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59 episodes with partial differential equations
2 episodes with partial differential equations
2 episodes with partial differential equations
4 episodes with partial differential equations
A 17-year-old student recently amazed millions of people worldwide by ranking 12th in the preliminary round of a global math competition, despite being up against formidable rivals from top universities, her own mentor and even artificial intelligence.最近，一名17岁的中专生姜萍在阿里巴巴全球数学竞赛初赛中排名第12位，引发网友热议。她的强大对手有来自顶尖大学的学生、她的导师甚至人工智能。Jiang Ping, who is majoring in fashion design at Lianshui Secondary Vocational School in Huai'an, Jiangsu province, taught herself advanced mathematics for about two years, and then made it to the finals of the 2024 Alibaba Global Mathematics Competition with a high score of 93 out of 120.江苏涟水中等专业学校服装设计专业的姜萍自学了两年的高等数学，以93分（满分120分）的高分入围2024年阿里巴巴全球数学竞赛的决赛。She became the first vocational school student to reach the finals and the only girl among the top 30 contestants. Most of the finalists, totaling around 800, are from prestigious institutions such as Peking University, Tsinghua University, MIT and Princeton University. 她成为第一位进入决赛的中专在读生，也是前30名参赛者中唯一的女生。大多数决赛选手来自北京大学、清华大学、麻省理工学院和普林斯顿大学等著名学府，总计约800人。Vocational schools like the one Jiang attends usually focus on providing students with practical knowledge and training related to specific professions. 姜萍所就读的这类中等职业学校通常专注于为学生提供特定专业的实践知识和培训。Jiang's story, which has touched and inspired countless internet users, clearly demonstrates that one's potential does not necessarily depend on one's educational background, and that dreams do come true if one is diligent and persistent.姜萍的故事感动和启发了无数网民，人的潜力并不一定取决于教育背景，只要勤奋和坚持，梦想终会成真。Jiang's keen understanding of numbers began in junior high school, when she was easily able to solve complex math problems. Most students in China learn advanced math in college.她对数字的敏锐理解始于初中，当时她能够轻松解决复杂的数学难题。而大部分中国学生大学才开始学习高等数学。"I lean toward subjects such as advanced math, as they spark my desire to explore," she said. "I enjoy the step-by-step process of mathematical deductions, and reaching the desired result brings me great joy." Jiang said she believes that regardless of which subject she is studying, fashion design or advanced mathematics, interest and effort are both crucial.“我倾向学习高等数学，因为它们激发了我的探索欲望。我喜欢循序渐进的数学演绎，得到我想证明的让我觉得很有成就感。”姜萍说，她认为无论学习是学习服装设计还是高等数学，兴趣和努力都是至关重要的。The teen, who spends most of her spare time solving math problems, keeps an English dictionary handy, so that language doesn't become a barrier when she's learning from Lawrence C. Evans' book Partial Differential Equations. 姜萍将课余时间花在解决数学问题上，她随身携带英语词典，在自学（PDE）《偏微分方程》时，英语不会成为她数学道路上的障碍。All of Jiang's books are full of notes, as she hopes to attend college someday and further explore her interest in math.她所有的书都写满了笔记，因为她希望能考上大学，进一步探索数学世界。Wang Runqiu, a teacher at the vocational school, recognized Jiang's talent in mathematics and recommended books to her on the subject. He offered her personal guidance and encouraged her to give the Alibaba competition a shot.Wang, who himself ranked 125th in the preliminary round of the contest, said, "I want to help young people as much as possible, and let them know they can have a different future."姜萍的数学老师王闰秋在本次竞赛初赛中排名第125，他认可了姜萍在数学方面的天赋，并向她推荐了这方面的书籍，提供辅导，并鼓励她去尝试阿里巴巴全球数学竞赛。“我想尽力帮助年轻人，他们能够拥有不同的未来。”王闰秋说。Multiple top universities congratulated Jiang on her success, and some even encouraged her to pursue higher studies at their campus. 许多顶尖大学都对姜萍表示祝贺，甚至鼓励她在进入本校深造。In a post addressed to Jiang on Sina Weibo on Friday, Shanghai-based Donghua University wrote: "Welcome to apply to Donghua University!" Two photos uploaded with the post showed the university's leading position in China in both fashion design and math.6月14日，东华大学在新浪微博上发给姜萍的帖子中写道：“欢迎申请东华大学！”随帖上传的两张图片显示了其服装设计和数学在中国大学名列前茅。Jiangsu University, based in Zhenjiang, Jiangsu, also welcomed Jiang through Weibo. Tongji University in Shanghai said on Weibo: "Every effort will bring its own rewards."位于江苏镇江的江苏大学也通过微博欢迎姜萍。同济大学在微博上发布：“每一次努力都会带来回报。”Jiang's story has also received widespread attention on social media. A hashtag item about her, which read "In a life not defined by others, anyone can be a dark horse", garnered more than 16 million views as of Sunday. 姜萍的故事在社交媒体上也受到了广泛关注。关于她的话题标签写着“#在不受他人定义的生活中，任何人都可以成为一匹黑马#”。截至6月16日，该话题已获得超过1600万的阅读量。The annual Alibaba math competition, which started in 2018, is open to all math enthusiasts regardless of age and background. Tens of thousands of people from across the globe participate in it every year.一年一度的阿里巴巴数学竞赛始于2018年，向所有数学爱好者开放，不分年龄和背景。每年，数万名来自世界各地的数学爱好者参与其中。This time, the competition was open to AI, but it failed to enter the finals.本次比赛也对人工智能（AI）开放，但AI未能进入决赛。The competition's organizing committee told JSTV.com that questions asked in the preliminary round were equivalent to undergraduate level, while those in the final round would be comparable to doctoral programs. The eight-hour final round is scheduled for Saturday, and all participants will answer online.比赛组委会告诉江苏卫视，初赛的问题相当于本科水平，而最后一轮的问题与博士课程相当。八小时的决赛将于6月22日线上进行，所有参赛者将线上作答。This year's competition will select five gold medalists, 10 silver medalists, 20 bronze medalists and 50 excellent award winners. The total prize money is $560,000, according to the competition's website. 今年的比赛将诞生金奖5人，银奖10人，铜奖20人 ，优秀奖50人。比赛的总奖金高达400万元。Alibaba Global Mathematics Competitionn.阿里巴巴全球数学竞赛vocational schooln.中等职业学校
