###### 6,963 Research products, page 1 of 697

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- Publication . Doctoral thesis . 2022Open Access GermanAuthors:Wewetzer, Jacob;Wewetzer, Jacob;Country: Germany
Die Arbeit beschäftigt sich mit der Frage der Vergütung von Amtsträgern in Betriebs- und Aufsichtsrat. Ein Schwerpunkt der Auseinandersetzung betrifft dabei die gesetzeskonforme Bestimmung der vergütungsrechtlichen Obergrenzen bei Betriebsratsmitgliedern. Es wird zugleich auf die Möglichkeit des Abschlusses einer Betriebsvereinbarung sowie auf sonstige Reformvorschläge auf dem Gebiet der Vergütung von Betriebsratsmitgliedern eingegangen. Ausgehend von der Grundfrage der Vergütung von Aufsichtsratsmitgliedern setzt sich die Arbeit im Anschluss mit den bei der Führung beider Ämter Personalunion auftretenden Wechselwirkungen auseinander.

- Publication . Report . 2022Open Access GermanAuthors:Grimm, Viktor; Heinlein, Alexander; Klawonn, Axel;Grimm, Viktor; Heinlein, Alexander; Klawonn, Axel;Country: Germany
The approach of using physics-based machine learning to solve PDEs has recently become very popular. A recent approach to solve PDEs based on CNNs uses finite difference stencils to include the residual of the partial differential equation into the loss function. In this work, the relation between the network training and the solution of a respective finite difference linear system of equations using classical numerical solvers is discussed. It turns out that many beneficial properties of the linear equation system are neglected in the network training. Finally, numerical results which underline the benefits of classical numerical solvers are presented.

- Publication . Other literature type . 2022Open Access EnglishAuthors:Pirzada, Mujeeb Ur Rehman;Pirzada, Mujeeb Ur Rehman;Publisher: Pirzada Mujeeb Ur RehmanCountry: Germany
- Publication . Doctoral thesis . 2022Open Access GermanAuthors:Zickenheiner, Frank;Zickenheiner, Frank;Country: Germany
This book provides a better understanding of rhetorical relations and their impact on the referent's abilities to serve as non-local antecedents for referential expressions. It argues that rhetorical relations rather relate speech acts than semantic denotations of sentences, which emphasizes the action theoretic character of communication and which makes it possible to describe and understand rhetorical relations for non-assertive speech acts. The arguments in this book are based on some cognitive and action theoretical assumptions which help to model a hearer's recognition of the speaker's communicative plans, which is believed to be the unrealized discourse structure. In particular, the common sense principle of inertia is used, which models the general assumption that agents believe that things in the world do not change if there is no reason for them to change. Based on this principle, it is possible to explain why acceptance moves in dialog are necessary for communication and, moreover, why by default, two speech acts are related by a support relation, i.e. by a relation where one speech act supports the goal of the speech act it is related to. Support relations are described as a further class of rhetorical relations that, roughly, correspond to subordinating relations but, in contrast to subordinating relations, have the advantage that it is possible to provide an intrinsic definition. Finally, it is shown how discourse structure as an impact on the prominence status of referents and speech acts, being roughly the ability of referents to serve as structural anchor for further communication.

- Publication . Report . 2022Open Access EnglishAuthors:Klawonn, Axel; Lanser, Martin;Klawonn, Axel; Lanser, Martin;Country: Germany
Nonlinear FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) is a nonlinear nonoverlapping domain decomposition method (DDM) which has a superior nonlinear convergence behavior compared with classical Newton-Krylov-DDMs - at least for many problems. Its fast and robust nonlinear convergence is strongly influenced by the choice of the second level or, in other words, the choice of the coarse constraints. Additionally, the convergence is also affected by the choice of an elimination set, that is, a set of degrees of freedom which are eliminated nonlinearly before linearization. In this article, an adaptive coarse space is combined with a problem-dependent and residual-based choice of the elimination set. An efficient implementation exploiting sparse local saddle point problems instead of an explicit transformation of basis is used. Unfortunately, this approach makes a further adaption of the elimination sets necessary, that is, edges and faces with coarse constraints have to be either included in the elimination set completely or not at all. Different strategies to fulfill this additional constraint are discussed and compared with a solely residual-based approach. The latter approach has to be implemented with an explicit transformation of basis. In general, the residual which is used to choose the elimination set has to be transformed to a space which basis functions explicitly contain the coarse constraints. This is computationally expensive. Here, for the first time, it is suggested to use an approximation of the transformed residual instead to compute the elimination set.

