Podcasts about PDES

Play Episode Listen Later Dec 13, 2018 51:04

In this episode Gudrun talks with her new colleague Xian Liao. In November 2018 Xian has been appointed as Junior Professor (with tenure track) at the KIT-Faculty of Mathematics. She belongs to the Institute of Analysis and works in the group Nonlinear Partial Differential Equations. She is very much interested in Dispersive Partial Differential Equations. These equations model, e.g., the behaviour of waves. For that it is a topic very much in the center of the CRC 1173 - Wave phenomena at our faculty. Her mathematical interest was always to better understand the solutions of partial differential equations. But she arrived at dispersive equations through several steps in her carreer. Originally she studied inhomogeneous incompressible fluids. This can for example mean that the fluid is a mixture of materials with different viscosities. If we have a look at the Navier-Stokes equations for materials like water or oil, one main assumption therein is, that the viscosity is a material constant. Nevertheless, the equations modelling their flows are already nonlinear and there are a few serious open questions. Studying flows of inhomogneous materials brings in further difficulties since there occur more and more complex nonlinearities in the equations. It is necessary to develop a frame in which one can characterise the central properties of the solutions and the flow. It turned out that for example finding and working with quantities which remain conserved in the dynamics of the process is a good guiding line - even if the physical meaning of the conserved quantitiy is not always clear. Coming from classical theory we know that it makes a lot of sense to have a look at the conservation of mass, energy and momentum, which translate to conserved quantities as combinations of velocity, its derivatives, pressure and density. Pressure and density are not independent in these simplified models but are independent in the models Xiao studies. In the complex world of inhomogeneous equations we lose the direct concept to translate between physics and mathematics but carry over the knowledge that scale invarance and conservation are central properties of the model. It is interesting to characterize how the complex system develops with a change of properties. To have a simple idea - if it is more developing in the direction of fast flowing air or slow flowing almost solid material. One number which helps to see what types of waves one has to expect is the Mach number. It helps to seperate sound waves from fluid waves. A mathematical/physical question then is to understand the process of letting the Mach number go to zero in the model. It is not that complicated to make this work in the formulae. But the hard work is done in proving that the solutions to the family of systems of PDEs with lower and lower Mach number really tend to the solutions of the derived limit system. For example in order to measure if solutions are similar to each other (i.e. they get nearer and nearer to each other) one needs to find the norms which measure the right properties. Xian was Undergraduate & Master student at the Nanjing University in China from 2004 to 2009, where she was working with Prof. Huicheng Yin on Partial Differential Equations. She succeeded in getting the scholarship from China Scholarship Council and did her PhD within the laboratory LAMA (with Prof. Raphaël Danchin on zero-Mach number system). She was member of the University Paris-Est but followed many master courses in the programs of other Parisian universities as well. In 2013 she spent 8 months at the Charles University in Prague as Postdoc within the research project MORE. There she collaborated with Prof. Eduard Feireisl and Prof. Josef Málek on understanding non-Newtonian fluids better. After that period she returned to China and worked two years at the Academy of Mathematics & Systems Science as Postdoc within the research center NCMIS. With Prof. Ping Zhang she was working on density patch problems. Before her appointment here in Karlsruhe she already returned to Europe. 2016-2018 she was Postdoc at the University Bonn within the CRC 1060. She was mainly working with Prof. Herbert Koch on Gross-Pitaevskii equations - a special topic within dispersive equations. References Short Interview with the CRC 1173 Wave phenomena X. Liao, R. Danchin: On the wellposedness of the full low-Mach number limit system in general Besov spaces. Commun. Contemp. Math.: 14(3), 1250022, 2012. X. Liao: A global existence result for a zero Mach number system. J. Math. Fluid Mech.: 16(1), 77-103, 2014. X. Liao, E. Feireisl and J. Málek: Global weak solutions to a class of non-Newtonian compressible fluids. Math. Methods Appl. Sci.: 38(16), 3482-3494, 2015. X. Liao: On the strong solutions of the nonhomogeneous incompressible Navier-Stokes equations in a thin domain. Differential Integral Equations: 29, 167-182, 2016. X. Liao, P. Zhang: Global regularities of 2-D density patches for viscous inhomogeneous incompressible flow with general density: high regularity case, 2016.

europe china master phd global institute academy pressure prof math wave analysis studying mathematics mach prague lama rapha parisian karlsruhe undergraduate commun crc postdoc fluids xiao liao newtonian gudrun xian charles university juniorprofessor josef m systems science nanjing university navier stokes pdes partial differential equations

Poroelastische Medien

Play Episode Listen Later Feb 22, 2018 28:47

Jonathan Fröhlich hat im Juli 2017 seine Masterarbeit zum Thema "Heterogeneous Multiscale Methods for Poroelastic Media" eingereicht. Sie wurde von Professor Christian Wieners in unserem Institut betreut. Strömungs- und Transportphänomene in sogenannten porösen Medien spielen eine wichtige Rolle in einem breiten Spektrum von Bereichen und Anwendungen, wie zum Beispiel in der Landwirtschaft, der Biomedizin, der Baugeologie und der Erdöltechnik. Betrachtet man beispielsweise den Boden, so stellt man fest, dass der Sand, das Gestein oder der Kies keine homogene Masse ist mit homogenen Materialeigenschaften, sondern aus unzähligen unterschiedlich großen und in den physikalischen Eigenschaften variierenden Teilen bestehen. Die hohe Heterogenität solcher Medien führt auf eine große Komplexität, die im Modell des porösen Mediums stark vereinfacht betrachtet wird. Es liegt deshalb die Frage nahe: Wie verallgemeinert man herkömmliche Modelle für poröse Medien so, dass nicht gleich die komplette Zusammensetzung benötigt wird, aber mehr von der Struktur berücksichtigt wird? Die vorliegende Arbeit und unser Gespräch konzentrieren sich auf einen Spezialfall, nämlich die einphasige Strömung durch poroelastische Medien. Sie sind gekennzeichnet durch die Wechselwirkung zwischen der Beanspruchung der intrinisischen Struktur und der Strömung der Flüssigkeit. Konkret erzwingt die Änderung des Flüssigkeitsdrucks eine Beanspruchung des Materials, wodurch es beschleunigt und bewegt wird. Ein Beispiel hierfür ist der Blutfluß durch Adern. Das Blut verändert im Fließen ständig die konkrete Geometrie der elastisch verformbaren Adern und gleichzeitig ändern die Adern die Fließrichtung und -geschwindigkeit des Blutes. Dieser Prozeß wird mit bestimmten partiellen Differentialgleichungen (PDEs) modelliert. Jonathan verwendete das von Biot (1941) eingeführte linearisierte Modell und erweitert es zu einem quasistatischen Konsolidationsmodell für die Bodenmechanik. Solche Probleme sind charakterisiert durch die enorme Größe des betrachteten Gebietes, beispielsweise mehrere Kilometer an Flussbett. Dies steht im Kontrast zu den sehr kleineskaligen geometrischen Informationen, wie Sandkorngrößen und -formen, die einen Einfluss auf das System haben. Die standardmäßige Finite-Elemente-Methode zur numerischen Lösung dieses Systems von PDEs wird nur dann gute Ergebnisse liefern, wenn die Auflösung des Netzes wirklich extrem hoch ist. Dies würde zu nicht realisierbaren Rechenzeiten führen. Deshalb wird eine Idee von E und Engquist benutzt, die sogenannte Finite Element Heterogene Multiskalen Methode (FE-HMM) von 2003. Sie entkoppelt den heterogenen Teil und löst ihn durch ein mikroskopisch modelliertes Problem. Das makroskopische Problem braucht dann nur ein viel gröberes Netz und benutzt die Informationen aus dem mikroskopischen Teil als Daten. Mathematisch gesehen verwendet die Theorie eine schwache Formulierung mit Hilfe von Bilinearformen und sucht nach Lösungen in Sobolev-Räumen. Die passende Numerik für das makroskopische Problem ist eine gemischte Finite-Elemente-Methode für ein gestörtes Sattelpunktproblem. Deshalb müssen für Existenz und Eindeutigkeit von schwachen Lösungen bestimmte Bedingungen erfüllt sein, die der klassischen LBB-Bedingung (auch inf-sup-Bedingung genannt) ähnlich sind. Das zu lösende mikroskopische Problem ist elliptisch und wird mithilfe klassischer Homogenisierungstheorie hergeleitet, wobei zusätzliche Bedingungen zur Sicherung der Zwei-Skalen Konvergenz erfüllt werden müssen. Literatur und weiterführende Informationen Maurice A. Biot: General Theory of Three‐Dimensional Consolidation Journal of Applied Physics 12, 155, 1941. E. Weinan, Björn Engquist: The Heterogeneous Multiscale MethodsCommunications in Mathematical Sciences, Volume 1, Number 1, 87-132, 2003. M. Sahimi: Flow and Transport in Porous Media and Fractured Rock Wiley Weinheim, 2011. Assyr Abdulle e.a.: The heterogeneous multiscale method Acta Numerica Volume 21, pp. 1-87, 2012. Podcasts J. Fröhlich: Getriebeauswahl, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 028, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2014. L.L.X. Augusto: Filters, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 112, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016.

