BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alex Townsend (Cornell University) DTSTART:20200427T200000Z DTEND:20200427T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/1 DESCRIPTION:Title: The ultraspherical spectral method\nby Alex Townsend (Cornell University) as part of CRM Applied Math Seminar\n\nLecture held in Webinar.\n\nAbstract\nPseudospectral methods\, based on high degree pol ynomials\, have spectral accuracy when solving differential equations but typically lead to dense and ill-conditioned matrices. The ultraspherical s pectral method is a numerical technique to solve ordinary and partial diff erential equations\, leading to almost banded well-conditioned linear syst ems while maintaining spectral accuracy. In this talk\, we introduce the u ltraspherical spectral method and develop it into a spectral element metho d using a modification to a hierarchical Poincaré-Steklov domain decompos ition method.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Thomas Bury (McGill University) DTSTART:20200511T200000Z DTEND:20200511T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/2 DESCRIPTION:Title: Detecting and distinguishing bifurcations from noisy tim e series data\nby Thomas Bury (McGill University) as part of CRM Appli ed Math Seminar\n\nLecture held in Webinar.\n\nAbstract\nNumerous systems in the natural sciences have the capacity to undergo an abrupt change in t heir dynamical behaviour as a threshold is crossed. Prominent examples inc lude the collapse of fisheries\, algal blooms and paleoclimatic transition s. Mathematical models reveal such transitions as the result of crossing a bifurcation and help to elucidate the underlying mechanisms. However\, th e number of unknowns is often large\, making it difficult to infer where t he bifurcation occurs in the real system.\nIn this talk\, we will look at methods for detecting bifurcations using data-driven approaches. These met hods exploit generic dynamical phenomena that occur prior to bifurcations\ , such as critical slowing down\, in order to infer their approach. We wil l show how the power spectrum of noisy time series data provides informati on on the type of bifurcation and validate this approach with empirical pr edator-prey experiment that undergoes a Hopf bifurcation. Finally\, we wil l explore deep learning methods for detection of bifurcations and make com parison to the more traditional statistical methods in their ability to de tect bifurcations.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Bamdad Hosseini (California Institute of Technology) DTSTART:20200622T200000Z DTEND:20200622T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/3 DESCRIPTION:Title: Data-driven supervised learning: Neural networks and unc ertainty quantification\nby Bamdad Hosseini (California Institute of T echnology) as part of CRM Applied Math Seminar\n\n\nAbstract\nIn this talk I will discuss some ideas at the intersection of machine learning and unc ertainty quantification with a particular focus on data-driven methods tha t do not require explicit knowledge of processes that generate the data. In the first half of the talk I will discuss supervised learning on Banach spaces for emulation of PDE based models and outline a method that combin es principal component analysis with neural network regression for mesh-in dependent approximation of PDE solutions. In the second half I will take a different approach to supervised learning viewing it as a conditional sa mpling problem. I will then introduce a measure transport framework based on generative adversarial networks (GANs) for data-driven conditional sam pling.​\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Theodore Kolokolnikov (Dalhousie University) DTSTART:20200629T200000Z DTEND:20200629T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/4 DESCRIPTION:Title: Simple agent-based models and their continuum limit\ nby Theodore Kolokolnikov (Dalhousie University) as part of CRM Applied Ma th Seminar\n\n\nAbstract\nWe discuss several very different ABM models and their continuum limits.\n\nFirst\, consider the following agent-based mod el of coronavirus spread: people move randomly and infection occurs with s ome nonzero probability when an infected individual comes within a certain ``infection radius'' of a susceptible individual. The question is how the infection radius affects the reproduction number. At low infection rates\ , this model leads to the classical S-I-R ODE model as its continuum limit . However higher infection rates lead to a saturation effect\, which we co mpute explicitly using basic probability theory. Its continuum limit It le ads to an S-I-R type model with a specific saturation term. We also show t hat this modified model gives a much better fit to the real-world data tha n the classical SIR model.