- Publication . Report . 2022Open Access EnglishAuthors:Klawonn, Axel; Lanser, Martin; Wasiak, Adam;Klawonn, Axel; Lanser, Martin; Wasiak, Adam;Country: Germany
The Virtual Element Method (VEM) is a discretization procedure for the solution of partial differential equations that allows for the use of nearly arbitrary polygonal/polyhedral grids. For the parallel scalable and iterative solution of large scale VE problems, the FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods have recently been applied. As for the case of finite element discretizations, a large global coarse problem that usually arises in large scale applications is a parallel scaling bottleneck of FETI-DP and BDDC. Nonetheless, the coarse problem/second level is usually necessary for the numerical robustness of the method. To alleviate this difficulty and to retain the scalability, the three-level BDDC method is applied to virtual element discretizations in this article. In this approach, to allow for a parallel solution of the coarse problem, the solution of it is only approximated by applying BDDC recursively, which automatically introduces a third level. Numerical results using several different configurations of the three-level approach and different polygonal meshes are presented and additionally compared with the classical two-level BDDC approach.

- Publication . Doctoral thesis . 2022Open Access EnglishAuthors:Yu, Dongli;Yu, Dongli;Country: Germany
Plant defense against microbial pathogens is mainly realized by pattern-triggered immunity (PTI) mediated by pattern recognition receptors (PRRs) at the cell surface, and effectortriggered immunity (ETI) mediated by nucleotide-binding leucine-rich repeat (NLR) immune receptors inside cells. Based on their N-terminal domains, plant NLRs can be divided into two categories: CC-NLRs (CNLs) with a coiled-coil (CC) domain and TIR-NLRs (TNLs) with a toll/interleukin 1 receptor (TIR) domain. Specific recognition of pathogen effectors induces oligomerization of NLRs, termed resistosomes, to transduce plant immune signaling. CNLs are able to form pentameric resistosomes upon activation and function as calcium (Ca2+)-permeable channels in the plasma membrane. Whether TNLs form resistosomes in response to pathogen infection remained an open question, although the TIR domain in TNLs has NADase activity that is required for TNL-mediated immunity. NADase activity, although essential, is not sufficient for TIR-triggered immune responses in plants, suggesting that other components may be required for TIR-mediated signaling. In my dissertation, I employed multiple approaches including biochemistry and structural biology to address these questions. The thesis contains three parts: In the first part, I present multiple lines of evidence showing that the Arabidopsis TNL RPP1 (for recognition of Peronospora parasitica 1) forms a tetrameric resistosome upon recognition of the cognate Hyaloperonospora arabidopsidis effector ATR1. Biochemical and structural data are summarized revealing the mechanism underlying the requirement of the RPP1 resistosome formation for NADase activity. The data from this study define the mechanism of direct effector recognition by a TNL, and demonstrate that the assembly of RPP1 resistosomes is required for TIR-encoded NADase activity and RPP1 function. In the second part, I describe biochemical evidence that TIR domain proteins also exhibit 2′,3′-cAMP/cGMP synthetase activity with RNA and probably DNA (RNA/DNA) as substrates. Then I present functional data supporting the physiological relevance of the synthetase activity in TIR-mediated immune responses. Structural data on a TIR domain protein bound by its dsDNA substrate are described, and the mechanisms of how TIR domain proteins encode both NADase and synthetase activities and how the two activities may act together to mediate TIR signaling are discussed. The data presented in this part reveal a novel enzymatic activity of plant TIR domain proteins and establish a role of 2′,3′-cAMP/cGMP in plant immunity. In the last section of my thesis, I describe experiments testing whether the RNase-like effector proteins associated with haustoria (RALPH effectors) have RNase activity.