Mind the Game #2 : Devil May Cast

Deez Podcasts

Play Episode Listen Later Oct 2, 2017 82:35

Second numéro de cette émission de Mind The Game. Second numéro, un peu moins de fébrilité malgré mon expérience podcastique et de nouveau un ami...Et de longue date pour ce ping-pong !Le jeu ?Devil May Cry, l'original de Capcom qui a quasi créé un sous-genre voire un genre pour les plus grands fans, plus de 4 épisodes et un Reboot. Un démon qui pue la classe armée d'une épée et d'une paire de guns qui ont chacun un nom...Badass on vous dit !C'est qui l'invité ?Il s'agit de Mathieu. Pas podcasteur à ses heures mais un compagnon de jeu vidéo, animation japonaise et bien d'autres délires. Un acharné de la série Devil May Cry...ça aide !Pourquoi ?Parce que c'était une super excuse lors de mon passage à l'époque à Strasbourg d'ajouter à un séjour déjà fort plaisant ! Donc pourquoi ne pas se faire un kiff supplémentaire sur un jeu qui nous a fait, et fait encore, vibrer aujourd'hui.Au final ?Une émission « stylish as hell ». Classe au possible en français ! :pDes bons délires, des blagues à deux balles qu'on adore tous les deux et des réflexions nouvelles auxquelles je n'aurai pas pensé sans lui !Au menu de ce podcast autour du jeu Devil May Cry :Partie 1 00:00:00 Intro de l'invité autour de Devil May Cry et du genre qu'il a créé ;« Quelques grammes d'exploration dans un monde de hack n'slash ? »Partie 2 00:20:55Level Design et autre construction de niveaux Partie 3 00:45:25Dante Must Die : toute une histoire !Partie 4 00:58:42Du démon à la sorcière : le pré-BayonettaLes références données durant l'émission :L'émission Speed game de Nesblog sur Devil May Cry : http://www.nesblog.com/lediabledemaipleur/http://www.nesblog.com/dmcaldante/La chaine Youtube de Saurian avec les Umbran arts : https://www.youtube.com/user/SaurLa chaine Youtube de n472a, c'est en japonais mais en fouillant avec les vignettes on s'en sort;) :https://www.youtube.com/user/n472aArticle de Moossye sur Bayonetta héroïne camp-roccoco : Un article fleuve est aussi dispo dans le numéro « Girl Power » de la collection « Les cahiers du jeu vidéo » aux éditions Pixn'loveEt enfin une intervention « épilogue » dans le podcast Badass (avec plein d'autres héroïnes cultes et de chouettes chroniqueuses!) : http://podtail.com/podcast/badass/l-heroine-qui-a-change-ma-vie/Crédits musiques (dans l'ordre d'apparition)OST Devil May CryLet's rock ! (Title)OST Devil May Cry HR HM ArrangeLock & LoadGM-03 (The Idol of Time and Space)OST Devil May Cry 3 OSTDevil Never Cry (staff roll)Tous les thèmes sont la propriété de leurs compositeurs et/ou sociétés respectives.Pour info : http://vgmdb.net/album/2903http://vgmdb.net/album/2901http://vgmdb.net/album/53080Le générique est composé de sons libres de droit (dont la base de données http://shtooka.net/index.php)Le morceau Dead pixels est composé par Lukash dont vous pouvez consulter en libre tous ses albums : http://lukhash.com/ET SURTOUT TOUT COMME MOI EVENTUELLEMENT LUI FAIRE UN DON SI VOUS APPRECIEZ SON TRAVAIL ! ;)

Cerebral Fluid Flow

Play Episode Listen Later May 25, 2017 35:37

This is one of two conversations which Gudrun Thäter recorded alongside the conference Women in PDEs which took place at our faculty in Karlsruhe on 27-28 April 2017. Marie Elisabeth Rognes was one of the seven invited speakers. Marie is Chief Research Scientist at the Norwegian research laboratory Simula near Oslo. She is Head of department for Biomedical Computing there. Marie got her university education with a focus on Applied Mathematics, Mechanics and Numerical Physics as well as her PhD in Applied mathematics at the Centre for Mathematics for Applications in the Department of Mathematics at the University of Oslo. Her work is devoted to providing robust methods to solve Partial Differential Equations (PDEs) for diverse applications. On the one hand this means that from the mathematical side she works on numerical analysis, optimal control, robust Finite Element software as well as Uncertainty quantification while on the other hand she is very much interested in the modeling with the help of PDEs and in particular Mathematical models of physiological processes. These models are useful to answer What if type-questions much more easily than with the help of laboratory experiments. In our conversation we discussed one of the many applications - Cerebral fluid flow, i.e. fluid flow in the context of the human brain. Medical doctors and biologists know that the soft matter cells of the human brain are filled with fluid. Also the space between the cells contains the water-like cerebrospinal fluid. It provides a bath for human brain. The brain expands and contracts with each heartbeat and appoximately 1 ml of fluid is interchanged between brain and spinal area. What the specialists do not know is: Is there a circulation of fluid? This is especially interesting since there is no traditional lymphatic system to transport away the biological waste of the brain (this process is at work everywhere else in our body). So how does the brain get rid of its litter? There are several hyotheses: Diffusion processes, Fast flow (and transport) along the space near blood vessel, Convection. The aim of Marie's work is to numerically test these (and other) hypotheses. Basic testing starts on very idalised geometries. For the overall picture one useful simplified geometry is the annulus i.e. a region bounded by two concentric circles. For the microlevel-look a small cube can be the chosen geometry. As material law the flow in a porous medium which is based on Darcy flow is the starting point - maybe taking into account the coupling with an elastic behaviour on the boundary. The difficult non-mathematical questions which have to be answered are: How to use clinical data for estabilishing and testing models How to prescribe the forces In the near future she hopes to better understand the multiscale character of the processes. Here especially for embedding 1d- into 3d-geometry there is almost no theory available. For the project Marie has been awarded a FRIPRO Young Research Talents Grant of the Research Council of Norway (3 years - starting April 2016) and the very prestegious ERC Starting Grant (5 years starting - 2017). References M.E. Rognes: Mathematics that cures us.TEDxOslo 3 May 2017 Young academy of Norway ERC Starting Grant: Mathematical and computational foundations for modeling cerebral fluid flow 5 years P.E. Farrell e.a.: Dolfin adjoint (Open source software project) FEniCS computing platform for PDEs (Open source software project) Wikipedia on FEniCS Collection of relevant literature implemented in FEniCS

women university head young phd open medical uncertainty wikipedia norway basic applications norwegian oslo mechanics mathematics applied farrell fluid cerebral karlsruhe diffusion mathematical applied mathematics simula research council convection chief research scientist pdes finite element erc starting grant gudrun th

Cerebral Fluid Flow

Play Episode Listen Later May 25, 2017 35:37

This is one of two conversations which Gudrun Thäter recorded alongside the conference Women in PDEs which took place at our faculty in Karlsruhe on 27-28 April 2017. Marie Elisabeth Rognes was one of the seven invited speakers. Marie is Chief Research Scientist at the Norwegian research laboratory Simula near Oslo. She is Head of department for Biomedical Computing there. Marie got her university education with a focus on Applied Mathematics, Mechanics and Numerical Physics as well as her PhD in Applied mathematics at the Centre for Mathematics for Applications in the Department of Mathematics at the University of Oslo. Her work is devoted to providing robust methods to solve Partial Differential Equations (PDEs) for diverse applications. On the one hand this means that from the mathematical side she works on numerical analysis, optimal control, robust Finite Element software as well as Uncertainty quantification while on the other hand she is very much interested in the modeling with the help of PDEs and in particular Mathematical models of physiological processes. These models are useful to answer What if type-questions much more easily than with the help of laboratory experiments. In our conversation we discussed one of the many applications - Cerebral fluid flow, i.e. fluid flow in the context of the human brain. Medical doctors and biologists know that the soft matter cells of the human brain are filled with fluid. Also the space between the cells contains the water-like cerebrospinal fluid. It provides a bath for human brain. The brain expands and contracts with each heartbeat and appoximately 1 ml of fluid is interchanged between brain and spinal area. What the specialists do not know is: Is there a circulation of fluid? This is especially interesting since there is no traditional lymphatic system to transport away the biological waste of the brain (this process is at work everywhere else in our body). So how does the brain get rid of its litter? There are several hyotheses: Diffusion processes, Fast flow (and transport) along the space near blood vessel, Convection. The aim of Marie's work is to numerically test these (and other) hypotheses. Basic testing starts on very idalised geometries. For the overall picture one useful simplified geometry is the annulus i.e. a region bounded by two concentric circles. For the microlevel-look a small cube can be the chosen geometry. As material law the flow in a porous medium which is based on Darcy flow is the starting point - maybe taking into account the coupling with an elastic behaviour on the boundary. The difficult non-mathematical questions which have to be answered are: How to use clinical data for estabilishing and testing models How to prescribe the forces In the near future she hopes to better understand the multiscale character of the processes. Here especially for embedding 1d- into 3d-geometry there is almost no theory available. For the project Marie has been awarded a FRIPRO Young Research Talents Grant of the Research Council of Norway (3 years - starting April 2016) and the very prestegious ERC Starting Grant (5 years starting - 2017). References M.E. Rognes: Mathematics that cures us.TEDxOslo 3 May 2017 Young academy of Norway ERC Starting Grant: Mathematical and computational foundations for modeling cerebral fluid flow 5 years P.E. Farrell e.a.: Dolfin adjoint (Open source software project) FEniCS computing platform for PDEs (Open source software project) Wikipedia on FEniCS Collection of relevant literature implemented in FEniCS

women university head young phd open medical uncertainty wikipedia norway basic applications norwegian oslo mechanics mathematics applied farrell fluid cerebral karlsruhe diffusion mathematical applied mathematics simula research council convection chief research scientist pdes finite element erc starting grant gudrun th