Summary All data systems are subject to the "garbage in, garbage out" problem. For machine learning applications bad data can lead to unreliable models and unpredictable results. Anomalo is a product designed to alert on bad data by applying machine learning models to various storage and processing systems. In this episode Jeremy Stanley discusses the various challenges that are involved in building useful and reliable machine learning models with unreliable data and the interesting problems that they are solving in the process. Announcements Hello and welcome to the Machine Learning Podcast, the podcast about machine learning and how to bring it from idea to delivery. Your host is Tobias Macey and today I'm interviewing Jeremy Stanley about his work at Anomalo, applying ML to the problem of data quality monitoring Interview Introduction How did you get involved in machine learning? Can you describe what Anomalo is and the story behind it? What are some of the ML approaches that you are using to address challenges with data quality/observability? What are some of the difficulties posed by your application of ML technologies on data sets that you don't control? How does the scale and quality of data that you are working with influence/constrain the algorithmic approaches that you are using to build and train your models? How have you implemented the infrastructure and workflows that you are using to support your ML applications? What are some of the ways that you are addressing data quality challenges in your own platform? What are the opportunities that you have for dogfooding your product? What are the most interesting, innovative, or unexpected ways that you have seen Anomalo used? What are the most interesting, unexpected, or challenging lessons that you have learned while working on Anomalo? When is Anomalo the wrong choice? What do you have planned for the future of Anomalo? Contact Info @jeremystan (https://twitter.com/jeremystan) on Twitter LinkedIn (https://www.linkedin.com/in/jeremystanley/) Parting Question From your perspective, what is the biggest barrier to adoption of machine learning today? Closing Announcements Thank you for listening! Don't forget to check out our other shows. The Data Engineering Podcast (https://www.dataengineeringpodcast.com) covers the latest on modern data management. Podcast.__init__ () covers the Python language, its community, and the innovative ways it is being used. Visit the site (https://www.themachinelearningpodcast.com) to subscribe to the show, sign up for the mailing list, and read the show notes. If you've learned something or tried out a project from the show then tell us about it! Email hosts@themachinelearningpodcast.com (mailto:hosts@themachinelearningpodcast.com)) with your story. To help other people find the show please leave a review on iTunes (https://podcasts.apple.com/us/podcast/the-machine-learning-podcast/id1626358243) and tell your friends and co-workers Links Anomalo (https://www.anomalo.com/) Data Engineering Podcast Episode (https://www.dataengineeringpodcast.com/anomalo-data-quality-platform-episode-256/) Partial Differential Equations (https://en.wikipedia.org/wiki/Partial_differential_equation) Neural Network (https://en.wikipedia.org/wiki/Neural_network) Neural Networks For Pattern Recognition (https://amzn.to/3k0Mpv8) by Christopher M. Bishop (affiliate link) Gradient Boosted Decision Trees (https://developers.google.com/machine-learning/decision-forests/intro-to-gbdt) Shapley Values (https://christophm.github.io/interpretable-ml-book/shapley.html) Sentry (https://sentry.io) dbt (https://www.getdbt.com/) Altair (https://altair-viz.github.io/) The intro and outro music is from Hitman's Lovesong feat. Paola Graziano (https://freemusicarchive.org/music/The_Freak_Fandango_Orchestra/Tales_Of_A_Dead_Fish/Hitmans_Lovesong/) by The Freak Fandango Orchestra (http://freemusicarchive.org/music/The_Freak_Fandango_Orchestra/)/CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/)
Andrés is an incoming 4th year PhD student in the Department of Mathematics. His research broadly consist in how to best numerically approximate solutions for Partial Differential Equations that appear in Mathematical-Physics. He investigates both theory and computational algorithms. He is also an international student coming from Colombia and the president of Comunidad Latinoamericana at MSU (CLA), CLA is a student organization that promotes culture and community for international students from across Latin America. (@comunidadmsu on Facebook and Instagram, and the website is https://comunidadmsu.wixsite.com/comunidad-msu) To keep up to date with WaMPS updates, you can follow @msuwamps on Instagram, Twitter, Facebook, or visit our website https://www.wamps.org. If you would like to learn more about graduate school in physics and astronomy at MSU, check out https://pa.msu.edu If you would like to leave comments, questions, or recommend someone to be interviewed on Journeys of Scientists, you can email WaMPS outreach coordinator Bryan at stanl142@msu.edu
Tune in to enjoy host Julie Mochan talk in depth to Nick Scalzo, CEO of RiskPro's parent company about the origin and development of this RegTech solution. Nick then shares quite a bit about his history growing up in Southern California,and the Orange County punk music scene. There are lots of shout outs to many people that have influenced and helped Nick all along the way to his current successes that he shares with his Co-CEO Megan Meade! Click for More About RiskPro Office: 949-259-6928 2077 West Coast Highway, Suite A Newport Beach, CA 92663 solutions@riskproadvisor.com Talk to Julie Here or email me: JulieM@tpfg.com Mentioned in this Podcast: Josh Emanuel, Chief Investment Officer for Wilshire Funds Management In his role as CIO, Mr. Emanuel leads the investment activities of Wilshire Funds Management, including asset allocation, manager research, portfolio management, and investment research. Mr. Emanuel also chairs the Wilshire Funds Management Investment Committee. Dr. Alfonso Agnew, Professor and Chair, Mathematics, California State University Fullerton Courses: Vector and Tensor Analysis, General Relativity, Partial Differential Equations, Linear Algebra and Ordinary Differential Equations Research Areas: Mathematical issues in classical and quantum gravity -- General Relativity Theory and gravitational radiation, curved space quantum field theory, twistor theory, Non-Hausdorff spaces This recording has been prepared and made available by RiskPro® to be used for information purposes only. RiskPro® is an investment risk profiling and portfolio construction software as a service platform developed by ProTools, LLC (“ProTools”). The information contained herein, including any expressions of opinion, has been obtained from or is based on sources believed to be reliable but its accuracy or completeness is not guaranteed and is subject to change without notice. Any expressions of opinions reflect the views of the speakers and are not necessarily those of ProTools or its affiliates. ProTools does not provide investment, tax or legal advice. Investors should consult their financial, tax or legal professionals before investing. Any third parties mentioned in the podcast have no affiliation with the Pacific Financial Group, Inc. or ProTools, LLC.