- Publication . Doctoral thesis . 2022Open Access GermanAuthors:Keil, Magali;Keil, Magali;Country: Germany
Die Rezeptorautoradiographie wurde bereits zur Untersuchung vieler anderer zytoarchitektonisch definierter Kortexareale genutzt. Diese Methode erlaubt es zum einen die neurochemische Struktur der Areale zu verstehen. Zum anderen ist es möglich innerhalb der Areale weitere Abgrenzungen vorzunehmen, welche zyto- oder myeloarchitektonisch nicht nachweisbar sind. Darüber hinaus spielen Rezeptoren eine Schlüsselrolle in der Signaltransduktion und ergänzen moderne Hirnkarten somit nicht nur um anatomisch-strukturelle, sondern auch funktionell relevante Informationen. Damit reiht sich dieser Ansatz in die Arbeit vieler anderer aktueller Hirnkartierungsprojekte ein und leistet einen essentiellen Beitrag im Rahmen neurowissenschaftlicher Grundlagenforschung zur Dekodierung der komplexen Funktionsweise des Gehirns. In der vorliegenden Arbeit wurde der mediale orbitofrontale Kortex rezeptorautoradiographisch untersucht und Vergleiche mit vorliegenden Karten desselben Kortexareals sowie mit der Rezeptorachitektur anderer Kortizes gezogen. Die Ergebnisse der rezeptorautoradiographischen Analyse spiegeln die zytoarchitektonische Einteilung des medialen orbitofrontalen Kortex nach Henssen et al. in die drei Areale Fo1, Fo2 und Fo3 wider. Darüber hinaus ergeben sich aufgrund der erhobenen Gradienten in den untersuchten Regionen innerhalb dieser Areale Hinweise darauf, dass eine weitere Unterteilung in Subareale möglich ist, welche mutmaßlich mit Unterschieden in der Funktion einhergehen. Insbesondere vor dem Hintergrund der komplexen Lern- und Entscheidungsprozesse, in die der orbitofrontale Kortex eingebunden ist, erscheint eine feinere funktionelle Trennung sinnvoll. Für eine definitive Abgrenzung mit entsprechendem bildmorphologischem Korrelat in den pseudo-farbkodierten Bildern sollten weitere Untersuchungen mit mehr Hemisphären erfolgen und die Abgrenzungen statistisch belegt werden. Mit Hilfe der Clusteranalyseverfahren konnte nachgewiesen werden, dass die Areale des medialen orbitofrontalen Kortex in ihrer Rezeptorarchitektur anderen Regionen des frontalen Assoziationskortex ähneln. Im Gegensatz dazu differieren sie in besonderem Maße von primären Kortizes des motorischen, somatosensorischen, visuellen und akustischen Systems.

- Publication . Report . 2022Open Access EnglishAuthors:Klawonn, Axel; Lanser, Martin; Weber, Janine;Klawonn, Axel; Lanser, Martin; Weber, Janine;Country: Germany
The highly parallel scalable three-level BDDC (Balancing Domain Decomposition by Constraints) method is a very successful approach to overcome the scaling bottleneck of directly solving a large coarse problem in classical two-level BDDC. As long as the problem is homogeneous on each subregion, three-level BDDC is also provably robust in many cases. For problems with complex microstructures, as, e.g., stationary diffusion problems with jumps in the diffusion coefficient function, in two-level BDDC methods, advanced adaptive or frugal coarse spaces have to be considered to obtain a robust preconditioner. Unfortunately, these approaches result in even larger coarse problems on the second level and, additionally, computing adaptive coarse constraints is computationally expensive. Therefore, in this article, the three-level approach is combined with a provably robust adaptive coarse space and the computationally cheaper frugal coarse space. Both coarse spaces are used on the second as well as the third level. All different possible combinations are investigated numerically for stationary diffusion problems with highly varying coefficient functions.

- Publication . Report . 2022Open Access EnglishAuthors:Klawonn, Axel; Lanser, Martin; Weber, Janine;Klawonn, Axel; Lanser, Martin; Weber, Janine;Country: Germany
Adaptive coarse spaces yield a robust convergence behavior for FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) and BDDC (Balancing Domain Decomposition by Constraints) methods for highly heterogeneous problems. However, the usage of such adaptive coarse spaces can be computationally expensive since, in general, it requires the setup and the solution of a relatively high amount of local eigenvalue problems on parts of the domain decomposition interface. In earlier works, see, e.g., [2], it has been shown that it is possible to train a neural network to make an automatic decision which of the eigenvalue problems in an adaptive FETI-DP method are actually necessary for robustness with a satisfactory accuracy. Moreover, these results have been extended in [6] by directly learning an approximation of the adaptive edge constraints themselves for regular, two-dimensional domain decompositions. In particular, this does not require the setup or the solution of any eigenvalue problems at all since the FETI-DP coarse space is, in this case, exclusively enhanced by the learned constraints obtained from the regression neural networks trained in an offline phase. Here, in contrast to [6], a regression neural network is trained with both, training data resulting from straight and irregular edges. Thus, it is possible to use the trained networks also for the approximation of adaptive constraints for irregular domain decompositions. Numerical results for a heterogeneous two-dimensional stationary diffusion problem are presented using both, a decomposition into regular and irregular subdomains.