Convolution Quadrature

Play Episode Listen Later May 18, 2017 30:32

This is one of two conversations which Gudrun Thäter recorded alongside the conference Women in PDEs which took place at our Department in Karlsruhe on 27-28 April 2017. Maria Lopez-Fernandez from the University La Sapienza in Rome was one of the seven invited speakers. She got her university degree at the University of Valladolid in Spain and worked as an academic researcher in Madrid and at the University of Zürich. Her field of research is numerical analyis and in particular the robust and efficient approximation of convolutions. The conversation is mainly focussed on its applications to wave scattering problems. The important questions for the numerical tools are: Consistency, stability and convergence analysis. The methods proposed by Maria are Convolution Quadrature type methods for the time discretization coupled with the boundary integral methods for the spatial discretization. Convolution Quadrature methods are based on Laplace transformation and numerical integration. They were initially mostly developed for parabolic problems and are now adapted to serve in the context of (hyperbolic) wave equations. Convolution quadrature methods introduce artificial dissipation in the computation, which stabilzes the numerics. However it would be physically more meaningful to work instead with schemes which conserve mass. She is mainly interested in fast algorithms with reduced memory requirements and adaptivity in time and space. The motivational example for her talk was the observation of severe acoustic problems inside a new building at the University of Zürich. Any conversation in the atrium made a lot of noise and if someone was speaking loud it was hard to understand by the others. An improvement was provided by specialised engineers who installed absorbing panels. From the mathematical point of view this is an nice application of the modelling and numerics of wave scattering problems. Of course, it would make a lot of sense to simulate the acoustic situation for such spaces before building them - if stable fast software for the distribution of acoustic pressure or the transport of signals was available. The mathematical challenges are high computational costs, high storage requirements and and stability problems. Due to the nonlocal nature of the equations it is also really hard to make the calculations in parallel to run faster. In addition time-adaptive methods for these types of problems were missing completely in the mathematical literature. In creating them one has to control the numerical errors with the help of a priori and a posteriori estimates which due to Maria's and others work during the last years is in principle known now but still very complicated. Also one easily runs into stability problems when changing the time step size. The acoustic pressure distribution for the new building in Zürich has been sucessfully simulated by co-workers in Zürich and Graz by using these results together with knowledge about the sound-source and deriving heuristic measures from that in order to find a sequence of time steps which keeps the problem stable and adapt to the computations effectively. There is a lot of hope to improve the performance of these tools by representing the required boundary element matrices by approximations with much sparser matrices. References M. López Fernández, S. Sauter: Generalized Convolution Quadrature with Variable Time Stepping. Part II: Algorithm and Numerical Results. Applied Numerical Mathematics, 94, pp. 88 - 105 (2015) M. López Fernández, S. Sauter: Generalized Convolution Quadrature based on Runge-Kutta Methods. Numerische Mathematik, 133 (4), pp. 734 - 779 (2016) S. Sauter, M. Schanz: Convolution Quadrature for the Wave Equation with Impedance Boundary Conditions. Journal of Computational Physics, Vol 334, pp. 442 - 459 (2017) Podcasts T. Arens: Lärmschutz, Gespräch mit S. Ritterbusch im Modellansatz Podcast, Folge 16, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2014. F. Sayas: Acoustic Scattering, Conversation with G. Thäter in the Modellansatz Podcast, Episode 58, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2016.

women university conversations spain rome journal consistency madrid vol mathematics laplace karlsruhe graz valladolid mathematik fakult sauter karlsruher institut quadrature convolution karlsruhe institute technologie kit computational physics technology kit pdes numerische mathematik gudrun th modellansatz podcast

Convolution Quadrature

Play Episode Listen Later May 18, 2017 30:32

This is one of two conversations which Gudrun Thäter recorded alongside the conference Women in PDEs which took place at our Department in Karlsruhe on 27-28 April 2017. Maria Lopez-Fernandez from the University La Sapienza in Rome was one of the seven invited speakers. She got her university degree at the University of Valladolid in Spain and worked as an academic researcher in Madrid and at the University of Zürich. Her field of research is numerical analyis and in particular the robust and efficient approximation of convolutions. The conversation is mainly focussed on its applications to wave scattering problems. The important questions for the numerical tools are: Consistency, stability and convergence analysis. The methods proposed by Maria are Convolution Quadrature type methods for the time discretization coupled with the boundary integral methods for the spatial discretization. Convolution Quadrature methods are based on Laplace transformation and numerical integration. They were initially mostly developed for parabolic problems and are now adapted to serve in the context of (hyperbolic) wave equations. Convolution quadrature methods introduce artificial dissipation in the computation, which stabilzes the numerics. However it would be physically more meaningful to work instead with schemes which conserve mass. She is mainly interested in fast algorithms with reduced memory requirements and adaptivity in time and space. The motivational example for her talk was the observation of severe acoustic problems inside a new building at the University of Zürich. Any conversation in the atrium made a lot of noise and if someone was speaking loud it was hard to understand by the others. An improvement was provided by specialised engineers who installed absorbing panels. From the mathematical point of view this is an nice application of the modelling and numerics of wave scattering problems. Of course, it would make a lot of sense to simulate the acoustic situation for such spaces before building them - if stable fast software for the distribution of acoustic pressure or the transport of signals was available. The mathematical challenges are high computational costs, high storage requirements and and stability problems. Due to the nonlocal nature of the equations it is also really hard to make the calculations in parallel to run faster. In addition time-adaptive methods for these types of problems were missing completely in the mathematical literature. In creating them one has to control the numerical errors with the help of a priori and a posteriori estimates which due to Maria's and others work during the last years is in principle known now but still very complicated. Also one easily runs into stability problems when changing the time step size. The acoustic pressure distribution for the new building in Zürich has been sucessfully simulated by co-workers in Zürich and Graz by using these results together with knowledge about the sound-source and deriving heuristic measures from that in order to find a sequence of time steps which keeps the problem stable and adapt to the computations effectively. There is a lot of hope to improve the performance of these tools by representing the required boundary element matrices by approximations with much sparser matrices. References M. López Fernández, S. Sauter: Generalized Convolution Quadrature with Variable Time Stepping. Part II: Algorithm and Numerical Results. Applied Numerical Mathematics, 94, pp. 88 - 105 (2015) M. López Fernández, S. Sauter: Generalized Convolution Quadrature based on Runge-Kutta Methods. Numerische Mathematik, 133 (4), pp. 734 - 779 (2016) S. Sauter, M. Schanz: Convolution Quadrature for the Wave Equation with Impedance Boundary Conditions. Journal of Computational Physics, Vol 334, pp. 442 - 459 (2017) Podcasts T. Arens: Lärmschutz, Gespräch mit S. Ritterbusch im Modellansatz Podcast, Folge 16, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2014. F. Sayas: Acoustic Scattering, Conversation with G. Thäter in the Modellansatz Podcast, Episode 58, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2016.

women university conversations spain rome journal consistency madrid vol mathematics laplace karlsruhe graz valladolid mathematik fakult sauter karlsruher institut quadrature convolution karlsruhe institute technologie kit computational physics technology kit pdes numerische mathematik gudrun th modellansatz podcast

