Remo Von Rickenbach (Universität Basel)
Abstract:
Many problems from physics or engineering result in partial differential equations. In both cases, the unknown function \\(u\\) can, in most practically relevant applications, only be approximated numerically. Therefore, it is essential to use efficient algorithms to approximate \\(u\\) at as low costs as possible. To build efficient algorithms, one first needs to understand how well a function of a given regularity can be approximated. For example, by using finite elements of grid size \\(h\\) on the unit interval \\(I\\), it is well established that the approximation error decays as \\begin{align*} \\inf_{v_h \\in V_h} \\|u - v_h\\|_{L^2(I)} \\leq C h^s \\|u\\|_{H^{s}(I)}, \\quad 0 \\leq s \\leq d, \\end{align*} where \\(d \\in \\mathbb{N}\\) is the polynomial order of the trial functions involved. However, what can we say about the approximation order if \\(u\\) is only piecewise regular, but admits a singularity and is therefore not in \\(H^{s}(I)\\)? For this, adaptive schemes, which rely on the concept of \\emph{nonlinear approximation}, need to be studied. In this talk, some basic concepts and examples of approximation theory will be introduced and we will characterise the approximation spaces with respect to some common basis systems. In particular, we will see that the approximation spaces with respect to nonlinear approximation by wavelet bases contain Besov spaces which are strictly larger than the corresponding, classical finite element approximation spaces, showing that adaptive schemes strictly outperform classical schemes in the case of limited regularity. abstract
12:15 • Universität Basel, Spiegelgasse 5, 05.001
Cécile Gachet (Bochum)
The nef, movable, and effective cones:From Mori Dream Spaces to Calabi—Yau pairs. abstract
Abstract:
To a complex projective variety come various convex cones attached, whose shapes encode knowledge of fibrations, birational contractions, symmetries of the initial variety, and more. The nef, movable, and effective cones all arise in this way. As they encode different, but related information, one can wonder how they overall relate: A classical result along those lines is due to Hu and Keel, and states that a variety whose nef and movable cone both are rational polyhedral also has, under additional assumptions, a rational polyhedral effective cone. This result made precise leads to the study of so-called Mori Dream Spaces, which are a source of motivation for the work I plan to present, and will be the focus of the first half of the talk.In the second half of the talk, I will report on recent joint work with H.-Y. Lin, I. Stenger, and L. Wang. Inspired by the Morrison-Kawamata cone conjecture, we introduce an effective cone conjecture, and relate it to the nef and movable cone conjectures. If time allows, we will explain how to harness implications between these three cone conjectures in concrete geometric situations.
13:15 • EPF Lausanne, CM 1517
Dr. Léo Mathis (Goethe University Frankfurt, DE)
Non centered Gaussian determinants with Gaussian zonoids abstract
Abstract:
A Theorem of Vitale, Molchanov and Wespi, states that the expected absolute determinant of a random matrix with independent columns is equal to the mixed volume of some convex bodies called zonoids. In the case where the columns of the matrix are centered Gaussian, the corresponding zonoids are ellipsoids and this was studied by Kabluchko and Zaporozhets. The case of non centered Gaussian vectors, however, remains relatively unstudied. The convex bodies we obtain in this case, which I call Gaussian zonoids, are not ellipsoids. In this talk, I will show you what a Gaussian zonoid looks like and how one can approximate it with an ellipsoid. At the level of random determinants, this allows to approximate expectation of non centered Gaussian determinants with centered ones. If time allows, I will show how this applies to Gaussian random fields. Namely, I will show how one can give a quantitative estimate of the concentration of a small Gaussian perturbation of a hypersurface.
