Podcasts about galerkin

In this paper, we apply the self-attention from the state-of-art Transformer in Attention Is All You Need [69] the first time to a data-driven operator learning problem related to partial differential equations. We put together an effort to explain the heuristics of, and improve the efficacy of the self-attention by demonstrating that the softmax normalization in the scaled dot-product attention is sufficient but not necessary, and have proved the approximation capacity of a linear variant as a Petrov-Galerkin projection. A new layer normalization scheme is proposed to allow a scaling to propagate through attention layers, which helps the model achieve remarkable accuracy in operator learning tasks with unnormalized data. Finally, we present three operator learning experiments, including the viscid Burgers' equation, an interface Darcy flow, and an inverse interface coefficient identification problem. All experiments validate the improvements of the newly proposed simple attention-based operator learner over their softmax-normalized counterparts. 2021: Shuhao Cao https://arxiv.org/pdf/2105.14995v2.pdf

Gudrun spricht mit Lydia Wagner über Elastoplastizität. Lydia hat im Rahmen ihrer im Mai 2019 abgeschlossenen Promotion Versetzungen in kristallinen Festkörpern numerisch simuliert. Elastizität beschreibt die (reversible) Verformung von Festkörpern unter Belastung. Bei zu großer Belastung reagieren Materialien nicht mehr elastisch, sondern es entstehen irreversible Deformationen. Das nennt man Plastizität. Im Rahmen der Kontinuumsmechanik wird die Deformation durch ein Kräftegleichgewicht basierend auf der Impuls- und Drehimpulserhaltung modelliert. Die konkreten Eigenschaften des Materials werden hierbei über eine spezifische Spannungs-Dehnungs-Relation berücksichtigt. Dabei tritt Plastizität auf, wenn im Material eine kritische Spannung erreicht wird. In klassischen phänomenologischen Plastizitätsmodellen der Kontinuumsmechanik wird dieses Verhalten über eine Fließbedingung in Abhängigkeit der Spannung modelliert. Diese wird durch eine Fließregel und ggf. eine Verfestigungsregel ergänzt, die das plastische Materialverhalten nach Erreichen der Fließgrenze beschreiben. Plastizität ist ein physikalischer Prozess, der auf Kristallebene stattfindet. Ein kristalliner Festkörper wird plastisch verformt, wenn sich eindimensionale Gitterfehler – Versetzungen – durch Belastung im Kristallgitter bewegen, d. h. wenn sich die atomaren Bindungen umordnen. Durch Mittelungsprozesse kann dieses diskrete Verhalten in einem Kontinuumsmodell, dem Continuum dislocation dynamics (CDD) Modell, beschrieben werden. Eine numerische Realisierung von diesem erweiterten Modell und die Evaluation im Vergleich zu diskreten Simulationen ist die Themenstellung der Dissertation von Lydia. Die Physik erarbeitete sich Lydia in Zusammenarbeit mit Materialwissenschaftlern und Ingenieuren in der DFG-Forschergruppe Dislocation based Plasticity am KIT. Literatur und weiterführende Informationen C. Wieners: Effiziente numerische Methoden in der Elasto-Plastizität, Vortragsfolien. P.M. Anderson, J.P. Hirth, J. Lothe: Theory of dislocations Cambridge University Press, New York, 2017, ISBN: 978-0-521-86436-7 T. Hochrainer et al.: Continuum dislocation dynamics: Towards a physical theory of crystal plasticity Journal of the Mechanics and Physics of Solids, 63, 167–178,2014. doi:10.1016/j.jmps.2013.09.012 K. Schulz, L. Wagner and C. Wieners: A mesoscale continuum approach of dislocation dynamics and the approximation by a Runge-Kutta discontinuous Galerkin method International Journal of Plasticity. doi:10.1016/j.ijplas.2019.05.003 Podcasts J. Fröhlich: Poroelastische Medien, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 156, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2018. A. Rick: Bézier Stabwerke, Gespräch mit S. Ritterbusch im Modellansatz Podcast, Folge 141, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2017. A. August: Materialschaum Gespräch mit S. Ritterbusch im Modellansatz Podcast, Folge 037, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2014.