Homogenization

Play Episode Listen Later Dec 1, 2016 56:00 Very Popular

Andrii Khrabustovskyi works at our faculty in the group Nonlinear Partial Differential Equations and is a member of the CRC Wave phenomena: analysis and numerics. He was born in Kharkiv in the Ukraine and finished his studies as well as his PhD at the Kharkiv National University and the Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine. He joined our faculty in 2012 as postdoc in the former Research Training Group 1294 Analysis, Simulation and Design of Nanotechnological Processes, which was active until 2014. Gudrun Thäter talked with him about one of his research interests Asymptotic analysis and homogenization of PDEs. Photonic crystals are periodic dielectric media in which electromagnetic waves from certain frequency ranges cannot propagate. Mathematically speaking this is due to gaps in the spectrum of the related differential operators. For that an interesting question is if there are gaps inbetween bands of the spectrum of operators related to wave propagation, especially on periodic geometries and with periodic coeffecicients in the operator. It is known that the spectrum of periodic selfadjoint operators has bandstructure. This means the spectrum is a locally finite union of compact intervals called bands. In general, the bands may overlap and the existence of gaps is therefore not guaranteed. A simple example for that is the spectrum of the Laplacian in which is the half axis . The classic approach to such problems in the whole space case is the Floquet–Bloch theory. Homogenization is a collection of mathematical tools which are applied to media with strongly inhomogeneous parameters or highly oscillating geometry. Roughly spoken the aim is to replace the complicated inhomogeneous by a simpler homogeneous medium with similar properties and characteristics. In our case we deal with PDEs with periodic coefficients in a periodic geometry which is considered to be infinite. In the limit of a characteristic small parameter going to zero it behaves like a corresponding homogeneous medium. To make this a bit more mathematically rigorous one can consider a sequence of operators with a small parameter (e.g. concerning cell size or material properties) and has to prove some properties in the limit as the parameter goes to zero. The optimal result is that it converges to some operator which is the right homogeneous one. If this limit operator has gaps in its spectrum then the gaps are present in the spectra of pre-limit operators (for small enough parameter). The advantages of the homogenization approach compared to the classical one with Floquet Bloch theory are: The knowledge of the limit operator is helpful and only available through homogenization. For finite domains Floquet Bloch does not work well. Though we always have a discrete spectrum we might want to have the gaps in fixed position independent of the size of our domain. Here the homogenization theory works in principle also for the bounded case (it is just a bit technical). An interesting geometry in this context is a domain with periodically distributed holes. The question arises: what happens if the sizes of holes and the period simultaneously go to zero? The easiest operator which we can study is the Laplace operator subject to the Dirichlet boundary conditions. There are three possible regimes: For holes of the same order as the period (even slightly smaller) the Dirichelet conditions on the boundary of holes dominate -- the solution for the corresponding Poisson equation tends to zero. For significantly smaller holes the influence on the holes is so small that the problem "forgets" about the influence of the holes as the parameter goes to zero. There is a borderline case which lies between cases 1 and 2. It represents some interesting effects and can explain the occurance of so-called strange terms. A traditional ansatz in homogenization works with the concept of so-called slow and fast variables. The name comes from the following observation. If we consider an infinite layer in cylindrical coordinates, then the variable r measures the distance from the origin when going "along the layer", the angle in that plane, and z is the variable which goes into the finite direction perpendicular to that plane. When we have functions then the derivative with respect to r changes the power to while the other derivatives leave that power unchanged. In the interesting case k is negative and the r-derivate makes it decreasing even faster. This leads to the name fast variable. The properties in this simple example translate as follows. For any function we will think of having a set of slow and fast variables (characteristic to the problem) and a small parameter eps and try to find u as where in our applications typically . One can formally sort through the -levels using the properties of the differential operator. The really hard part then is to prove that this formal result is indeed true by finding error estimates in the right (complicated) spaces. There are many more tools available like the technique of Tartar/Murat, who use a weak formulation with special test functions depending on the small parameter. The weak point of that theory is that we first have to know the resulat as the parameter goes to zero before we can to construct the test function. Also the concept of Gamma convergence or the unfolding trick of Cioranescu are helpful. An interesting and new application to the mathematical results is the construction of wave guides. The corresponding domain in which we place a waveguide is bounded in two directions and unbounded in one (e.g. an unbounded cylinder). Serguei Nazarov proposed to make holes in order to make gaps into the line of the spectrum for a specified wave guide. Andrii Khrabustovskyi suggests to distribute finitely many traps, which do not influence the essential spectrum but add eigenvalues. One interesting effect is that in this way one can find terms which are nonlocal in time or space and thus stand for memory effects of the material. References P. Exner and A. Khrabustovskyi: On the spectrum of narrow Neumann waveguide with periodically distributed δ′ traps, Journal of Physics A: Mathematical and Theoretical, 48 (31) (2015), 315301. A. Khrabustovskyi: Opening up and control of spectral gaps of the Laplacian in periodic domains, Journal of Mathematical Physics, 55 (12) (2014), 121502. A. Khrabustovskyi: Periodic elliptic operators with asymptotically preassigned spectrum, Asymptotic Analysis, 82 (1-2) (2013), 1-37. S.A. Nazarov, G. Thäter: Asymptotics at infinity of solutions to the Neumann problem in a sieve-type layer, Comptes Rendus Mecanique 331(1) (2003) 85-90. S.A. Nazarov: Asymptotic Theory of Thin Plates and Rods: Vol.1. Dimension Reduction and Integral Estimates. Nauchnaya Kniga: Novosibirsk, 2002.

design phd ukraine institute journal engineering analysis sciences simulation national academy neumann gamma laplace theoretical kharkiv poisson mathematically photonic homogenization asymptotic mathematical physics laplacian pdes dirichlet gudrun th

Homogenization

Play Episode Listen Later Dec 1, 2016 56:00

Andrii Khrabustovskyi works at our faculty in the group Nonlinear Partial Differential Equations and is a member of the CRC Wave phenomena: analysis and numerics. He was born in Kharkiv in the Ukraine and finished his studies as well as his PhD at the Kharkiv National University and the Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine. He joined our faculty in 2012 as postdoc in the former Research Training Group 1294 Analysis, Simulation and Design of Nanotechnological Processes, which was active until 2014. Gudrun Thäter talked with him about one of his research interests Asymptotic analysis and homogenization of PDEs. Photonic crystals are periodic dielectric media in which electromagnetic waves from certain frequency ranges cannot propagate. Mathematically speaking this is due to gaps in the spectrum of the related differential operators. For that an interesting question is if there are gaps inbetween bands of the spectrum of operators related to wave propagation, especially on periodic geometries and with periodic coeffecicients in the operator. It is known that the spectrum of periodic selfadjoint operators has bandstructure. This means the spectrum is a locally finite union of compact intervals called bands. In general, the bands may overlap and the existence of gaps is therefore not guaranteed. A simple example for that is the spectrum of the Laplacian in which is the half axis . The classic approach to such problems in the whole space case is the Floquet–Bloch theory. Homogenization is a collection of mathematical tools which are applied to media with strongly inhomogeneous parameters or highly oscillating geometry. Roughly spoken the aim is to replace the complicated inhomogeneous by a simpler homogeneous medium with similar properties and characteristics. In our case we deal with PDEs with periodic coefficients in a periodic geometry which is considered to be infinite. In the limit of a characteristic small parameter going to zero it behaves like a corresponding homogeneous medium. To make this a bit more mathematically rigorous one can consider a sequence of operators with a small parameter (e.g. concerning cell size or material properties) and has to prove some properties in the limit as the parameter goes to zero. The optimal result is that it converges to some operator which is the right homogeneous one. If this limit operator has gaps in its spectrum then the gaps are present in the spectra of pre-limit operators (for small enough parameter). The advantages of the homogenization approach compared to the classical one with Floquet Bloch theory are: The knowledge of the limit operator is helpful and only available through homogenization. For finite domains Floquet Bloch does not work well. Though we always have a discrete spectrum we might want to have the gaps in fixed position independent of the size of our domain. Here the homogenization theory works in principle also for the bounded case (it is just a bit technical). An interesting geometry in this context is a domain with periodically distributed holes. The question arises: what happens if the sizes of holes and the period simultaneously go to zero? The easiest operator which we can study is the Laplace operator subject to the Dirichlet boundary conditions. There are three possible regimes: For holes of the same order as the period (even slightly smaller) the Dirichelet conditions on the boundary of holes dominate -- the solution for the corresponding Poisson equation tends to zero. For significantly smaller holes the influence on the holes is so small that the problem "forgets" about the influence of the holes as the parameter goes to zero. There is a borderline case which lies between cases 1 and 2. It represents some interesting effects and can explain the occurance of so-called strange terms. A traditional ansatz in homogenization works with the concept of so-called slow and fast variables. The name comes from the following observation. If we consider an infinite layer in cylindrical coordinates, then the variable r measures the distance from the origin when going "along the layer", the angle in that plane, and z is the variable which goes into the finite direction perpendicular to that plane. When we have functions then the derivative with respect to r changes the power to while the other derivatives leave that power unchanged. In the interesting case k is negative and the r-derivate makes it decreasing even faster. This leads to the name fast variable. The properties in this simple example translate as follows. For any function we will think of having a set of slow and fast variables (characteristic to the problem) and a small parameter eps and try to find u as where in our applications typically . One can formally sort through the -levels using the properties of the differential operator. The really hard part then is to prove that this formal result is indeed true by finding error estimates in the right (complicated) spaces. There are many more tools available like the technique of Tartar/Murat, who use a weak formulation with special test functions depending on the small parameter. The weak point of that theory is that we first have to know the resulat as the parameter goes to zero before we can to construct the test function. Also the concept of Gamma convergence or the unfolding trick of Cioranescu are helpful. An interesting and new application to the mathematical results is the construction of wave guides. The corresponding domain in which we place a waveguide is bounded in two directions and unbounded in one (e.g. an unbounded cylinder). Serguei Nazarov proposed to make holes in order to make gaps into the line of the spectrum for a specified wave guide. Andrii Khrabustovskyi suggests to distribute finitely many traps, which do not influence the essential spectrum but add eigenvalues. One interesting effect is that in this way one can find terms which are nonlocal in time or space and thus stand for memory effects of the material. References P. Exner and A. Khrabustovskyi: On the spectrum of narrow Neumann waveguide with periodically distributed δ′ traps, Journal of Physics A: Mathematical and Theoretical, 48 (31) (2015), 315301. A. Khrabustovskyi: Opening up and control of spectral gaps of the Laplacian in periodic domains, Journal of Mathematical Physics, 55 (12) (2014), 121502. A. Khrabustovskyi: Periodic elliptic operators with asymptotically preassigned spectrum, Asymptotic Analysis, 82 (1-2) (2013), 1-37. S.A. Nazarov, G. Thäter: Asymptotics at infinity of solutions to the Neumann problem in a sieve-type layer, Comptes Rendus Mecanique 331(1) (2003) 85-90. S.A. Nazarov: Asymptotic Theory of Thin Plates and Rods: Vol.1. Dimension Reduction and Integral Estimates. Nauchnaya Kniga: Novosibirsk, 2002.