\n\nNext\, we will look at a very simple stochas tic model of bacterial aggregation which leads to a novel fourth-order non linear PDE in its continuum limit. This PDE admits soliton-type solutions corresponding to bacterial aggregation patterns\, which we explicitly cons truct. \n\nIf time allows\, we will consider a spatial model of wealth exc hange which leads to novel integro-differential equations.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Stephen Becker (University of Colorado Boulder\, USA) DTSTART:20200921T183000Z DTEND:20200921T193000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/5 DESCRIPTION:Title: Algorithmic stability for generalization guarantees in m achine learning\nby Stephen Becker (University of Colorado Boulder\, U SA) as part of CRM Applied Math Seminar\n\n\nAbstract\nInspired by the pra ctical success of deep learning\, the broader math community has been ener gized recently to find theoretical justification for these methods. There is a large amount of theory from the computer science community\, dating t o the 1980s and earlier\, but usually the quantitative guarantees are too loose to be helpful in practice\, and it is rare that theory can predict s omething useful (such as what iteration to perform early-stopping in order to prevent over-fitting). \nMany of these theories are less well-known in side applied math\, so we briefly review essential results before focusing on the notion of algorithmic stability\, popularized in the early 2000s\, which is an alternative to the more mainstream VC dimension approach\, an d is one avenue that might give sharper theoretical guarantees. Algorithmi c stability is appealing to applied mathematicians\, and in particular ana lysts\, since a lot of the technical work is similar to analysis used for convergence proofs. \nWe give an overview of the fundamental results of al gorithmic stability\, focusing on the stochastic gradient descent (SGD) me thod in the context of a nonconvex loss function\, and give the latest sta te-of-the-art bounds\, including some of our own work (joint with L. Madde n and E. Dall'Anese) which is one of the first results that suggests when to do early-stopping.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Yuji Nakatsukasa (Oxford University\, UK) DTSTART:20200928T200000Z DTEND:20200928T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/6 DESCRIPTION:Title: Fast and stable randomized low-rank matrix approximation \nby Yuji Nakatsukasa (Oxford University\, UK) as part of CRM Applied Math Seminar\n\n\nAbstract\nRandomized SVD has become an extremely success ful approach for efficiently computing a low-rank approximation of matrice s. In particular the paper by Halko\, Martinsson\, and Tropp (SIREV 2011) contains extensive analysis\, and has made it a very popular method. The t ypical complexity for a rank-r approximation of mxn matrices is O(mnlog n+ (m+n)r^2) for dense matrices. The classical Nystrom method is much faster\ , but only applicable to positive semidefinite matrices. This work studies a generalization of Nystrom's method applicable to general matrices\, and shows that (i) it has near-optimal approximation quality comparable to co mpeting methods\, (ii) the computational cost is the near-optimal O(mnlog n+r^3) for dense matrices\, with small hidden constants\, and (iii) crucia lly\, it can be implemented in a numerically stable fashion despite the pr esence of an ill-conditioned pseudoinverse. Numerical experiments illustra te that generalized Nystrom can significantly outperform state-of-the-art methods\, especially when r>>1\, achieving up to a 10-fold speedup. The me thod is also well suited to updating and downdating the matrix.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/6/ END:VEVENT BEGIN:VEVENT SUMMARY:David Russell Luke (Universität Göttingen\, Germany) DTSTART:20201005T200000Z DTEND:20201005T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/7 DESCRIPTION:Title: Optimization on Spheres : Models and Proximal Algorithms with Computational Performance Comparisons\nby David Russell Luke (Un iversität Göttingen\, Germany) as part of CRM Applied Math Seminar\n\n\n Abstract\nWe present a unified treatment of the abstract problem of findin g the best approximation between a cone and spheres in the image of affine transformations. Prominent instances of this problem are phase retrieval and source localization. The common geometry binding these problems permit s a generic application of algorithmic ideas and abstract convergence resu lts for nonconvex optimization. We organize variational models for this pr oblem into three different classes and derive the main algorithmic approac hes within these classes (13 in all). We identify the central ideas underl ying these methods and provide thorough numerical benchmarks comparing the ir performance on synthetic and laboratory data. The software and data of our experiments are all publicly accessible. We also introduce one new alg orithm\, a cyclic relaxed Douglas-Rachford algorithm\, which outperforms a ll other algorithms by every measure: speed\, stability and accuracy. The analysis of this algorithm remains open.