###### 6,963 Research products, page 1 of 697

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- Publication . Doctoral thesis . 2022Open Access GermanAuthors:Wewetzer, Jacob;Wewetzer, Jacob;Country: Germany
Die Arbeit beschäftigt sich mit der Frage der Vergütung von Amtsträgern in Betriebs- und Aufsichtsrat. Ein Schwerpunkt der Auseinandersetzung betrifft dabei die gesetzeskonforme Bestimmung der vergütungsrechtlichen Obergrenzen bei Betriebsratsmitgliedern. Es wird zugleich auf die Möglichkeit des Abschlusses einer Betriebsvereinbarung sowie auf sonstige Reformvorschläge auf dem Gebiet der Vergütung von Betriebsratsmitgliedern eingegangen. Ausgehend von der Grundfrage der Vergütung von Aufsichtsratsmitgliedern setzt sich die Arbeit im Anschluss mit den bei der Führung beider Ämter Personalunion auftretenden Wechselwirkungen auseinander.

- Publication . Report . 2022Open Access GermanAuthors:Grimm, Viktor; Heinlein, Alexander; Klawonn, Axel;Grimm, Viktor; Heinlein, Alexander; Klawonn, Axel;Country: Germany
The approach of using physics-based machine learning to solve PDEs has recently become very popular. A recent approach to solve PDEs based on CNNs uses finite difference stencils to include the residual of the partial differential equation into the loss function. In this work, the relation between the network training and the solution of a respective finite difference linear system of equations using classical numerical solvers is discussed. It turns out that many beneficial properties of the linear equation system are neglected in the network training. Finally, numerical results which underline the benefits of classical numerical solvers are presented.

- Publication . Other literature type . 2022Open Access EnglishAuthors:Pirzada, Mujeeb Ur Rehman;Pirzada, Mujeeb Ur Rehman;Publisher: Pirzada Mujeeb Ur RehmanCountry: Germany
- Publication . Doctoral thesis . 2022Open Access GermanAuthors:Zickenheiner, Frank;Zickenheiner, Frank;Country: Germany
This book provides a better understanding of rhetorical relations and their impact on the referent's abilities to serve as non-local antecedents for referential expressions. It argues that rhetorical relations rather relate speech acts than semantic denotations of sentences, which emphasizes the action theoretic character of communication and which makes it possible to describe and understand rhetorical relations for non-assertive speech acts. The arguments in this book are based on some cognitive and action theoretical assumptions which help to model a hearer's recognition of the speaker's communicative plans, which is believed to be the unrealized discourse structure. In particular, the common sense principle of inertia is used, which models the general assumption that agents believe that things in the world do not change if there is no reason for them to change. Based on this principle, it is possible to explain why acceptance moves in dialog are necessary for communication and, moreover, why by default, two speech acts are related by a support relation, i.e. by a relation where one speech act supports the goal of the speech act it is related to. Support relations are described as a further class of rhetorical relations that, roughly, correspond to subordinating relations but, in contrast to subordinating relations, have the advantage that it is possible to provide an intrinsic definition. Finally, it is shown how discourse structure as an impact on the prominence status of referents and speech acts, being roughly the ability of referents to serve as structural anchor for further communication.

- Publication . Report . 2022Open Access EnglishAuthors:Klawonn, Axel; Lanser, Martin;Klawonn, Axel; Lanser, Martin;Country: Germany
Nonlinear FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) is a nonlinear nonoverlapping domain decomposition method (DDM) which has a superior nonlinear convergence behavior compared with classical Newton-Krylov-DDMs - at least for many problems. Its fast and robust nonlinear convergence is strongly influenced by the choice of the second level or, in other words, the choice of the coarse constraints. Additionally, the convergence is also affected by the choice of an elimination set, that is, a set of degrees of freedom which are eliminated nonlinearly before linearization. In this article, an adaptive coarse space is combined with a problem-dependent and residual-based choice of the elimination set. An efficient implementation exploiting sparse local saddle point problems instead of an explicit transformation of basis is used. Unfortunately, this approach makes a further adaption of the elimination sets necessary, that is, edges and faces with coarse constraints have to be either included in the elimination set completely or not at all. Different strategies to fulfill this additional constraint are discussed and compared with a solely residual-based approach. The latter approach has to be implemented with an explicit transformation of basis. In general, the residual which is used to choose the elimination set has to be transformed to a space which basis functions explicitly contain the coarse constraints. This is computationally expensive. Here, for the first time, it is suggested to use an approximation of the transformed residual instead to compute the elimination set.