In den nächsten Wochen bis zum 20.2.2020 möchte Anna Hein, Studentin der Wissenschaftskommunikation am KIT, eine Studie im Rahmen ihrer Masterarbeit über den Podcast Modellansatz durchführen. Dazu möchte sie gerne einige Interviews mit Ihnen, den Hörerinnen und Hörern des Podcast Modellansatz führen, um herauszufinden, wer den Podcast hört und wie und wofür er genutzt wird. Die Interviews werden anonymisiert und werden jeweils circa 15 Minuten in Anspruch nehmen. Für die Teilnahme an der Studie können Sie sich bis zum 20.2.2020 unter der Emailadresse studie.modellansatz@web.de bei Anna Hein melden. Wir würden uns sehr freuen, wenn sich viele Interessenten melden würden. In the coming weeks until February 20, 2020, Anna Hein, student of science communication at KIT, intends to conduct a study on the Modellansatz Podcast within her master's thesis. For this purpose, she would like to conduct some interviews with you, the listeners of the Modellansatz Podcast, to find out who listens to the podcast and how and for what purpose it is used. The interviews will be anonymous and will take about 15 minutes each. To participate in the study, you can register with Anna Hein until 20.2.2020 at studie.modellansatz@web.de . We would be very pleased if many interested parties would contact us. This is the second of three conversation recorded Conference on mathematics of wave phenomena 23-27 July 2018 in Karlsruhe. Gudrun is in conversation with Mariana Haragus about Benard-Rayleigh problems. On the one hand this is a much studied model problem in Partial Differential Equations. There it has connections to different fields of research due to the different ways to derive and read the stability properties and to work with nonlinearity. On the other hand it is a model for various applications where we observe an interplay between boyancy and gravity and for pattern formation in general. An everyday application is the following: If one puts a pan with a layer of oil on the hot oven (in order to heat it up) one observes different flow patterns over time. In the beginning it is easy to see that the oil is at rest and not moving at all. But if one waits long enough the still layer breaks up into small cells which makes it more difficult to see the bottom clearly. This is due to the fact that the oil starts to move in circular patterns in these cells. For the problem this means that the system has more than one solutions and depending on physical parameters one solution is stable (and observed in real life) while the others are unstable. In our example the temperature difference between bottom and top of the oil gets bigger as the pan is heating up. For a while the viscosity and the weight of the oil keep it still. But if the temperature difference is too big it is easier to redistribute the different temperature levels with the help of convection of the oil. The question for engineers as well as mathematicians is to find the point where these convection cells evolve in theory in order to keep processes on either side of this switch. In theory (not for real oil because it would start to burn) for even bigger temperature differences the original cells would break up into even smaller cells to make the exchange of energy faster. In 1903 Benard did experiments similar to the one described in the conversation which fascinated a lot of his colleagues at the time. The equations where derived a bit later and already in 1916 Lord Rayleigh found the 'switch', which nowadays is called the critical Rayleigh number. Its size depends on the thickness of the configuration, the viscositiy of the fluid, the gravity force and the temperature difference. Only in the 1980th it became clear that Benards' experiments and Rayleigh's analysis did not really cover the same problem since in the experiment the upper boundary is a free boundary to the surrounding air while Rayleigh considered fixed boundaries. And this changes the size of the critical Rayleigh number. For each person doing experiments it is also an observation that the shape of the container with small perturbations in the ideal shape changes the convection patterns. Maria does study the dynamics of nonlinear waves and patterns. This means she is interested in understanding processes which change over time. Her main questions are: Existence of observed waves as solutions of the equations The stability of certain types of solutions How is the interaction of different waves She treats her problems with the theory of dynamical systems and bifurcations. The simplest tools go back to Poincaré when understanding ordinary differential equations. One could consider the partial differential equations to be the evolution in an infinite dimensional phase space. Here, in the 1980s, Klaus Kirchgässner had a few crucial ideas how to construct special solutions to nonlinear partial differential equations. It is possible to investigate waterwave problems which are dispersive equations as well as flow problems which are dissipative. Together with her colleagues in Besancon she is also very keen to match experiments for optical waves with her mathematical analysis. There Mariana is working with a variant of the Nonlinear Schrödinger equation called Lugiato-Lefever Equation. It has many different solutions, e.g. periodic solutions and solitons. Since 2002 Mariana has been Professor in Besancon (University of Franche-Comté, France). Before that she studied and worked in a lot of different places, namely in Bordeaux, Stuttgart, Bucharest, Nice, and Timisoara. References V.A. Getling: Rayleigh-Bénard Convection Structures and Dynamics, Advanced Series in Nonlinear Dynamics, Volume 11, World Scientific, Oxford (1998) P. H. Rabinowitz: Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rational Mech. Anal. (1968) 29: 32. M. Haragus and G. Iooss: Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems. Universitext. Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011. Newell, Alan C. Solitons in mathematics and physics. CBMS-NSF Regional Conference Series in Applied Mathematics, 48. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985. Y. K. Chembo, D. Gomila, M. Tlidi, C. R. Menyuk: Topical Issue: Theory and Applications of the Lugiato-Lefever Equation. Eur. Phys. J. D 71 (2017). Podcasts S. Fliss, G. Thäter: Transparent Boundaries. Conversation in the Modellansatz Podcast episode 75, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2015. M. Kray, G. Thäter: Splitting Waves. Conversation in the Modellansatz Podcast episode 62, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2015. F. Sayas, G. Thäter: Acoustic scattering. Conversation in the Modellansatz Podcast episode 58, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2015.