design phd ukraine institute journal engineering analysis sciences simulation national academy neumann gamma laplace theoretical kharkiv poisson mathematically photonic homogenization asymptotic mathematical physics laplacian pdes dirichlet gudrun th

PDEs (5.4)

Play Episode Listen Later Oct 17, 2016 51:10

In this lecture, Professor Trefethen discusses finite differencing in general grids and multiple space dimensions.

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PDEs (5.1)

Play Episode Listen Later Oct 17, 2016 51:18

In this lecture, Professor Trefethen discusses PDEs in science and engineering, and explicit 1D finite differences.

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PDEs (5.8)

Play Episode Listen Later Oct 17, 2016 53:32

In this concluding lecture, Professor Nick Trefethen discusses the question Who invented the great numerical algorithms?

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PDEs (5.7)

Play Episode Listen Later Oct 17, 2016 50:50

In this lecture, Professor Trefethen discusses Chebyshev spectral discretization.

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PDEs (5.6)

Play Episode Listen Later Oct 17, 2016 50:50

In this lecture, Professor Trefethen discusses Fourier, Laurent, and Chebyshev. Then, Chebyshev series and interpolants

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PDEs (5.5)

Play Episode Listen Later Oct 17, 2016 50:02

In this lecture, Professor Trefethen discusses Fourier spectral discretization and Fourier spectral discretization via FFT.

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PDEs (5.3)

Play Episode Listen Later Oct 17, 2016 52:08

In this lecture, Professor Trefethen discusses order of accuracy and reaction-diffusion equations and other stiff PDEs.

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PDEs (5.2)

Play Episode Listen Later Oct 17, 2016 49:19

In this lecture, Professor Trefethen discusses numerical instability and implicit 1D finite differences.

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Crime Prevention

Play Episode Listen Later Oct 13, 2016 40:34 Very Popular

This is the last of four conversation Gudrun had during the British Applied Mathematics Colloquium which took place 5th – 8th April 2016 in Oxford. Andrea Bertozzi from the University of California in Los Angeles (UCLA) held a public lecture on The Mathematics of Crime. She has been Professor of Mathematics at UCLA since 2003 and Betsy Wood Knapp Chair for Innovation and Creativity (since 2012). From 1995-2004 she worked mostly at Duke University first as Associate Professor of Mathematics and then as Professor of Mathematics and Physics. As an undergraduate at Princeton University she studied physics and astronomy alongside her major in mathematics and went through a Princeton PhD-program. For her thesis she worked in applied analysis and studied fluid flow. As postdoc she worked with Peter Constantin at the University of Chicago (1991-1995) on global regularity for vortex patches. But even more importantly, this was the moment when she found research problems that needed knowledge about PDEs and flow but in addition both numerical analysis and scientific computing. She found out that she really likes to collaborate with very different specialists. Today hardwork can largely be carried out on a desktop but occasionally clusters or supercomputers are necessary. The initial request to work on Mathematics in crime came from a colleague, the social scientist Jeffrey Brantingham. He works in Anthropology at UCLA and had well established contacts with the police in LA. He was looking for mathematical input on some of his problems and raised that issue with Andrea Bertozzi. Her postdoc George Mohler came up with the idea to adapt an earthquake model after a discussion with Frederic Paik Schoenberg, a world expert in that field working at UCLA. The idea is to model crimes of opportunity as being triggered by crimes that already happend. So the likelihood of new crimes can be predicted as an excitation in space and time like the shock of an earthquake. Of course, here statistical models are necessary which say how the excitement is distributed and decays in space and time. Mathematically this is a self-exciting point process. The traditional Poisson process model has a single parameter and thus, no memory - i.e. no connections to other events can be modelled. The Hawkes process builds on the Poisson process as background noise but adds new events which then are triggering events according to an excitation rate and the exponential decay of excitation over time. This is a memory effect based on actual events (not only on a likelihood) and a three parameter model. It is not too difficult to process field data, fit data to that model and make an extrapolation in time. Meanwhile the results of that idea work really well in the field. Results of field trials both in the UK and US have just been published and there is a commercial product available providing services to the police. In addition to coming up with useful ideas and having an interdisciplinary group of people committed to make them work it was necessery to find funding in order to support students to work on that topic. The first grant came from the National Science Foundation and from this time on the group included George Tita (UC Irvine) a criminology expert in LA-Gangs and Lincoln Chayes as another mathematician in the team. The practical implementation of this crime prevention method for the police is as follows: Before the policemen go out on a shift they ususally meet to divide their teams over the area they are serving. The teams take the crime prediction for that shift which is calculated by the computer model on the basis of whatever data is available up to shift. According to expected spots of crimes they especially assign teams to monitor those areas more closely. After introducing this method in the police work in Santa Cruz (California) police observed a significant reduction of 27% in crime. Of course this is a wonderful success story. Another success story involves the career development of the students and postdocs who now have permanent positions. Since this was the first group in the US to bring mathematics to police work this opened a lot of doors for young people involved. Another interesting topic in the context of Mathematics and crime are gang crime data. As for the the crime prediction model the attack of one gang on a rival gang usually triggers another event soon afterwards. A well chosen group of undergraduates already is mathematically educated enough to study the temporary distribution of gang related crime in LA with 30 street gangs and a complex net of enemies. We are speaking about hundreds of crimes in one year related to the activity of gangs. The mathematical tool which proved to be useful was a maximum liklihood penalization model again for the Hawkes process applied on the expected retaliatory behaviour. A more complex problem, which was treated in a PhD-thesis, is to single out gangs which would be probably responsable for certain crimes. This means to solve the inverse problem: We know the time and the crime and want to find out who did it. The result was published in Inverse Problems 2011. The tool was a variational model with an energy which is related to the data. The missing information is guessed and then put into the energy . In finding the best guess related to the chosen energy model a probable candidate for the crime is found. For a small number of unsolved crimes one can just go through all possible combinations. For hundreds or even several hundreds of unsolved crimes - all combinations cannot be handled. We make it easier by increasing the number of choices and formulate a continuous instead of the discrete problem, for which the optimization works with a standard gradient descent algorithm. A third topic and a third tool is Compressed sensing. It looks at sparsitiy in data like the probability distribution for crime in different parts of the city. Usually the crime rate is high in certain areas of a city and very low in others. For these sharp changes one needs different methods since we have to allow for jumps. Here the total variation enters the model as the -norm of the gradient. It promotes sparsity of edges in the solution. Before coming up with this concept it was necessary to cross-validate quite a number of times, which is computational very expensive. So instead of in hours the result is obtained in a couple minutes now. When Andrea Bertozzi was a young child she spent a lot of Sundays in the Science museum in Boston and wanted to become a scientist when grown up. The only problem was, that she could not decide which science would be the best choice since she liked everything in the museum. Today she says having chosen applied mathematics indeed she can do all science since mathematics works as a connector between sciences and opens a lot of doors. References Press coverage of Crime prevention collected Website of Mathematical and Simulation Modeling of Crime Examples for work of undergraduates M. Allenby, e.a.: A Point Process Model for Simulating Gang-on-Gang Violence, Project Report, 2010. K. Louie: Statistical Modeling of Gang Violence in Los Angeles, talk at AMS Joint meetings San Francisco, AMS Session on Mathematics in the Social Sciences, 2010] Publications of A. Bertozzi and co-workers on Crime prevention G.O. Mohler e.a.: Randomized controlled field trials of predictive policing, J. Am. Stat. Assoc., 111(512), 1399-1411, 2015. J. T. Woodworth e.a.: Nonlocal Crime Density Estimation Incorporating Housing Information, Phil. Trans. Roy. Soc. A, 372(2028), 20130403, 2014. J. Zipkin, M. B. Short & A. L. Bertozzi: Cops on the dots in a mathematical model of urban crime and police response, Discrete and Continuous Dynamical Systems B, 19(5), pp. 1479-1506, 2014. H. Hu e.a.: A Method Based on Total Variation for Network Modularity Optimization using the MBO Scheme, SIAM J. Appl. Math., 73(6), pp. 2224-2246, 2013. L.M. Smith e.a.: Adaptation of an Ecological Territorial Model to Street Gang Spatial Patterns in Los Angeles Discrete and Continuous Dynamical Systems A, 32(9), pp. 3223 - 3244, 2012. G. Mohler e.a.. (2011): Self- exciting point process modeling of crime, Journal of the American Statistical Association, 106(493):100–108, 2011. A. Stomakhin, M. Short, and A. Bertozzi: Reconstruction of missing data in social networks based on temporal patterns of interactions. Inverse Problems, 27, 2011. N. Rodriguez & A. Bertozzi: Local Existence and Uniqueness of Solutions to a PDE model for Criminal Behavior , M3AS, special issue on Mathematics and Complexity in Human and Life Sciences, Vol. 20, Issue supp01, pp. 1425-1457, 2010. Related Podcasts AMS - Mathematical Moments Podcast: MM97 - Forecasting Crime British Applied Mathematics Colloquium 2016 Special J.Dodd: Crop Growth, Conversation with G. Thäter in the Modellansatz Podcast episode 89, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2016. http://modellansatz.de/crop-growth H. Wilson: Viscoelastic Fluids, Conversation with G. Thäter in the Modellansatz Podcast episode 92, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2016. http://modellansatz.de/viscoelastic-fluids A. Hosoi: Robots, Conversation with G. Thäter in the Modellansatz Podcast, episode 108, Department for Mathematics, Karlsruhe Institute of Technologie (KIT), 2016. http://modellansatz.de/robot A. Bertozzi: Crime Prevention, Conversation with G. Thäter in the Modellansatz Podcast, episode 109, Department for Mathematics, Karlsruhe Institute of Technologie (KIT), 2016. http://modellansatz.de/crime-prevention