- Publication . Report . 2022Open Access EnglishAuthors:Klawonn, Axel; Lanser, Martin; Wasiak, Adam;Klawonn, Axel; Lanser, Martin; Wasiak, Adam;Country: Germany
The Virtual Element Method (VEM) is a discretization procedure for the solution of partial differential equations that allows for the use of nearly arbitrary polygonal/polyhedral grids. For the parallel scalable and iterative solution of large scale VE problems, the FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods have recently been applied. As for the case of finite element discretizations, a large global coarse problem that usually arises in large scale applications is a parallel scaling bottleneck of FETI-DP and BDDC. Nonetheless, the coarse problem/second level is usually necessary for the numerical robustness of the method. To alleviate this difficulty and to retain the scalability, the three-level BDDC method is applied to virtual element discretizations in this article. In this approach, to allow for a parallel solution of the coarse problem, the solution of it is only approximated by applying BDDC recursively, which automatically introduces a third level. Numerical results using several different configurations of the three-level approach and different polygonal meshes are presented and additionally compared with the classical two-level BDDC approach.

- Publication . Doctoral thesis . 2022Open Access EnglishAuthors:Yu, Dongli;Yu, Dongli;Country: Germany
Plant defense against microbial pathogens is mainly realized by pattern-triggered immunity (PTI) mediated by pattern recognition receptors (PRRs) at the cell surface, and effectortriggered immunity (ETI) mediated by nucleotide-binding leucine-rich repeat (NLR) immune receptors inside cells. Based on their N-terminal domains, plant NLRs can be divided into two categories: CC-NLRs (CNLs) with a coiled-coil (CC) domain and TIR-NLRs (TNLs) with a toll/interleukin 1 receptor (TIR) domain. Specific recognition of pathogen effectors induces oligomerization of NLRs, termed resistosomes, to transduce plant immune signaling. CNLs are able to form pentameric resistosomes upon activation and function as calcium (Ca2+)-permeable channels in the plasma membrane. Whether TNLs form resistosomes in response to pathogen infection remained an open question, although the TIR domain in TNLs has NADase activity that is required for TNL-mediated immunity. NADase activity, although essential, is not sufficient for TIR-triggered immune responses in plants, suggesting that other components may be required for TIR-mediated signaling. In my dissertation, I employed multiple approaches including biochemistry and structural biology to address these questions. The thesis contains three parts: In the first part, I present multiple lines of evidence showing that the Arabidopsis TNL RPP1 (for recognition of Peronospora parasitica 1) forms a tetrameric resistosome upon recognition of the cognate Hyaloperonospora arabidopsidis effector ATR1. Biochemical and structural data are summarized revealing the mechanism underlying the requirement of the RPP1 resistosome formation for NADase activity. The data from this study define the mechanism of direct effector recognition by a TNL, and demonstrate that the assembly of RPP1 resistosomes is required for TIR-encoded NADase activity and RPP1 function. In the second part, I describe biochemical evidence that TIR domain proteins also exhibit 2′,3′-cAMP/cGMP synthetase activity with RNA and probably DNA (RNA/DNA) as substrates. Then I present functional data supporting the physiological relevance of the synthetase activity in TIR-mediated immune responses. Structural data on a TIR domain protein bound by its dsDNA substrate are described, and the mechanisms of how TIR domain proteins encode both NADase and synthetase activities and how the two activities may act together to mediate TIR signaling are discussed. The data presented in this part reveal a novel enzymatic activity of plant TIR domain proteins and establish a role of 2′,3′-cAMP/cGMP in plant immunity. In the last section of my thesis, I describe experiments testing whether the RNase-like effector proteins associated with haustoria (RALPH effectors) have RNase activity.