In den nächsten Wochen bis zum 20.2.2020 möchte Anna Hein, Studentin der Wissenschaftskommunikation am KIT, eine Studie im Rahmen ihrer Masterarbeit über den Podcast Modellansatz durchführen. Dazu möchte sie gerne einige Interviews mit Ihnen, den Hörerinnen und Hörern des Podcast Modellansatz führen, um herauszufinden, wer den Podcast hört und wie und wofür er genutzt wird. Die Interviews werden anonymisiert und werden jeweils circa 15 Minuten in Anspruch nehmen. Für die Teilnahme an der Studie können Sie sich bis zum 20.2.2020 unter der Emailadresse studie.modellansatz@web.de bei Anna Hein melden. Wir würden uns sehr freuen, wenn sich viele Interessenten melden würden. In the coming weeks until February 20, 2020, Anna Hein, student of science communication at KIT, intends to conduct a study on the Modellansatz Podcast within her master's thesis. For this purpose, she would like to conduct some interviews with you, the listeners of the Modellansatz Podcast, to find out who listens to the podcast and how and for what purpose it is used. The interviews will be anonymous and will take about 15 minutes each. To participate in the study, you can register with Anna Hein until 20.2.2020 at studie.modellansatz@web.de . We would be very pleased if many interested parties would contact us. This is the second of three conversation recorded Conference on mathematics of wave phenomena 23-27 July 2018 in Karlsruhe. Gudrun is in conversation with Mariana Haragus about Benard-Rayleigh problems. On the one hand this is a much studied model problem in Partial Differential Equations. There it has connections to different fields of research due to the different ways to derive and read the stability properties and to work with nonlinearity. On the other hand it is a model for various applications where we observe an interplay between boyancy and gravity and for pattern formation in general. An everyday application is the following: If one puts a pan with a layer of oil on the hot oven (in order to heat it up) one observes different flow patterns over time. In the beginning it is easy to see that the oil is at rest and not moving at all. But if one waits long enough the still layer breaks up into small cells which makes it more difficult to see the bottom clearly. This is due to the fact that the oil starts to move in circular patterns in these cells. For the problem this means that the system has more than one solutions and depending on physical parameters one solution is stable (and observed in real life) while the others are unstable. In our example the temperature difference between bottom and top of the oil gets bigger as the pan is heating up. For a while the viscosity and the weight of the oil keep it still. But if the temperature difference is too big it is easier to redistribute the different temperature levels with the help of convection of the oil. The question for engineers as well as mathematicians is to find the point where these convection cells evolve in theory in order to keep processes on either side of this switch. In theory (not for real oil because it would start to burn) for even bigger temperature differences the original cells would break up into even smaller cells to make the exchange of energy faster. In 1903 Benard did experiments similar to the one described in the conversation which fascinated a lot of his colleagues at the time. The equations where derived a bit later and already in 1916 Lord Rayleigh found the 'switch', which nowadays is called the critical Rayleigh number. Its size depends on the thickness of the configuration, the viscositiy of the fluid, the gravity force and the temperature difference. Only in the 1980th it became clear that Benards' experiments and Rayleigh's analysis did not really cover the same problem since in the experiment the upper boundary is a free boundary to the surrounding air while Rayleigh considered fixed boundaries. And this changes the size of the critical Rayleigh number. For each person doing experiments it is also an observation that the shape of the container with small perturbations in the ideal shape changes the convection patterns. Maria does study the dynamics of nonlinear waves and patterns. This means she is interested in understanding processes which change over time. Her main questions are: Existence of observed waves as solutions of the equations The stability of certain types of solutions How is the interaction of different waves She treats her problems with the theory of dynamical systems and bifurcations. The simplest tools go back to Poincaré when understanding ordinary differential equations. One could consider the partial differential equations to be the evolution in an infinite dimensional phase space. Here, in the 1980s, Klaus Kirchgässner had a few crucial ideas how to construct special solutions to nonlinear partial differential equations. It is possible to investigate waterwave problems which are dispersive equations as well as flow problems which are dissipative. Together with her colleagues in Besancon she is also very keen to match experiments for optical waves with her mathematical analysis. There Mariana is working with a variant of the Nonlinear Schrödinger equation called Lugiato-Lefever Equation. It has many different solutions, e.g. periodic solutions and solitons. Since 2002 Mariana has been Professor in Besancon (University of Franche-Comté, France). Before that she studied and worked in a lot of different places, namely in Bordeaux, Stuttgart, Bucharest, Nice, and Timisoara. References V.A. Getling: Rayleigh-Bénard Convection Structures and Dynamics, Advanced Series in Nonlinear Dynamics, Volume 11, World Scientific, Oxford (1998) P. H. Rabinowitz: Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rational Mech. Anal. (1968) 29: 32. M. Haragus and G. Iooss: Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems. Universitext. Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011. Newell, Alan C. Solitons in mathematics and physics. CBMS-NSF Regional Conference Series in Applied Mathematics, 48. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985. Y. K. Chembo, D. Gomila, M. Tlidi, C. R. Menyuk: Topical Issue: Theory and Applications of the Lugiato-Lefever Equation. Eur. Phys. J. D 71 (2017). Podcasts S. Fliss, G. Thäter: Transparent Boundaries. Conversation in the Modellansatz Podcast episode 75, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2015. M. Kray, G. Thäter: Splitting Waves. Conversation in the Modellansatz Podcast episode 62, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2015. F. Sayas, G. Thäter: Acoustic scattering. Conversation in the Modellansatz Podcast episode 58, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2015.
Gudrun Talks to Sema Coşkun who at the moment of the conversation in 2018 is a Post Doc researcher at the University Kaiserslautern in the group of financial mathematics. She constructs models for the behaviour of energy markets. In short the conversation covers the questions How are classical markets modelled? In which way are energy markets different and need new ideas? The seminal work of Black and Scholes (1973) established the modern financial theory. In a Black-Scholes setting, it is assumed that the stock price follows a Geometric Brownian Motion with a constant drift and constant volatility. The stochastic differential equation for the stock price process has an explicit solution. Therefore, it is possible to obtain the price of a European call option in a closed-form formula. Nevertheless, there exist drawbacks of the Black-Scholes assumptions. The most criticized aspect is the constant volatility assumption. It is considered an oversimplification. Several improved models have been introduced to overcome those drawbacks. One significant example of such new models is the Heston stochastic volatility model (Heston, 1993). In this model, volatility is indirectly modeled by a separate mean reverting stochastic process, namely. the Cox-Ingersoll-Ross (CIR) process. The CIR process captures the dynamics of the volatility process well. However, it is not easy to obtain option prices in the Heston model since the model has more complicated dynamics compared to the Black-Scholes model. In financial mathematics, one can use several methods to deal with these problems. In general, various stochastic processes are used to model the behavior of financial phenomena. One can then employ purely stochastic approaches by using the tools from stochastic