university california chicago conversations science los angeles san francisco phd professor innovation creativity united kingdom crime journal human oxford math ucla associate professor trans vol results rodriguez physics duke university complexity mathematics adaptation anthropology princeton university social sciences stat uniqueness life sciences publications hu national science foundation soc assoc mathematical poisson randomized hawkes gang violence gudrun crime prevention mohler discrete mathematically compressed woodworth los angeles ucla appl santa cruz california criminal behavior pde allenby american statistical association karlsruhe institute technologie kit la gangs technology kit pdes princeton phd inverse problems simulation modeling modellansatz podcast

Crime Prevention

Play Episode Listen Later Oct 13, 2016 40:34 Very Popular

This is the last of four conversation Gudrun had during the British Applied Mathematics Colloquium which took place 5th – 8th April 2016 in Oxford. Andrea Bertozzi from the University of California in Los Angeles (UCLA) held a public lecture on The Mathematics of Crime. She has been Professor of Mathematics at UCLA since 2003 and Betsy Wood Knapp Chair for Innovation and Creativity (since 2012). From 1995-2004 she worked mostly at Duke University first as Associate Professor of Mathematics and then as Professor of Mathematics and Physics. As an undergraduate at Princeton University she studied physics and astronomy alongside her major in mathematics and went through a Princeton PhD-program. For her thesis she worked in applied analysis and studied fluid flow. As postdoc she worked with Peter Constantin at the University of Chicago (1991-1995) on global regularity for vortex patches. But even more importantly, this was the moment when she found research problems that needed knowledge about PDEs and flow but in addition both numerical analysis and scientific computing. She found out that she really likes to collaborate with very different specialists. Today hardwork can largely be carried out on a desktop but occasionally clusters or supercomputers are necessary. The initial request to work on Mathematics in crime came from a colleague, the social scientist Jeffrey Brantingham. He works in Anthropology at UCLA and had well established contacts with the police in LA. He was looking for mathematical input on some of his problems and raised that issue with Andrea Bertozzi. Her postdoc George Mohler came up with the idea to adapt an earthquake model after a discussion with Frederic Paik Schoenberg, a world expert in that field working at UCLA. The idea is to model crimes of opportunity as being triggered by crimes that already happend. So the likelihood of new crimes can be predicted as an excitation in space and time like the shock of an earthquake. Of course, here statistical models are necessary which say how the excitement is distributed and decays in space and time. Mathematically this is a self-exciting point process. The traditional Poisson process model has a single parameter and thus, no memory - i.e. no connections to other events can be modelled. The Hawkes process builds on the Poisson process as background noise but adds new events which then are triggering events according to an excitation rate and the exponential decay of excitation over time. This is a memory effect based on actual events (not only on a likelihood) and a three parameter model. It is not too difficult to process field data, fit data to that model and make an extrapolation in time. Meanwhile the results of that idea work really well in the field. Results of field trials both in the UK and US have just been published and there is a commercial product available providing services to the police. In addition to coming up with useful ideas and having an interdisciplinary group of people committed to make them work it was necessery to find funding in order to support students to work on that topic. The first grant came from the National Science Foundation and from this time on the group included George Tita (UC Irvine) a criminology expert in LA-Gangs and Lincoln Chayes as another mathematician in the team. The practical implementation of this crime prevention method for the police is as follows: Before the policemen go out on a shift they ususally meet to divide their teams over the area they are serving. The teams take the crime prediction for that shift which is calculated by the computer model on the basis of whatever data is available up to shift. According to expected spots of crimes they especially assign teams to monitor those areas more closely. After introducing this method in the police work in Santa Cruz (California) police observed a significant reduction of 27% in crime. Of course this is a wonderful success story. Another success story involves the career development of the students and postdocs who now have permanent positions. Since this was the first group in the US to bring mathematics to police work this opened a lot of doors for young people involved. Another interesting topic in the context of Mathematics and crime are gang crime data. As for the the crime prediction model the attack of one gang on a rival gang usually triggers another event soon afterwards. A well chosen group of undergraduates already is mathematically educated enough to study the temporary distribution of gang related crime in LA with 30 street gangs and a complex net of enemies. We are speaking about hundreds of crimes in one year related to the activity of gangs. The mathematical tool which proved to be useful was a maximum liklihood penalization model again for the Hawkes process applied on the expected retaliatory behaviour. A more complex problem, which was treated in a PhD-thesis, is to single out gangs which would be probably responsable for certain crimes. This means to solve the inverse problem: We know the time and the crime and want to find out who did it. The result was published in Inverse Problems 2011. The tool was a variational model with an energy which is related to the data. The missing information is guessed and then put into the energy . In finding the best guess related to the chosen energy model a probable candidate for the crime is found. For a small number of unsolved crimes one can just go through all possible combinations. For hundreds or even several hundreds of unsolved crimes - all combinations cannot be handled. We make it easier by increasing the number of choices and formulate a continuous instead of the discrete problem, for which the optimization works with a standard gradient descent algorithm. A third topic and a third tool is Compressed sensing. It looks at sparsitiy in data like the probability distribution for crime in different parts of the city. Usually the crime rate is high in certain areas of a city and very low in others. For these sharp changes one needs different methods since we have to allow for jumps. Here the total variation enters the model as the -norm of the gradient. It promotes sparsity of edges in the solution. Before coming up with this concept it was necessary to cross-validate quite a number of times, which is computational very expensive. So instead of in hours the result is obtained in a couple minutes now. When Andrea Bertozzi was a young child she spent a lot of Sundays in the Science museum in Boston and wanted to become a scientist when grown up. The only problem was, that she could not decide which science would be the best choice since she liked everything in the museum. Today she says having chosen applied mathematics indeed she can do all science since mathematics works as a connector between sciences and opens a lot of doors. References Press coverage of Crime prevention collected Website of Mathematical and Simulation Modeling of Crime Examples for work of undergraduates M. Allenby, e.a.: A Point Process Model for Simulating Gang-on-Gang Violence, Project Report, 2010. K. Louie: Statistical Modeling of Gang Violence in Los Angeles, talk at AMS Joint meetings San Francisco, AMS Session on Mathematics in the Social Sciences, 2010] Publications of A. Bertozzi and co-workers on Crime prevention G.O. Mohler e.a.: Randomized controlled field trials of predictive policing, J. Am. Stat. Assoc., 111(512), 1399-1411, 2015. J. T. Woodworth e.a.: Nonlocal Crime Density Estimation Incorporating Housing Information, Phil. Trans. Roy. Soc. A, 372(2028), 20130403, 2014. J. Zipkin, M. B. Short & A. L. Bertozzi: Cops on the dots in a mathematical model of urban crime and police response, Discrete and Continuous Dynamical Systems B, 19(5), pp. 1479-1506, 2014. H. Hu e.a.: A Method Based on Total Variation for Network Modularity Optimization using the MBO Scheme, SIAM J. Appl. Math., 73(6), pp. 2224-2246, 2013. L.M. Smith e.a.: Adaptation of an Ecological Territorial Model to Street Gang Spatial Patterns in Los Angeles Discrete and Continuous Dynamical Systems A, 32(9), pp. 3223 - 3244, 2012. G. Mohler e.a.. (2011): Self- exciting point process modeling of crime, Journal of the American Statistical Association, 106(493):100–108, 2011. A. Stomakhin, M. Short, and A. Bertozzi: Reconstruction of missing data in social networks based on temporal patterns of interactions. Inverse Problems, 27, 2011. N. Rodriguez & A. Bertozzi: Local Existence and Uniqueness of Solutions to a PDE model for Criminal Behavior , M3AS, special issue on Mathematics and Complexity in Human and Life Sciences, Vol. 20, Issue supp01, pp. 1425-1457, 2010. Related Podcasts AMS - Mathematical Moments Podcast: MM97 - Forecasting Crime British Applied Mathematics Colloquium 2016 Special J.Dodd: Crop Growth, Conversation with G. Thäter in the Modellansatz Podcast episode 89, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2016. http://modellansatz.de/crop-growth H. Wilson: Viscoelastic Fluids, Conversation with G. Thäter in the Modellansatz Podcast episode 92, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2016. http://modellansatz.de/viscoelastic-fluids A. Hosoi: Robots, Conversation with G. Thäter in the Modellansatz Podcast, episode 108, Department for Mathematics, Karlsruhe Institute of Technologie (KIT), 2016. http://modellansatz.de/robot A. Bertozzi: Crime Prevention, Conversation with G. Thäter in the Modellansatz Podcast, episode 109, Department for Mathematics, Karlsruhe Institute of Technologie (KIT), 2016. http://modellansatz.de/crime-prevention

