification of shapes is an important problem in shape analysis and of high relevance for many possible applications. In this paper, we consider the use of very fast and easy to compute statistical techniques for assessing shapes, which may for instance be useful for a first similarity search in a shape database. To this end, we construct shape signatures at hand of stochastic sampling of distances between points of interest in a given shape. By employing the Kolmogorov-Smirnov statistics we then propose to formulate the problem of shape classification as a statistical hypothesis test that enables to assess the similarity of the signature distributions. In order to illustrate some important properties of our approach, we explore the use of simple sampling techniques. At hand of experiments conducted with a variety of shapes in two dimensions, we give a discussion of potentially interesting features of the method.}, language = {en} } @misc{SridharBreuss, author = {Sridhar, Vivek and Breuß, Michael}, title = {An Exact Fast Fourier Method for Morphological Dilation and Erosion Using the Umbra Technique}, series = {19th Conference on Robots and Vision (CRV), 2022}, journal = {19th Conference on Robots and Vision (CRV), 2022}, publisher = {IEEE}, isbn = {978-1-6654-9774-9}, doi = {10.1109/CRV55824.2022.00032}, pages = {190 -- 196}, abstract = {In this paper we consider the fundamental operations dilation and erosion of mathematical morphology. It is well known that many powerful image filtering operations can be constructed by their combinations. We propose a fast and novel algorithm based on the Fast Fourier Transform to compute grey-value morphological operations on an image. The novel method may deal with non-flat filters and incorporates no restrictions on shape and size of the filtering window, in contrast to many other fast methods in the field. Unlike fast Fourier techniques from previous works, the novel method gives exact results and is not an approximation. The key aspect which allows to achieve this is to explore here for the first time in this context the umbra formulation of images and filters. We show that the new method is in practice particularly suitable for filtering images with small tonal range or when employing large filter sizes.}, language = {en} } @misc{SchneidereitBreuss, author = {Schneidereit, Toni and Breuß, Michael}, title = {Computational characteristics of feedforward neural networks for solving a stiff differential equation}, series = {Neural Computing and Applications}, volume = {34}, journal = {Neural Computing and Applications}, number = {10}, doi = {10.1007/s00521-022-06901-6}, pages = {7975 -- 7989}, abstract = {Feedforward neural networks offer a possible approach for solving differential equations. However, the reliability and accuracy of the approximation still represent delicate issues that are not fully resolved in the current literature. Computational approaches are in general highly dependent on a variety of computational parameters as well as on the choice of optimisation methods, a point that has to be seen together with the structure of the cost function. The intention of this paper is to make a step towards resolving these open issues. To this end, we study here the solution of a simple but fundamental stiff ordinary differential equation modelling a damped system. We consider two computational approaches for solving differential equations by neural forms. These are the classic but still actual method of trial solutions defining the cost function, and a recent direct construction of the cost function related to the trial solution method. Let us note that the settings we study can easily be applied more generally, including solution of partial differential equations. By a very detailed computational study, we show that it is possible to identify preferable choices to be made for parameters and methods. We also illuminate some interesting effects that are observable in the neural network simulations. Overall we extend the current literature in the field by showing what can be done in order to obtain useful and accurate results by the neural network approach. By doing this we illustrate the importance of a careful choice of the computational setup.}, language = {en} } @misc{BaehrBreuss, author = {B{\"a}hr, Martin and Breuß, Michael}, title = {Efficient Long-Term Simulation of the Heat Equation with Application in Geothermal Energy Storage}, series = {Mathematics}, volume = {10}, journal = {Mathematics}, number = {13}, issn = {2227-7390}, doi = {10.3390/math10132309}, abstract = {Long-term evolutions of parabolic partial differential equations, such as the heat equation, are the subject of interest in many applications. There are several numerical solvers marking the state-of-the-art in diverse scientific fields that may be used with benefit for the numerical simulation of such long-term scenarios. We show how to adapt some of the currently most efficient numerical approaches for solving the fundamental problem of long-term linear heat evolution with internal and external boundary conditions as well as source terms. Such long-term simulations are required for the optimal dimensioning of geothermal energy storages and their profitability assessment, for which we provide a comprehensive analytical and numerical model. Implicit methods are usually considered the best choice for resolving long-term simulations of linear parabolic problems; however, in practice the efficiency of such schemes in terms of the combination of computational load and obtained accuracy may be a delicate issue, as it depends very much on the properties of the underlying model. For example, one of the challenges in long-term simulation may arise by the presence of time-dependent boundary conditions, as in our application. In order to provide both a computationally efficient and accurate enough simulation, we give a thorough discussion of the various numerical solvers along with many technical details and own adaptations. By our investigation, we focus on two largely competitive approaches for our application, namely the fast explicit diffusion method originating in image processing and an adaptation of the Krylov subspace model order reduction method. We validate our numerical findings via several experiments using synthetic and real-world data. We show that we can obtain fast and accurate long-term simulations of typical geothermal energy storage facilities. We conjecture that our techniques can be highly useful for tackling long-term heat evolution in many applications.}, language = {en} }