calculus or probabilistic approaches by using the tools from probability theory. On the other hand, it is also possible to use Partial Differential Equations (the PDE approach). The correspondence between the stochastic problem and its related PDE representation is established by the help of Feynman-Kac theorem. Also in their original paper, Black and Scholes transferred the stochastic representation of the problem into its corresponding PDE, the heat equation. After solving the heat equation, they transformed the solution back into the relevant option price. As a third type of methods, one can employ numerical methods such as Monte Carlo methods. Monte Carlo methods are especially useful to compute the expected value of a random variable. Roughly speaking, instead of examining the probabilistic evolution of this random variable, we focus on the possible outcomes of it. One generates random numbers with the same distribution as the random variable and then we simulate possible outcomes by using those random numbers. Then we replace the expected value of the random variable by taking the arithmetic average of the possible outcomes obtained by the Monte Carlo simulation. The idea of Monte Carlo is simple. However, it takes its strength from two essential theorems, namely Kolmogorov’s strong law of large numbers which ensures convergence of the estimates and the central limit theorem, which refers to the error distribution of our estimates. Electricity markets exhibit certain properties which we do not observe in other markets. Those properties are mainly due to the unique characteristics of the production and consumption of electricity. Most importantly one cannot physically store electricity. This leads to several differences compared to other financial markets. For example, we observe spikes in electricity prices. Spikes refer to sudden upward or downward jumps which are followed by a fast reversion to the mean level. Therefore, electricity prices show extreme variability compared to other commodities or stocks. For example, in stock markets we observe a moderate volatility level ranging between 1% and 1.5%, commodities like crude oil or natural gas have relatively high volatilities ranging between 1.5% and 4% and finally the electricity energy has up to 50% volatility (Weron, 2000). Moreover, electricity prices show strong seasonality which is related to day to day and month to month variations in the electricity consumption. In other words, electricity consumption varies depending on the day of the week and month of the year. Another important property of the electricity prices is that they follow a mean reverting process. Thus, the Ornstein-Uhlenbeck (OU) process which has a Gaussian distribution is widely used to model electricity prices. In order to incorporate the spike behavior of the electricity prices, a jump or a Levy component is merged into the OU process. These models are known as generalized OU processes (Barndorff-Nielsen & Shephard, 2001; Benth, Kallsen & Meyer-Brandis, 2007). There exist several models to capture those properties of electricity prices. For example, structural models which are based on the equilibrium of supply and demand (Barlow, 2002), Markov jump diffusion models which combine the OU process with pure jump diffusions (Geman & Roncoroni, 2006), regime-switching models which aim to distinguish the base and spike regimes of the electricity prices and finally the multi-factor models which have a deterministic component for seasonality, a mean reverting process for the base signal and a jump or Levy process for spikes (Meyer-Brandis & Tankov, 2008). The German electricity market is one of the largest in Europe. The energy strategy of Germany follows the objective to phase out the nuclear power plants by 2021 and gradually introduce renewable energy ressources. For electricity production, the share of renewable ressources will increase up to 80% by 2050. The introduction of renewable ressources brings also some challenges for electricity trading. For example, the forecast errors regarding the electricity production might cause high risk for market participants. However, the developed market structure of Germany is designed to reduce this risk as much as possible. There are two main electricity spot price markets where the market participants can trade electricity. The first one is the day-ahead market in which the trading takes place around noon on the day before the delivery. In this market, the trades are based on auctions. The second one is the intraday market in which the trading starts at 3pm on the day before the delivery and continues up until 30 minutes before the delivery. Intraday market allows continuous trading of electricity which indeed helps the market participants to adjust their positions more precisely in the market by reducing the forecast errors. References S. Coskun and R. Korn: Pricing Barrier Options in the Heston Model Using the Heath-Platen estimator. Monte Carlo Methods and Applications. 24 (1) 29-42, 2018. S. Coskun: Application of the Heath–Platen Estimator in Pricing Barrier and Bond Options. PhD thesis, Department of Mathematics, University of Kaiserslautern, Germany, 2017. S. Desmettre and R. Korn: 10 Computationally challenging problems in Finance. FPGA Based Accelerators for Financial Applications, Springer, Heidelberg, 1–32, 2015. F. Black and M. Scholes: The pricing of options and corporate liabilities. The Journal of Political Economy, 81(3):637-654, 1973. S.L. Heston: A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2):327–343, 1993. R. Korn, E. Korn and G. Kroisandt: Monte Carlo Methods and Models in Finance and Insurance. Chapman & Hall/CRC Financ. Math. Ser., CRC Press, Boca Raton, 2010. P. Glasserman, Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability, Appl. Math. (New York) 53, Springer, New York, 2004. M.T. Barlow: A diffusion model for electricity prices. Mathematical Finance, 12(4):287-298, 2002. O.E. Barndorff-Nielsen and N. Shephard: Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. Journal of the Royal Statistical Society B, 63(2):167-241, 2001. H. Geman and A. Roncoroni: Understanding the fine structure of electricity prices. The Journal of Business, 79(3):1225-1261, 2006. T. Meyer-Brandis and P. Tankov: Multi-factor jump-diffusion models of electricity prices. International Journal of Theoretical and Applied Finance, 11(5):503-528, 2008. R. Weron: Energy price risk management. Physica A, 285(1-2):127–134, 2000. Podcasts G. Thäter, M. Hofmanová: Turbulence, conversation in the Modellansatz Podcast, episode 155, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2018. http://modellansatz.de/turbulence G. Thäter, M. J. Amtenbrink: Wasserstofftankstellen, Gespräch im Modellansatz Podcast, Folge 163, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2018. http://modellansatz.de/wasserstofftankstellen S. Ajuvo, S. Ritterbusch: Finanzen damalsTM, Gespräch im Modellansatz Podcast, Folge 97, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016. http://modellansatz.de/finanzen-damalstm K. Cindric, G. Thäter: Kaufverhalten, Gespräch im Modellansatz Podcast, Folge 45, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2015. http://modellansatz.de/kaufverhalten V. Riess, G. Thäter: Gasspeicher, Gespräch im Modellansatz Podcast, Folge 23, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2015. http://modellansatz.de/gasspeicher F. Schueth, T. Pritlove: Energieforschung, Episode 12 im Forschergeist Podcast, Stifterverband/Metaebene, 2015. https://forschergeist.de/podcast/fg012-energieforschung/