university california chicago conversations science los angeles san francisco phd professor innovation creativity united kingdom crime journal human oxford math ucla associate professor trans vol results rodriguez physics duke university complexity mathematics adaptation anthropology princeton university social sciences stat uniqueness life sciences publications hu national science foundation soc assoc mathematical poisson randomized hawkes gang violence gudrun crime prevention mohler discrete mathematically compressed woodworth los angeles ucla appl santa cruz california criminal behavior pde allenby american statistical association karlsruhe institute technologie kit la gangs technology kit pdes princeton phd inverse problems simulation modeling modellansatz podcast

Complex Geometries

Play Episode Listen Later Mar 24, 2016 32:39 Very Popular

Sandra May works at the Seminar for Applied Mathematics at ETH Zurich and visited Karlsruhe for a talk at the CRC Wave phenomena. Her research is in numerical analysis, more specifically in numerical methods for solving PDEs. The focus is on hyperbolic PDEs and systems of conservation laws. She is both interested in theoretical aspects (such as proving stability of a certain method) and practical aspects (such as working on high-performance implementations of algorithms). Sandra May graduated with a PhD in Mathematics from the Courant Institute of Mathematical Sciences (part of New York University) under the supervision of Marsha Berger. She likes to look back on the multicultural working and learning experience there. We talked about the numerical treatment of complex geometries. The main problem is that it is difficult to automatically generate grids for computations on the computer if the shape of the boundary is complex. Examples for such problems are the simulation of airflow around airplanes, trucks or racing cars. Typically, the approach for these flow simulations is to put the object in the middle of the grid. Appropriate far-field boundary conditions take care of the right setting of the finite computational domain on the outer boundary (which is cut from an infinite model). Typically in such simulations one is mainly interested in quantities close to the boundary of the object. Instead of using an unstructured or body-fitted grid, Sandra May is using a Cartesian embedded boundary approach for the grid generation: the object with complex geometry is cut out of a Cartesian background grid, resulting in so called cut cells where the grid intersects the object and Cartesian cells otherwise. This approach is fairly straightforward and fully automatic, even for very complex geometries. The price to pay comes in shape of the cut cells which need special treatment. One particular challenge is that the cut cells can become arbitrarily small since a priori their size is not bounded from below. Trying to eliminate cut cells that are too small leads to additional problems which conflict with the goal of a fully automatic grid generation in 3d, which is why Sandra May keeps these potentially very small cells and develops specific strategies instead. The biggest challenge caused by the small cut cells is the small cell problem: easy to implement (and therefore standard) explicit time stepping schemes are only stable if a CFL condition is satisfied; this condition essentially couples the time step length to the spatial size of the cell. Therefore, for the very small cut cells one would need to choose tiny time steps, which is computationally not feasible. Instead, one would like to choose a time step appropriate for the Cartesian cells and use this same time step on cut cells as well. Sandra May and her co-workers have developed a mixed explicit implicit scheme for this purpose: to guarantee stability on cut cells, an implicit time stepping method is used on cut cells. This idea is similar to the approach of using implicit time stepping schemes for solving stiff systems of ODEs. As implicit methods are computationally more expensive than explicit methods, the implicit scheme is only used where needed (namely on cut cells and their direct neighbors). In the remaining part of the grid (the vast majority of the grid cells), a standard explicit scheme is used. Of course, when using different schemes on different cells, one needs to think about a suitable way of coupling them. The mixed explicit implicit scheme has been developed in the context of Finite Volume methods. The coupling has been designed with the goals of mass conservation and stability and is based on using fluxes to couple the explicit and the implicit scheme. This way, mass conservation is guaranteed by construction (no mass is lost). In terms of stability of the scheme, it can be shown that using a second-order explicit scheme coupled to a first-order implicit scheme by flux bounding results in a TVD stable method. Numerical results for coupling a second-order explicit scheme to a second-order implicit scheme show second-order convergence in the L^1 norm and between first- and second-order convergence in the maximum norm along the surface of the object in two and three dimensions. We also talked about the general issue of handling shocks in numerical simulations properly: in general, solutions to nonlinear hyperbolic systems of conservation laws such as the Euler equations contain shocks and contact discontinuities, which in one dimension express themselves as jumps in the solution. For a second-order finite volume method, typically slopes are reconstructed on each cell. If one reconstructed these slopes using e.g. central difference quotients in one dimension close to shocks, this would result in oscillations and/or unphysical results (like negative density). To avoid this, so called slope limiters are typically used. There are two main ingredients to a good slope limiter (which is applied after an initial polynomial based on interpolation has been generated): first, the algorithm (slope limiter) needs to detect whether the solution in this cell is close to a shock or whether the solution is smooth in the neighborhood of this cell. If the algorithm thinks that the solution is close to a shock, the algorithm reacts and adjusts the reconstruted polynomial appropriately. Otherwise, it sticks with the polynomial based on interpolation. One commonly used way in one dimension to identify whether one is close to a shock or not is to compare the values of a right-sided and a left-sided difference quotient. If they differ too much the solution is (probably) not smooth there. Good reliable limiters are really difficult to find. Literature and additional material S. May, M. Berger: An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes, Preprint available as SAM report, number 2015-44, 2015. S. May, M. Berger: A mixed explicit implicit time stepping scheme for Cartesian embedded boundary meshes, Finite Volumes for Complex Applications VII - Methods and Theoretical Aspects, pp. 393-400, Springer, 2014. S. May, M. Berger: Two-dimensional slope limiters for finite volume schemes on non-coordinate-aligned meshes, SIAM J. Sci. Comput. 35 (5) pp. A2163-A2187, 2013.

phd complex literature new york university mathematics seminar springer cfl geometry karlsruhe applied mathematics eth zurich euler cartesian odes numerical mathematical sciences tvd courant institute comput pdes