- Publication . Doctoral thesis . 2022Open Access GermanAuthors:Keil, Magali;Keil, Magali;Country: Germany
Die Rezeptorautoradiographie wurde bereits zur Untersuchung vieler anderer zytoarchitektonisch definierter Kortexareale genutzt. Diese Methode erlaubt es zum einen die neurochemische Struktur der Areale zu verstehen. Zum anderen ist es möglich innerhalb der Areale weitere Abgrenzungen vorzunehmen, welche zyto- oder myeloarchitektonisch nicht nachweisbar sind. Darüber hinaus spielen Rezeptoren eine Schlüsselrolle in der Signaltransduktion und ergänzen moderne Hirnkarten somit nicht nur um anatomisch-strukturelle, sondern auch funktionell relevante Informationen. Damit reiht sich dieser Ansatz in die Arbeit vieler anderer aktueller Hirnkartierungsprojekte ein und leistet einen essentiellen Beitrag im Rahmen neurowissenschaftlicher Grundlagenforschung zur Dekodierung der komplexen Funktionsweise des Gehirns. In der vorliegenden Arbeit wurde der mediale orbitofrontale Kortex rezeptorautoradiographisch untersucht und Vergleiche mit vorliegenden Karten desselben Kortexareals sowie mit der Rezeptorachitektur anderer Kortizes gezogen. Die Ergebnisse der rezeptorautoradiographischen Analyse spiegeln die zytoarchitektonische Einteilung des medialen orbitofrontalen Kortex nach Henssen et al. in die drei Areale Fo1, Fo2 und Fo3 wider. Darüber hinaus ergeben sich aufgrund der erhobenen Gradienten in den untersuchten Regionen innerhalb dieser Areale Hinweise darauf, dass eine weitere Unterteilung in Subareale möglich ist, welche mutmaßlich mit Unterschieden in der Funktion einhergehen. Insbesondere vor dem Hintergrund der komplexen Lern- und Entscheidungsprozesse, in die der orbitofrontale Kortex eingebunden ist, erscheint eine feinere funktionelle Trennung sinnvoll. Für eine definitive Abgrenzung mit entsprechendem bildmorphologischem Korrelat in den pseudo-farbkodierten Bildern sollten weitere Untersuchungen mit mehr Hemisphären erfolgen und die Abgrenzungen statistisch belegt werden. Mit Hilfe der Clusteranalyseverfahren konnte nachgewiesen werden, dass die Areale des medialen orbitofrontalen Kortex in ihrer Rezeptorarchitektur anderen Regionen des frontalen Assoziationskortex ähneln. Im Gegensatz dazu differieren sie in besonderem Maße von primären Kortizes des motorischen, somatosensorischen, visuellen und akustischen Systems.

- Publication . Report . 2022Open Access EnglishAuthors:Klawonn, Axel; Lanser, Martin; Weber, Janine;Klawonn, Axel; Lanser, Martin; Weber, Janine;Country: Germany
The highly parallel scalable three-level BDDC (Balancing Domain Decomposition by Constraints) method is a very successful approach to overcome the scaling bottleneck of directly solving a large coarse problem in classical two-level BDDC. As long as the problem is homogeneous on each subregion, three-level BDDC is also provably robust in many cases. For problems with complex microstructures, as, e.g., stationary diffusion problems with jumps in the diffusion coefficient function, in two-level BDDC methods, advanced adaptive or frugal coarse spaces have to be considered to obtain a robust preconditioner. Unfortunately, these approaches result in even larger coarse problems on the second level and, additionally, computing adaptive coarse constraints is computationally expensive. Therefore, in this article, the three-level approach is combined with a provably robust adaptive coarse space and the computationally cheaper frugal coarse space. Both coarse spaces are used on the second as well as the third level. All different possible combinations are investigated numerically for stationary diffusion problems with highly varying coefficient functions.

- Publication . Report . 2022Open Access EnglishAuthors:Klawonn, Axel; Lanser, Martin; Weber, Janine;Klawonn, Axel; Lanser, Martin; Weber, Janine;Country: Germany
Adaptive coarse spaces yield a robust convergence behavior for FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) and BDDC (Balancing Domain Decomposition by Constraints) methods for highly heterogeneous problems. However, the usage of such adaptive coarse spaces can be computationally expensive since, in general, it requires the setup and the solution of a relatively high amount of local eigenvalue problems on parts of the domain decomposition interface. In earlier works, see, e.g., [2], it has been shown that it is possible to train a neural network to make an automatic decision which of the eigenvalue problems in an adaptive FETI-DP method are actually necessary for robustness with a satisfactory accuracy. Moreover, these results have been extended in [6] by directly learning an approximation of the adaptive edge constraints themselves for regular, two-dimensional domain decompositions. In particular, this does not require the setup or the solution of any eigenvalue problems at all since the FETI-DP coarse space is, in this case, exclusively enhanced by the learned constraints obtained from the regression neural networks trained in an offline phase. Here, in contrast to [6], a regression neural network is trained with both, training data resulting from straight and irregular edges. Thus, it is possible to use the trained networks also for the approximation of adaptive constraints for irregular domain decompositions. Numerical results for a heterogeneous two-dimensional stationary diffusion problem are presented using both, a decomposition into regular and irregular subdomains.