Gudrun Talks to Sema Coşkun who at the moment of the conversation in 2018 is a Post Doc researcher at the University Kaiserslautern in the group of financial mathematics. She constructs models for the behaviour of energy markets. In short the conversation covers the questions How are classical markets modelled? In which way are energy markets different and need new ideas? The seminal work of Black and Scholes (1973) established the modern financial theory. In a Black-Scholes setting, it is assumed that the stock price follows a Geometric Brownian Motion with a constant drift and constant volatility. The stochastic differential equation for the stock price process has an explicit solution. Therefore, it is possible to obtain the price of a European call option in a closed-form formula. Nevertheless, there exist drawbacks of the Black-Scholes assumptions. The most criticized aspect is the constant volatility assumption. It is considered an oversimplification. Several improved models have been introduced to overcome those drawbacks. One significant example of such new models is the Heston stochastic volatility model (Heston, 1993). In this model, volatility is indirectly modeled by a separate mean reverting stochastic process, namely. the Cox-Ingersoll-Ross (CIR) process. The CIR process captures the dynamics of the volatility process well. However, it is not easy to obtain option prices in the Heston model since the model has more complicated dynamics compared to the Black-Scholes model. In financial mathematics, one can use several methods to deal with these problems. In general, various stochastic processes are used to model the behavior of financial phenomena. One can then employ purely stochastic approaches by using the tools from stochastic calculus or probabilistic approaches by using the tools from probability theory. On the other hand, it is also possible to use Partial Differential Equations (the PDE approach). The correspondence between the stochastic problem and its related PDE representation is established by the help of Feynman-Kac theorem. Also in their original paper, Black and Scholes transferred the stochastic representation of the problem into its corresponding PDE, the heat equation. After solving the heat equation, they transformed the solution back into the relevant option price. As a third type of methods, one can employ numerical methods such as Monte Carlo methods. Monte Carlo methods are especially useful to compute the expected value of a random variable. Roughly speaking, instead of examining the probabilistic evolution of this random variable, we focus on the possible outcomes of it. One generates random numbers with the same distribution as the random variable and then we simulate possible outcomes by using those random numbers. Then we replace the expected value of the random variable by taking the arithmetic average of the possible outcomes obtained by the Monte Carlo simulation. The idea of Monte Carlo is simple. However, it takes its strength from two essential theorems, namely Kolmogorov’s strong law of large numbers which ensures convergence of the estimates and the central limit theorem, which refers to the error distribution of our estimates. Electricity markets exhibit certain properties which we do not observe in other markets. Those properties are mainly due to the unique characteristics of the production and consumption of electricity. Most importantly one cannot physically store electricity. This leads to several differences compared to other financial markets. For example, we observe spikes in electricity prices. Spikes refer to sudden upward or downward jumps which are followed by a fast reversion to the mean level. Therefore, electricity prices show extreme variability compared to other commodities or stocks. For example, in stock markets we observe a moderate volatility level ranging between 1% and 1.5%, commodities like crude oil or natural gas have relatively high volatilities ranging between 1.5% and 4% and finally the electricity energy has up to 50% volatility (Weron, 2000). Moreover, electricity prices show strong seasonality which is related to day to day and month to month variations in the electricity consumption. In other words, electricity consumption varies depending on the day of the week and month of the year. Another important property of the electricity prices is that they follow a mean reverting process. Thus, the Ornstein-Uhlenbeck (OU) process which has a Gaussian distribution is widely used to model electricity prices. In order to incorporate the spike behavior of the electricity prices, a jump or a Levy component is merged into the OU process. These models are known as generalized OU processes (Barndorff-Nielsen & Shephard, 2001; Benth, Kallsen & Meyer-Brandis, 2007). There exist several models to capture those properties of electricity prices. For example, structural models which are based on the equilibrium of supply and demand (Barlow, 2002), Markov jump diffusion models which combine the OU process with pure jump diffusions (Geman & Roncoroni, 2006), regime-switching models which aim to distinguish the base and spike regimes of the electricity prices and finally the multi-factor models which have a deterministic component for seasonality, a mean reverting process for the base signal and a jump or Levy process for spikes (Meyer-Brandis & Tankov, 2008). The German electricity market is one of the largest in Europe. The energy strategy of Germany follows the objective to phase out the nuclear power plants by 2021 and gradually introduce renewable energy ressources. For electricity production, the share of renewable ressources will increase up to 80% by 2050. The introduction of renewable ressources brings also some challenges for electricity trading. For example, the forecast errors regarding the electricity production might cause high risk for market participants. However, the developed market structure of Germany is designed to reduce this risk as much as possible. There are two main electricity spot price markets where the market participants can trade electricity. The first one is the day-ahead market in which the trading takes place around noon on the day before the delivery. In this market, the trades are based on auctions. The second one is the intraday market in which the trading starts at 3pm on the day before the delivery and continues up until 30 minutes before the delivery. Intraday market allows continuous trading of electricity which indeed helps the market participants to adjust their positions more precisely in the market by reducing the forecast errors. References S. Coskun and R. Korn: Pricing Barrier Options in the Heston Model Using the Heath-Platen estimator. Monte Carlo Methods and Applications. 24 (1) 29-42, 2018. S. Coskun: Application of the Heath–Platen Estimator in Pricing Barrier and Bond Options. PhD thesis, Department of Mathematics, University of Kaiserslautern, Germany, 2017. S. Desmettre and R. Korn: 10 Computationally challenging problems in Finance. FPGA Based Accelerators for Financial Applications, Springer, Heidelberg, 1–32, 2015. F. Black and M. Scholes: The pricing of options and corporate liabilities. The Journal of Political Economy, 81(3):637-654, 1973. S.L. Heston: A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2):327–343, 1993. R. Korn, E. Korn and G. Kroisandt: Monte Carlo Methods and Models in Finance and Insurance. Chapman & Hall/CRC Financ. Math. Ser., CRC Press, Boca Raton, 2010. P. Glasserman, Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability, Appl. Math. (New York) 53, Springer, New York, 2004. M.T. Barlow: A diffusion model for electricity prices. Mathematical Finance, 12(4):287-298, 2002. O.E. Barndorff-Nielsen and N. Shephard: Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. Journal of the Royal Statistical Society B, 63(2):167-241, 2001. H. Geman and A. Roncoroni: Understanding the fine structure of electricity prices. The Journal of Business, 79(3):1225-1261, 2006. T. Meyer-Brandis and P. Tankov: Multi-factor jump-diffusion models of electricity prices. International Journal of Theoretical and Applied Finance, 11(5):503-528, 2008. R. Weron: Energy price risk management. Physica A, 285(1-2):127–134, 2000. Podcasts G. Thäter, M. Hofmanová: Turbulence, conversation in the Modellansatz Podcast, episode 155, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2018. http://modellansatz.de/turbulence G. Thäter, M. J. Amtenbrink: Wasserstofftankstellen, Gespräch im Modellansatz Podcast, Folge 163, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2018. http://modellansatz.de/wasserstofftankstellen S. Ajuvo, S. Ritterbusch: Finanzen damalsTM, Gespräch im Modellansatz Podcast, Folge 97, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016. http://modellansatz.de/finanzen-damalstm K. Cindric, G. Thäter: Kaufverhalten, Gespräch im Modellansatz Podcast, Folge 45, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2015. http://modellansatz.de/kaufverhalten V. Riess, G. Thäter: Gasspeicher, Gespräch im Modellansatz Podcast, Folge 23, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2015. http://modellansatz.de/gasspeicher F. Schueth, T. Pritlove: Energieforschung, Episode 12 im Forschergeist Podcast, Stifterverband/Metaebene, 2015. https://forschergeist.de/podcast/fg012-energieforschung/
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.