Complex Geometries

Play Episode Listen Later Mar 24, 2016 32:39

Sandra May works at the Seminar for Applied Mathematics at ETH Zurich and visited Karlsruhe for a talk at the CRC Wave phenomena. Her research is in numerical analysis, more specifically in numerical methods for solving PDEs. The focus is on hyperbolic PDEs and systems of conservation laws. She is both interested in theoretical aspects (such as proving stability of a certain method) and practical aspects (such as working on high-performance implementations of algorithms). Sandra May graduated with a PhD in Mathematics from the Courant Institute of Mathematical Sciences (part of New York University) under the supervision of Marsha Berger. She likes to look back on the multicultural working and learning experience there. We talked about the numerical treatment of complex geometries. The main problem is that it is difficult to automatically generate grids for computations on the computer if the shape of the boundary is complex. Examples for such problems are the simulation of airflow around airplanes, trucks or racing cars. Typically, the approach for these flow simulations is to put the object in the middle of the grid. Appropriate far-field boundary conditions take care of the right setting of the finite computational domain on the outer boundary (which is cut from an infinite model). Typically in such simulations one is mainly interested in quantities close to the boundary of the object. Instead of using an unstructured or body-fitted grid, Sandra May is using a Cartesian embedded boundary approach for the grid generation: the object with complex geometry is cut out of a Cartesian background grid, resulting in so called cut cells where the grid intersects the object and Cartesian cells otherwise. This approach is fairly straightforward and fully automatic, even for very complex geometries. The price to pay comes in shape of the cut cells which need special treatment. One particular challenge is that the cut cells can become arbitrarily small since a priori their size is not bounded from below. Trying to eliminate cut cells that are too small leads to additional problems which conflict with the goal of a fully automatic grid generation in 3d, which is why Sandra May keeps these potentially very small cells and develops specific strategies instead. The biggest challenge caused by the small cut cells is the small cell problem: easy to implement (and therefore standard) explicit time stepping schemes are only stable if a CFL condition is satisfied; this condition essentially couples the time step length to the spatial size of the cell. Therefore, for the very small cut cells one would need to choose tiny time steps, which is computationally not feasible. Instead, one would like to choose a time step appropriate for the Cartesian cells and use this same time step on cut cells as well. Sandra May and her co-workers have developed a mixed explicit implicit scheme for this purpose: to guarantee stability on cut cells, an implicit time stepping method is used on cut cells. This idea is similar to the approach of using implicit time stepping schemes for solving stiff systems of ODEs. As implicit methods are computationally more expensive than explicit methods, the implicit scheme is only used where needed (namely on cut cells and their direct neighbors). In the remaining part of the grid (the vast majority of the grid cells), a standard explicit scheme is used. Of course, when using different schemes on different cells, one needs to think about a suitable way of coupling them. The mixed explicit implicit scheme has been developed in the context of Finite Volume methods. The coupling has been designed with the goals of mass conservation and stability and is based on using fluxes to couple the explicit and the implicit scheme. This way, mass conservation is guaranteed by construction (no mass is lost). In terms of stability of the scheme, it can be shown that using a second-order explicit scheme coupled to a first-order implicit scheme by flux bounding results in a TVD stable method. Numerical results for coupling a second-order explicit scheme to a second-order implicit scheme show second-order convergence in the L^1 norm and between first- and second-order convergence in the maximum norm along the surface of the object in two and three dimensions. We also talked about the general issue of handling shocks in numerical simulations properly: in general, solutions to nonlinear hyperbolic systems of conservation laws such as the Euler equations contain shocks and contact discontinuities, which in one dimension express themselves as jumps in the solution. For a second-order finite volume method, typically slopes are reconstructed on each cell. If one reconstructed these slopes using e.g. central difference quotients in one dimension close to shocks, this would result in oscillations and/or unphysical results (like negative density). To avoid this, so called slope limiters are typically used. There are two main ingredients to a good slope limiter (which is applied after an initial polynomial based on interpolation has been generated): first, the algorithm (slope limiter) needs to detect whether the solution in this cell is close to a shock or whether the solution is smooth in the neighborhood of this cell. If the algorithm thinks that the solution is close to a shock, the algorithm reacts and adjusts the reconstruted polynomial appropriately. Otherwise, it sticks with the polynomial based on interpolation. One commonly used way in one dimension to identify whether one is close to a shock or not is to compare the values of a right-sided and a left-sided difference quotient. If they differ too much the solution is (probably) not smooth there. Good reliable limiters are really difficult to find. Literature and additional material S. May, M. Berger: An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes, Preprint available as SAM report, number 2015-44, 2015. S. May, M. Berger: A mixed explicit implicit time stepping scheme for Cartesian embedded boundary meshes, Finite Volumes for Complex Applications VII - Methods and Theoretical Aspects, pp. 393-400, Springer, 2014. S. May, M. Berger: Two-dimensional slope limiters for finite volume schemes on non-coordinate-aligned meshes, SIAM J. Sci. Comput. 35 (5) pp. A2163-A2187, 2013.

phd complex literature new york university mathematics seminar springer cfl geometry karlsruhe applied mathematics eth zurich euler cartesian odes numerical mathematical sciences tvd courant institute comput pdes

Electrodynamics

Play Episode Listen Later Oct 29, 2015 29:16 Very Popular

This episode discusses the Born-Infeld model for Electromagnetodynamics. Here, the standard model are the Maxwell equations coupling the interaction of magnetic and electric field with the help of a system of partial differential equations. This is a well-understood classical system. But in this classical model, one serious drawback is that the action of a point charge (which is represented by a Dirac measure on the right-hand side) leads to an infinite energy in the electric field which physically makes no sense. On the other hand, it should be possible to study the electric field of point charges since this is how the electric field is created. One solution for this challenge is to slightly change the point of view in a way similar to special relativity theory of Einstein. There, instead of taking the momentum () as preserved quantity and Lagrange parameter the Lagrangian is changed in a way that the bound for the velocity (in relativity the speed of light) is incorporated in the model. In the electromagnetic model, the Lagrangian would have to restrict the intensity of the fields. This was the idea which Borne and Infeld published already at the beginning of the last century. For the resulting system it is straightforward to calculate the fields for point charges. But unfortunately it is impossible to add the fields for several point charges (no superposition principle) since the resulting theory (and the PDE) are nonlinear. Physically this expresses, that the point charges do not act independently from each other but it accounts for certain interaction between the charges. Probably this interaction is really only important if charges are near enough to each other and locally it should be only influenced by the charge nearest. But it has not been possible to prove that up to now. The electrostatic case is elliptic but has a singularity at each point charge. So no classical regularity results are directly applicable. On the other hand, there is an interesting interplay with geometry since the PDE occurs as the mean curvature equation of hypersurfaces in the Minkowski space in relativity. The evolution problem is completely open. In the static case we have existence and uniqueness without really looking at the PDEs from the way the system is built. The PDE should provide at least qualitative information on the electric field. So if, e.g., there is a positive charge there could be a maximum of the field (for negative charges a minimum - respectively), and we would expect the field to be smooth outside these singular points. So a Lipschitz regular solution would seem probable. But it is open how to prove this mathematically. A special property is that the model has infinitely many inherent scales, namely all even powers of the gradient of the field. So to understand maybe asymptotic limits in theses scales could be a first interesting step. Denis Bonheure got his mathematical education at the Free University of Brussels and is working there as Professor of Mathematics at the moment. Literature and additional material M. Kiessling: Electromagnetic Field Theory Without Divergence Problems 1, The Born Legacy, Journal of Statistical Physics, Volume 116, Issue 1, pp 1057-1122, 2004. M. Kiessling: Electromagnetic Field Theory Without Divergence Problems 2, A Least Invasively Quantized Theory, Journal of Statistical Physics, Volume 116, Issue 1, pp 1123-1159, 2004. M. Kiessling: On the motion of point defects in relativistic fields, in Quantum Field Theory and Gravity, Conceptual and Mathematical Advances in the Search for a Unified Framework, Finster e.a. (ed.), Springer, 2012. Y. Brenier: Some Geometric PDEs Related to Hydrodynamics and Electrodynamics, ICM Vol. III pp 761--772, 2002.

Electrodynamics

Play Episode Listen Later Oct 29, 2015 29:16

This episode discusses the Born-Infeld model for Electromagnetodynamics. Here, the standard model are the Maxwell equations coupling the interaction of magnetic and electric field with the help of a system of partial differential equations. This is a well-understood classical system. But in this classical model, one serious drawback is that the action of a point charge (which is represented by a Dirac measure on the right-hand side) leads to an infinite energy in the electric field which physically makes no sense. On the other hand, it should be possible to study the electric field of point charges since this is how the electric field is created. One solution for this challenge is to slightly change the point of view in a way similar to special relativity theory of Einstein. There, instead of taking the momentum () as preserved quantity and Lagrange parameter the Lagrangian is changed in a way that the bound for the velocity (in relativity the speed of light) is incorporated in the model. In the electromagnetic model, the Lagrangian would have to restrict the intensity of the fields. This was the idea which Borne and Infeld published already at the beginning of the last century. For the resulting system it is straightforward to calculate the fields for point charges. But unfortunately it is impossible to add the fields for several point charges (no superposition principle) since the resulting theory (and the PDE) are nonlinear. Physically this expresses, that the point charges do not act independently from each other but it accounts for certain interaction between the charges. Probably this interaction is really only important if charges are near enough to each other and locally it should be only influenced by the charge nearest. But it has not been possible to prove that up to now. The electrostatic case is elliptic but has a singularity at each point charge. So no classical regularity results are directly applicable. On the other hand, there is an interesting interplay with geometry since the PDE occurs as the mean curvature equation of hypersurfaces in the Minkowski space in relativity. The evolution problem is completely open. In the static case we have existence and uniqueness without really looking at the PDEs from the way the system is built. The PDE should provide at least qualitative information on the electric field. So if, e.g., there is a positive charge there could be a maximum of the field (for negative charges a minimum - respectively), and we would expect the field to be smooth outside these singular points. So a Lipschitz regular solution would seem probable. But it is open how to prove this mathematically. A special property is that the model has infinitely many inherent scales, namely all even powers of the gradient of the field. So to understand maybe asymptotic limits in theses scales could be a first interesting step. Denis Bonheure got his mathematical education at the Free University of Brussels and is working there as Professor of Mathematics at the moment. Literature and additional material M. Kiessling: Electromagnetic Field Theory Without Divergence Problems 1, The Born Legacy, Journal of Statistical Physics, Volume 116, Issue 1, pp 1057-1122, 2004. M. Kiessling: Electromagnetic Field Theory Without Divergence Problems 2, A Least Invasively Quantized Theory, Journal of Statistical Physics, Volume 116, Issue 1, pp 1123-1159, 2004. M. Kiessling: On the motion of point defects in relativistic fields, in Quantum Field Theory and Gravity, Conceptual and Mathematical Advances in the Search for a Unified Framework, Finster e.a. (ed.), Springer, 2012. Y. Brenier: Some Geometric PDEs Related to Hydrodynamics and Electrodynamics, ICM Vol. III pp 761--772, 2002.

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