14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Stefan Wager (Stanford University)
Joint talk ETH-FDS Seminar - Research Seminar on Statistics: "Optimal Mechanisms for Demand Response: An Indifference Set Approach" abstract
Abstract:
The time at which renewable (e.g., solar or wind) energy resources produce electricity cannot generally be controlled. In many settings, consumers have some flexibility in their energy consumption needs, and there is growing interest in demand-response programs that leverage this flexibility to shift energy consumption to better match renewable production -- thus enabling more efficient utilization of these resources. We study optimal demand response in a model where consumers operate home energy management systems (HEMS) that can compute the "indifference set" of energy-consumption profiles that meet pre-specified consumer objectives, receive demand-response signals from the grid, and control consumer devices within the indifference set. For example, if a consumer asks for the indoor temperature to remain between certain upper and lower bounds, a HEMS could time use of air conditioning or heating to align with high renewable production when possible. Here, we show that while price-based mechanisms do not in general achieve optimal demand response, i.e., dynamic pricing cannot induce HEMS to choose optimal demand consumption profiles within the available indifference sets, pricing is asymptotically optimal in a mean-field limit with a growing number of consumers. Furthermore, we show that large-sample optimal dynamic prices can be efficiently derived via an algorithm that only requires querying HEMS about their planned consumption schedules given different prices. We demonstrate our approach in a grid simulation powered by OpenDSS, and show that it achieves meaningful demand response without creating grid instability.Mohammad Mehrabi, Omer Karaduman, Stefan Wagerhttps://arxiv.org/abs/2409.07655
16:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room E 3
Stefan Wager (Stanford University)
Joint talk ETH-FDS Seminar - Research Seminar on Statistics: "Optimal Mechanisms for Demand Response: An Indifference Set Approach" abstract
Abstract:
The time at which renewable (e.g., solar or wind) energy resources produce electricity cannot generally be controlled. In many settings, consumers have some flexibility in their energy consumption needs, and there is growing interest in demand-response programs that leverage this flexibility to shift energy consumption to better match renewable production -- thus enabling more efficient utilization of these resources. We study optimal demand response in a model where consumers operate home energy management systems (HEMS) that can compute the "indifference set" of energy-consumption profiles that meet pre-specified consumer objectives, receive demand-response signals from the grid, and control consumer devices within the indifference set. For example, if a consumer asks for the indoor temperature to remain between certain upper and lower bounds, a HEMS could time use of air conditioning or heating to align with high renewable production when possible. Here, we show that while price-based mechanisms do not in general achieve optimal demand response, i.e., dynamic pricing cannot induce HEMS to choose optimal demand consumption profiles within the available indifference sets, pricing is asymptotically optimal in a mean-field limit with a growing number of consumers. Furthermore, we show that large-sample optimal dynamic prices can be efficiently derived via an algorithm that only requires querying HEMS about their planned consumption schedules given different prices. We demonstrate our approach in a grid simulation powered by OpenDSS, and show that it achieves meaningful demand response without creating grid instability. Mohammad Mehrabi, Omer Karaduman, Stefan Wager https://arxiv.org/abs/2409.07655
16:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room E 3
Patricia Cahn (Smith College)
Rep-Tiles abstract
Abstract:
An n-dimensional rep-tile is a PL submanifold of ℝn that can be decomposed into isometric re-scaled copies of itself, with non-overlapping interiors. We give a complete isotopy classification of rep-tiles in all dimensions. This is joint work with Blair, Kjuchukova, and Schwartz.
Yash Lodha (University of Hawaii at Manoa)
Mini course: Orderability and groups of dynamical origin abstract
Abstract:
The study of bi-orderable groups has a long history that goes back to seminal work of Dedekind, H older and Hilbert. The study of left orderable groups has more modern origins, and the notion provides a remarkable characterization of whether a countable group admits a faithful action by orientation preserving homeomorphisms on the real line.In this mini course we will introduce and study these notions.The first lecture will be an introduction to left orderable and bi-orderable groups,and shall describe an algebraic point of view.The second lecture will feature various examples of groups of dynamical origin that naturally emerge as groups of homeomorphisms of the real line and the circle.The third lecture will describe connections with the analytic notions of second bounded cohomology, rotation numbers, stable commutator length and quasi-morphisms.The fourth lecture will focus on the space of left orders on a countable group and the Witte-Morris theorem.Throughout the mini course, I will discuss open problems and new directions of research.
16:15 • Université de Genève, Conseil Général 7-9, Room 1-15
Dr. Ayman Said (CNRS)
A classification theorem for steady Euler flows abstract
Abstract:
In this talk I am going to present a recent result in collaboration with Tarek Elgindi, Yupei Huang and Chujing Xie where we show that all analytic steady solutions to the Euler equations in a simply connected domain are either radial or global solution to a semi-linear elliptic equation of the \\(\\Delta \\psi= F(\\psi)\\).
16:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 46
Sascha Günther (Université de Lausanne)
Efficiently computing annuity conversion factors via feed-forward neural networks abstract
Abstract:
Many pension plans and private retirement products contain annuity factors, converting the funds at some future time into lifelong income. In general model settings like for example the Li-Lee mortality model, analytical values for the annuity factors are not available and one has to rely on numerical techniques. Their computation typically requires nested simulations as they depend on the interest rate level and the mortality tables at the time of retirement. We exploit the flexibility and efficiency of feed-forward neural networks to value the annuity factors at the time of retirement. In a numerical study, we compare our deep learning approach to (least-squares) Monte-Carlo (LSMC) which can be represented as a special case of the neural network (NN).
17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43