Das Gespräch mit Susanne Höllbacher von der Simulationsgruppe an der Frankfurter Goethe-Universität war ein Novum in unserer Podcastgeschichte. Das erste mal hatte sich eine Hörerin gemeldet, die unser Interesse an Partikeln in Strömungen teilte, was sofort den Impuls in Gudrun auslöste, sie zu einem Podcastgespräch zu diesem Thema einzuladen. Susanne hat in der Arbeitsgruppe von Gabriel Wittum in Frankfurt promoviert. Dort werden Finite-Volumen-Verfahren zur Lösung von Partiellen Differentialgleichungen benutzt. Das Verfahren betrifft hier insbesondere die räumliche Diskretisierung: Das Rechengebiet wird in Kontrollvolumen aufgeteilt, in denen durch das Verfahren sichergestellt wird, dass bestimmte Größen erhalten bleiben (z.B. die Masse). Diese Verfahren stammen aus dem Umfeld hyperbolischer Probleme, die vor allem als Erhaltungsgesetze modelliert sind. Diese Gleichungen haben die Eigenschaft, dass Fehler nicht automatisch geglättet werden und abklingen sondern potentiell aufgeschaukelt werden können. Trotzdem ist es möglich, diese numerischen Verfahren ähnlich wie Finite-Elemente-Verfahren als Variationsprobleme zu formulieren und die beiden Familien in der Analyse etwas näher zusammenrücken zu lassen. Gemeinsam ist ihnen ja ohnehin, dass sie auf große Gleichungssysteme führen, die anschließend gelöst werden müssen. Hier ist eine billige und doch wirkungsvolle Vorkonditionierung entscheidend für die Effizienz und sogar dafür, ob die Lösungen durch das numerische Verfahren überhaupt gefunden werden. Hier hilft es, schon auf Modell-Ebene die Eigenschaften des diskreten Systems zu berücksichtigen, da ein konsistentes Modell bereits als guter Vorkonditionierer fungiert. Das Promotionsprojekt von Susanne war es, eine Methode zur direkten numerischen Simulation (DNS) von Partikeln in Fluiden auf Basis eines finite Volumen-Verfahrens zu entwickeln. Eine grundsätzliche Frage ist dabei, wie man die Partikel darstellen möchte und kann, die ja winzige Festkörper sind und sich anders als die Strömung verhalten. Sie folgen anderen physikalischen Gesetzen und man ist geneigt, sie als Kräfte in die Strömung zu integrieren. Susanne hat die Partikel jedoch als Teil des Fluides modelliert, indem die Partikel als finite (und nicht infinitesimal kleine) Volumen mit zusätzlicher Rotation als Freiheitsgrad in die diskreten Gleichungen integriert werden. Damit fügen sich die Modelle für die Partikel natürlich und konsistent in das diskrete System für die Strömung ein. Vorhandene Symmetrien bleiben erhalten und ebenso die Kopplung der Kräfte zwischen Fluid und Partikel ist gewährleistet. Die Nebenbedingungen an das System werden so formuliert, dass eine Sattelpunkt-Formulierung vermieden wird. Die grundlegende Strategie dabei ist, die externen Kräfte, welche bedingt durch die Partikel und deren Ränder wirken, direkt in die Funktionenräume des zugrundeliegenden Operators zu integrieren. In biologischen Systemen mit hoher Viskotität des Fluides fungiert die Wirkung der Partikel auf das Fluid als Informationstransport zwischen den Partikeln und ist sehr wichtig. In der Umsetzung dieser Idee verhielten sich die Simulationen des Geschwindigkeitsfeldes sehr gutartig, aber Susanne beobachtete Oszillationen im Druck. Da sie sich nicht physikalisch erklären ließen, musste es sich um numerische Artekfakte handeln. Bei näherem Hinsehen zeigte sich, dass es vor allem daran lag, dass die Richtungen von Kraftwirkungen auf dem Rand der Partikel im diskreten System nicht sinnvoll approximiert wurden. In den berechneten Lösungen für das Geschwindigkeitsfeld hat sich dies kaum messbar niedergeschlagen. Im Druck zeigte sich jedoch, dass es sich lohnt, hier das numerische Verfahren zu ändern, so dass die Normalenrichtungen auf dem Rand jeweils korrekt sind. Mathematisch heißt das, dass die Ansatzfunktionen so geändert werden, dass deren Freiheitsgrade auf dem Rand liegen. Der Aufwand dafür ist vergleichsweise gering und die Resultate sind überzeugend. Die Oszillationen verschwinden komplett. Der Nachweis der Stabilität des entstehenden Gleichungssystems lässt sich über die inf-sup-Bedingung des orginalen Verfahrens erbringen, da die Konstruktion den Raum in der passenden Weise erweitert. Literatur und weiterführende Informationen S. V. Apte, M. Martin, N. A. Patankar: A numerical method for fully resolved simulation (FRS) of rigid particle–flow interactions in complex flows, Journal of Computational Physics 228, S. 2712–2738, 2009. R. E. Bank, D. J. Rose: Some Error Estimates for the Box Method, SIAM Journal on Numerical Analysis 24, S. 777–787, 1987. Glowinski, R.: Finite element methods for incompressible viscous flow, P. G. Ciarlet, J. L. Lions (Eds.), Handbook of Numerical Analysis IX (North-Holland, Amsterdam), S. 3–1176, 2003. Strang, G.: Wissenschaftlisches Rechnen, Springer-Verlag Berlin Heidelberg, 2010. A. Vogel, S. Reiter, M. Rupp, A. Naegel, G. Wittum: UG 4: A novel flexible software system for simulating PDE based models on high performance computers, Computing and Visualization in Science 16, S. 165–179, 2013. G. J. Wagner, N. Moes, W. K. Liu, T. Belytschko: The extended finite element method for rigid particles in Stokes flow, International Journal for Numerical Methods in Engineering 51, S. 293–313, 2001. D. Wan, S. Turek: Fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows, Journal of Computational Physics 222, S. 28–56, 2007. P. Wessling: Principles of Computational Fluid Dynamics, Springer, Series in Computational Mathematics, 2001. J. Xu, Q. Zou: Analysis of linear and quadratic simplicial finite volume methods for elliptic equations, Numerische Mathematik 111, S. 469–492, 2009. X. Ye: On the Relationship Between Finite Volume and Finite Element Methods Applied to the Stokes Equations, Numerical Methods for Partial Differential Equations 17, S. 440–453, 2001. Podcasts T. Henn: Partikelströmungen, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 115, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016. http://modellansatz.de/partikelstroemungen L.L.X. Augusto: Filters, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 112, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016. http://modellansatz.de/filters L. Adlung: Systembiologie, Gespräch mit G. Thäter und S. Ritterbusch im Modellansatz Podcast, Folge 39, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2014. http://modellansatz.de/systembiologie
Professor José Francisco Rodrigues, Lisbon/CMAF, delivers the ASC Complexity Cluster Lecture entitled 'Some Mathematical Aspects of Planet Earth' at Keble College. The Planet Earth System is composed of several sub-systems including the atmosphere, the liquid oceans and the icecaps, the internal structure and the biosphere. In all of them Mathematics, enhanced by the supercomputers, has currently a key role through the "universal method" for their study, which consists of mathematical modeling, analysis, simulation and control, as it was re-stated by Jacques-Louis Lions at the end of 20th century. Much before the advent of computers, the representation of the Earth, navigation and cartography have contributed in a decisive form to the mathematical sciences. Nowadays new global challenges contribute to stimulate several mathematical research topics. In this lecture, we present a brief historical introduction to some of the essential mathematics for understanding the Planet Earth, stressing the importance of Mathematical Geography and its role in the Scientific Revolution(s), the modeling efforts of Winds, Heating, Earthquakes, Climate and their influence on basic aspects of the theory of Partial Differential Equations. As a special topic to illustrate the wide scope of these (Geo)physical problems we describe briefly some examples from History and from current research and advances in Free Boundary Problems arising in the Planet Earth. Finally we conclude by referring the potential impact of the international initiative Mathematics of Planet Earth (http://www.mpe2013.org) in Raising Public Awareness of Mathematics, in Research and in the Communication of the Mathematical Sciences to the new generations.