Stephanie Wollherr hat ihr Mathestudium am Karlsruher Institut für Technologie (KIT) absolviert und in unserer Arbeitsgruppe die Abschlussarbeit im Kontext von numerischen Methoden für Wellengleichungen geschrieben. Damals hat Gudrun Thäter sie aus dem Podcastgespräch verabschiedet mit dem Wunsch, in einigen Jahren zu hören, was sie in der Zwischenzeit mathematisches tut. Was wie eine Floskel klingen mag, hat nun zum ersten Mal tatsächlich stattgefunden - ein Gespräch zur Arbeit von Stephanie in der Seismologie an der Ludwig-Maximillians-Universität (LMU) in München. In der Geophysik an der LMU wurde in den letzten 10 Jahren eine Software zur Wellenausbreitung entwickelt und benutzt, die immer weiter um simulierbare Phänomene ergänzt wird. Stephanie arbeitet an Dynamic Rupture Problemen - also der Simulation der Bruchdynamik als Quelle von Erdbeben. Hier geht es vor allem darum, weitere physikalische Eigenschaften wie z.B. Plastizität (bisher wurde meist vorausgesetzt, dass sich das Gestein elastisch verformt) und neue Reibungsgesetze zu implementieren und in Simulationen auf ihre Wirkung zu testen. Als Basis der Simulationen stehen zum einen Beobachtungen von Erdbeben aus der Vergangenheit zur Verfügung, zum anderen versucht man auch durch Laborexperimente, die aber leider ganz andere Größenskalen als die Realität haben, mögliche Eigenschaften der Bruchdynamik miteinzubeziehen. Die Daten der Seimsologischen Netzwerke sind zum Teil sogar öffentlich zugänglich. Im Bereich Dynamic Rupture Simulationen kann man eine gewisse Konzentration an Forschungskompetenz in Kalifornien feststellen, weil dort die möglicherweise katastrophalen Auswirkungen von zu erwartenden Erdbeben recht gegenwärtig sind. Das South California Earthquake Center unterstützt zum Beispiel unter anderem Softwares, die diese Art von Problemen simulieren, indem sie synthetische Testprobleme zur Verfügung stellen, die man benutzen kann, um die Ergebnisse seiner Software mit anderen zu vergleichen. Prinzipiell sind der Simulation von Bruchzonen bei Erdbeben gewissen Grenzen mit traditionellen Methoden gesetzt, da die Stetigkeit verloren geht. Der momentan gewählte Ausweg ist, im vornherein festzulegen, wo die Bruchzone verläuft, zutreffende Reibungsgesetze als Randbedingung zu setzen und mit Discontinuous Galerkin Methoden numerisch zu lösen. Diese unstetig angesetzten Verfahren eignen sich hervorragend, weil sie zwischen den Elementen Sprünge zulassen. Im Moment liegt der Fokus darauf, schon stattgefundene Erbeben zu simulieren. Leider sind auch hier die Informationen stets unvollständig: zum Beispiel können schon vorhandenen Bruchzonen unterhalb der Oberfläche unentdeckt bleiben und auch das regionale Spannungsfeld ist generell nicht sehr gut bestimmt. Eine weitere Herausforderung ist, dass die Prozesse an der Verwerfungszone mit sehr hoher Auflösung (bis auf ein paar 100m) gerechnet werden müssen, während der Vergleich mit Werten von Messstationen, die vielleicht einige 100 km entfernt sind einen sehr großen Simulationsbereich erfordert, was schnell zu einer hohen Anzahl an Elementen führt. Die Rechnungen laufen auf dem SuperMUC Supercomputer am LRZ in Garching und wurden durch eine Kooperation mit der Informatik an der TUM deutlich verbessert. Discontinuous Galerkin Verfahren haben den großen Vorteil, dass keine großen, globalen Matrizen entstehen, was eine Parallelisierung relativ einfach macht. Auf der anderen Seite kommen durch die element-lokale Kommunikation viele kleinere Matrix-Vektor Produkte vor, die grundlegend optimiert wurden. Ein weiterer Aspekt der Zusammenarbeit mit der TUM beschäftigte sich zum Beispiel mit der zu verteilenden Last, wenn für einige Elemente nur die Wellengleichung und für andere Elemente zusätzlich noch die Bruchdynamik gelöst werden muss. Auch bei der Input/Output Optimierung konnten die Informatiker willkommene Beiträge leisten. Dieser Beitrag zeigt die Notwendigkeit von interdisziplinärer Zusammenarbeit zwischen Mathematiker, Geophysikern und Informatikern, um Erdbeben und die Dynamik ihrer Quelle besser zu verstehen. Literatur und weiterführende Informationen A. Heinecke, A. Breuer, S. Rettenberger, M. Bader, A. Gabriel, C. Pelties, X.-K. Liao: Petascale High Order Dynamic Rupture Earthquake Simulations on Heterogeneous Supercomputers, proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis SC14, 3–15, 2014. A.-A. Gabriel, E. H. Madden, T. Ulrich, S. Wollherr: Earthquake scenarios from Sumatra to Iceland - High-resolution simulations of source physics on natural fault systems, Poster, Department of Earth and Environmental Sciences, LMU Munich, Germany. S. Wollherr, A.-A. Gabriel, H. Igel: Realistic Physics for Dynamic Rupture Scenarios: The Example of the 1992 Landers Earthquake, Poster, Department of Earth and Environmental Sciences, LMU Munich. J. S. Hesthaven, T. Warburton: Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, Springer Science & Business Media, 2007. M. Dumbser, M. Käser: An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes—II. The three-dimensional isotropic case, Geophysical Journal International, 167(1), 319-336, 2006. C. Pelties, J. de la Puente, J.-P. Ampuero, G. B. Brietzke, M. Käser, M: Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes, Journal of Geophysical Research, 117(B2), B02309, 2012. A.-A. Gabriel: Physics of dynamic rupture pulses and macroscopic earthquake source properties in elastic and plastic media. Diss. ETH No. 20567, 2013. K. C. Duru, A.-A. Gabriel, H. Igel: A new discontinuous Galerkin spectral element method for elastic waves with physically motivated numerical fluxes, in WAVES17 International Conference on Mathematical and Numerical Aspects of Wave Propagation, 2016. Weingärtner, Mirjam, Alice-Agnes Gabriel, and P. Martin Mai: Dynamic Rupture Earthquake Simulations on complex Fault Zones with SeisSol at the Example of the Husavik-Flatey Fault in Proceedings of the International Workshop on Earthquakes in North Iceland, Husavik, North Iceland, 31 May - 3 June 2016. Gabriel, Alice-Agnes, Jean-Paul Ampuero, Luis A. Dalguer, and P. Martin Mai: Source Properties of Dynamic Rupture Pulses with Off-Fault Plasticity, J. Geophys. Res., 118(8), 4117–4126, 2013. Miloslav Feistauer and Vit Dolejsi: Discontinuous Galerkin Method: Analysis and Applications to compressible flow Springer, 2015. Podcasts S. Wollherr: Erdbeben und Optimale Versuchsplanung, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 012, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2013.