Professor José Francisco Rodrigues, Lisbon/CMAF, delivers the ASC Complexity Cluster Lecture entitled 'Some Mathematical Aspects of Planet Earth' at Keble College. The Planet Earth System is composed of several sub-systems including the atmosphere, the liquid oceans and the icecaps, the internal structure and the biosphere. In all of them Mathematics, enhanced by the supercomputers, has currently a key role through the "universal method" for their study, which consists of mathematical modeling, analysis, simulation and control, as it was re-stated by Jacques-Louis Lions at the end of 20th century. Much before the advent of computers, the representation of the Earth, navigation and cartography have contributed in a decisive form to the mathematical sciences. Nowadays new global challenges contribute to stimulate several mathematical research topics. In this lecture, we present a brief historical introduction to some of the essential mathematics for understanding the Planet Earth, stressing the importance of Mathematical Geography and its role in the Scientific Revolution(s), the modeling efforts of Winds, Heating, Earthquakes, Climate and their influence on basic aspects of the theory of Partial Differential Equations. As a special topic to illustrate the wide scope of these (Geo)physical problems we describe briefly some examples from History and from current research and advances in Free Boundary Problems arising in the Planet Earth. Finally we conclude by referring the potential impact of the international initiative Mathematics of Planet Earth (http://www.mpe2013.org) in Raising Public Awareness of Mathematics, in Research and in the Communication of the Mathematical Sciences to the new generations.
Professor Gui-Qiang G. Chen presents in his inaugural lecture several examples to illustrate the origins, developments, and roles of partial differential equations in our changing world. While calculus is a mathematical theory concerned with change, differential equations are the mathematician's foremost aid for describing change. In the simplest case, a process depends on one variable alone, for example time. More complex phenomena depend on several variables - perhaps time and, in addition, one, two or three space variables. Such processes require the use of partial differential equations. The behaviour of every material object in nature, with timescales ranging from picoseconds to millennia and length scales ranging from sub-atomic to astronomical, can be modeled by nonlinear partial differential equations or by equations with similar features. The roles of partial differential equations within mathematics and in the other sciences become increasingly significant. The mathematical theory of partial differential equations has a long history. In the recent decades, the subject has experienced a vigorous growth, and research is marching on at a brisk pace.
Professor Gui-Qiang G. Chen presents in his inaugural lecture several examples to illustrate the origins, developments, and roles of partial differential equations in our changing world. While calculus is a mathematical theory concerned with change, differential equations are the mathematician's foremost aid for describing change. In the simplest case, a process depends on one variable alone, for example time. More complex phenomena depend on several variables - perhaps time and, in addition, one, two or three space variables. Such processes require the use of partial differential equations. The behaviour of every material object in nature, with timescales ranging from picoseconds to millennia and length scales ranging from sub-atomic to astronomical, can be modeled by nonlinear partial differential equations or by equations with similar features. The roles of partial differential equations within mathematics and in the other sciences become increasingly significant. The mathematical theory of partial differential equations has a long history. In the recent decades, the subject has experienced a vigorous growth, and research is marching on at a brisk pace.
Professor Gui-Qiang G. Chen presents in his inaugural lecture several examples to illustrate the origins, developments, and roles of partial differential equations in our changing world. While calculus is a mathematical theory concerned with change, differential equations are the mathematician's foremost aid for describing change. In the simplest case, a process depends on one variable alone, for example time. More complex phenomena depend on several variables - perhaps time and, in addition, one, two or three space variables. Such processes require the use of partial differential equations. The behaviour of every material object in nature, with timescales ranging from picoseconds to millennia and length scales ranging from sub-atomic to astronomical, can be modeled by nonlinear partial differential equations or by equations with similar features. The roles of partial differential equations within mathematics and in the other sciences become increasingly significant. The mathematical theory of partial differential equations has a long history. In the recent decades, the subject has experienced a vigorous growth, and research is marching on at a brisk pace.
Jonathan Cohen, a Senior Research Scientist at NVIDIA Research, talks about solving partial differential equations with CUDA. (May 4, 2010)
As a model for multiscale systems under random influences on physical boundary, a stochastic partial differential equation under a fast random dynamical boundary condition is investigated. An effective equation is derived and justified by reducing the random dynamical boundary condition to a usual random boundary condition. The effective system is still a stochastic partial differential equation, but is more tractable. Furthermore, the quantitative comparison between the solution of the original stochastic system and the effective solution is provided by estimating deviations. Jinqiao DUAN. Illinois Institute of Technology. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1182790011791 (pdf) Bande son disponible au format mp3 Durée : 42 mn
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