Metodo degli elementi finti: metodo dei residui pesati di Galerkin.

Metodo degli elementi finti: metodo dei residui pesati di Galerkin.

Prof. Francisco Sayas from the Department of Mathematical Sciences of the University of Delaware in Newark has been visiting our faculty in June 2015. He is an expert in the simulation of scattering of transient acoustic waves. Scattering is a phenomenon in the propagation of waves. An interesting example from our everyday experience is when sound waves hit obstacles the wave field gets distorted. So, in a way, we can "hear" the obstacle. Sound waves are scalar, namely, changes in pressure. Other wave types scatter as well but can have a more complex structure. For example, seismic waves are elastic waves and travel at two different speeds as primary and secondary waves. As mathematician, one can abandon the application completely and say a wave is defined as a solution of a Wave Equation. Hereby, we also mean finding these solution in different appropriate function spaces (which represent certain properties of the class of solutions), but it is a very global look onto different wave properties and gives a general idea about waves. The equations are treated as an entity of their own right. Only later in the process it makes sense to compare the results with experiments and to decide if the equations fit or are too simplified. Prof. Sayas startet out in a "save elliptic world" with well-established and classical theories such as the mapping property between data and solutions. But for the study of wave equations, today there is no classical or standard method, but very many different tools are used to find different types of results, such as the preservation of energy. Sometimes it is obvious, that the results cannot be optimal (or sharp) if e.g. properties like convexity of obstacles do not play any role in getting results. And many questions are still wide open. Also, the numerical methods must be well designed. Up to now, transient waves are the most challenging and interesting problem for Prof. Sayas. They include all frequencies and propagate in time. So it is difficult to find the correct speed of propagation and also dispersion enters the configuration. On the one hand, the existence and regularity together with other properties of solutions have to be shown, but on the other hand, it is necessary to calculate the propagation process for simulations - i.e. the solutions - numerically.There are many different numerical schemes for bounded domains. Prof. Sayas prefers FEM and combines them with boundary integral equations as representative for the outer domain effects. The big advantage of the boundary integral representation is that it is physical correct but unfortunately, it is very complicated and all points on the boundary are interconnected. Finite Elements fit well to a black box approach which leads to its popularity among engineers. The regularity of the boundary can be really low if one chooses Galerkin methods. The combination of both tools is a bit tricky since the solver for the Wave Equations needs data on the boundary which it has to get from the Boundary element code and vice versa. Through this coupling it is already clear that in the coding the integration of the different tools is an important part and has to be done in a way that all users of the code which will improve it in the future can understand what is happening. Prof. Sayas is fascinated by his research field. This is also due to its educational aspect: the challenging mathematics, the set of tools still mainly unclear together with the intensive computational part of his work. The area is still wide open and one has to explain mathematics to other people interested in the results. In his carreer he started out with studying Finite Elements at the University in Zaragoza and worked on boundary elements with his PhD-supervisor from France. After some time he was looking for a challenging new topic and found his field in which he can combine both fields. He has worked three years at the University of Minnesota (2007-2010) and decided to find his future at a University in the U.S.. In this way he arrived at the University of Delaware and is very satisfied with the opportunities in his field of research and the chances for young researchers. Literature and additional material deltaBEM - Easy to Implement Boundary Integral Equations, open source software developed by Team Pancho at the Department of Mathematical Sciences in Delaware. A. R. Laliena, F. J. Sayas: Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves, Numerische Mathematik, 112(4), 637-678, 2009. F. J. Sayas: Energy estimates for Galerkin semidiscretizations of time domain boundary integral equations, Numerische Mathematik, 124(1), 121-149, 2013. Modellansatz Podcast 003: Unsichtbarkeit (in German)

Prof. Francisco Sayas from the Department of Mathematical Sciences of the University of Delaware in Newark has been visiting our faculty in June 2015. He is an expert in the simulation of scattering of transient acoustic waves. Scattering is a phenomenon in the propagation of waves. An interesting example from our everyday experience is when sound waves hit obstacles the wave field gets distorted. So, in a way, we can "hear" the obstacle. Sound waves are scalar, namely, changes in pressure. Other wave types scatter as well but can have a more complex structure. For example, seismic waves are elastic waves and travel at two different speeds as primary and secondary waves. As mathematician, one can abandon the application completely and say a wave is defined as a solution of a Wave Equation. Hereby, we also mean finding these solution in different appropriate function spaces (which represent certain properties of the class of solutions), but it is a very global look onto different wave properties and gives a general idea about waves. The equations are treated as an entity of their own right. Only later in the process it makes sense to compare the results with experiments and to decide if the equations fit or are too simplified. Prof. Sayas startet out in a "save elliptic world" with well-established and classical theories such as the mapping property between data and solutions. But for the study of wave equations, today there is no classical or standard method, but very many different tools are used to find different types of results, such as the preservation of energy. Sometimes it is obvious, that the results cannot be optimal (or sharp) if e.g. properties like convexity of obstacles do not play any role in getting results. And many questions are still wide open. Also, the numerical methods must be well designed. Up to now, transient waves are the most challenging and interesting problem for Prof. Sayas. They include all frequencies and propagate in time. So it is difficult to find the correct speed of propagation and also dispersion enters the configuration. On the one hand, the existence and regularity together with other properties of solutions have to be shown, but on the other hand, it is necessary to calculate the propagation process for simulations - i.e. the solutions - numerically.There are many different numerical schemes for bounded domains. Prof. Sayas prefers FEM and combines them with boundary integral equations as representative for the outer domain effects. The big advantage of the boundary integral representation is that it is physical correct but unfortunately, it is very complicated and all points on the boundary are interconnected. Finite Elements fit well to a black box approach which leads to its popularity among engineers. The regularity of the boundary can be really low if one chooses Galerkin methods. The combination of both tools is a bit tricky since the solver for the Wave Equations needs data on the boundary which it has to get from the Boundary element code and vice versa. Through this coupling it is already clear that in the coding the integration of the different tools is an important part and has to be done in a way that all users of the code which will improve it in the future can understand what is happening. Prof. Sayas is fascinated by his research field. This is also due to its educational aspect: the challenging mathematics, the set of tools still mainly unclear together with the intensive computational part of his work. The area is still wide open and one has to explain mathematics to other people interested in the results. In his carreer he started out with studying Finite Elements at the University in Zaragoza and worked on boundary elements with his PhD-supervisor from France. After some time he was looking for a challenging new topic and found his field in which he can combine both fields. He has worked three years at the University of Minnesota (2007-2010) and decided to find his future at a University in the U.S.. In this way he arrived at the University of Delaware and is very satisfied with the opportunities in his field of research and the chances for young researchers. Literature and additional material deltaBEM - Easy to Implement Boundary Integral Equations, open source software developed by Team Pancho at the Department of Mathematical Sciences in Delaware. A. R. Laliena, F. J. Sayas: Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves, Numerische Mathematik, 112(4), 637-678, 2009. F. J. Sayas: Energy estimates for Galerkin semidiscretizations of time domain boundary integral equations, Numerische Mathematik, 124(1), 121-149, 2013. Modellansatz Podcast 003: Unsichtbarkeit (in German)

Wed, 20 Jun 2012 12:00:00 +0100 https://edoc.ub.uni-muenchen.de/14524/ https://edoc.ub.uni-muenchen.de/14524/1/Pelties_Christian.pdf Pelties, Christian

Lecture 19 explains the hemicube estimates for patch-to-patch form factors for "Galerkin" piecewise constant radiosity and for vertex-to-vertex form factors for "point collocation" piecewise linear radiosity. It also continues the proof that the global lines form factor computation is correct. (At 26:48 minutes, "cos(theta) = length/1" should be "cos(theta) = 1/length" and at 40:40 minutes, the sound track saying "is two at these other vertices" should say "is zero at these other vertices").

Ainsworth, M (Strathclyde) Monday 26 March 2007, 15:30-16:15