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Sheehan Olver (Imperial College London\, UK) DTSTART:20201019T200000Z DTEND:20201019T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/8 DESCRIPTION:Title: Sparse Spectral Methods for Power-Law Interactions\n by Sheehan Olver (Imperial College London\, UK) as part of CRM Applied Mat h Seminar\n\n\nAbstract\nAttractive-repulsive power law equilbriums are an important tool in modelling phenomena in collective behaviour: picture a flock of birds which simultaneously group together\, but not too closely ( i.e.\, they practice social distancing)\, until an equilibrium distributio n is reached. In this talk we show that orthogonal polynomials have sparse recurrence relationships for power law (Riesz) kernels. This leads to hig hly structured and efficiently solvable linear systems for the attractive- repulsive case with two such kernels of opposite sign\, giving an effectiv e numerical method for computing such equilibrium distributions. This link s to and builds on related work in logarithmic potential theory\, singular integral equations\, and fractional differential equations.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Johannes Royset (Naval Postgraduate School\, USA) DTSTART:20201026T200000Z DTEND:20201026T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/9 DESCRIPTION:Title: Variational Perspectives on Mathematical Optimization\nby Johannes Royset (Naval Postgraduate School\, USA) as part of CRM App lied Math Seminar\n\n\nAbstract\nThe mathematical tools for building optim ization models and algorithms grow out of linear algebra\, differential ca lculus and real analysis. However\, the needs of applications have led to a new area of mathematics that can handle systems of inequalities and func tions that are neither smooth nor well-defined in a traditional sense. Var iational analysis is the broad term for this area of mathematics. In this presentation\, we show its crucial role in the development of optimization models and algorithms in finite dimensions. First\, we examine variationa l geometry and definitions of normal and tangent vectors that extend the c lassical notions for smooth manifolds. This in turn leads to subdifferenti ability\, a wide range of calculus rules and optimality conditions for arb itrary functions. Second\, we develop an approximation theory for optimiza tion problems that leads to consistent approximations\, error bounds and r ates of convergence even in the nonconvex and nonsmooth setting.\n\nDr. Jo hannes O. Royset is Professor of Operations Research at the Naval Postgrad uate School. Dr. Royset's research focuses on formulating and solving stoc hastic and deterministic optimization problems arising in data analytics\, sensor management\, and reliability engineering. He was awarded a Nationa l Research Council postdoctoral fellowship in 2003\, a Young Investigator Award from the Air Force Office of Scientific Research in 2007\, and the B archi Prize as well as the MOR Journal Award from the Military Operations Research Society in 2009. He received the Carl E. and Jessie W. Menneken F aculty Award for Excellence in Scientific Research in 2010 and the Goodeve Medal from the Operational Research Society in 2019. Dr. Royset was a ple nary speaker at the International Conference on Stochastic Programming in 2016 and at the SIAM Conference on Uncertainty Quantification in 2018. He has a Doctor of Philosophy degree from the University of California at Ber keley (2002). Dr. Royset has been an associate or guest editor of Operatio ns Research\, Mathematical Programming\, Journal of Optimization Theory an d Applications\, Journal of Convex Analysis\, Set-Valued and Variational A nalysis\, Naval Research Logistics\, and Computational Optimization and Ap plications.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Heather Harrington (Oxford University\, UK) DTSTART:20201116T210000Z DTEND:20201116T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/10 DESCRIPTION:Title: Algebraic Systems Biology\nby Heather Harrington (O xford University\, UK) as part of CRM Applied Math Seminar\n\n\nAbstract\n Signalling pathways in molecular biology can be modelled by polynomial dyn amical systems. I will present models describing two biological systems in volved in development and cancer. I will overview approaches to analyse th ese models with data using computational algebraic geometry\, differential algebra and statistics. Finally\, I will present how topological data ana lysis can provide additional information to distinguish wild-type and muta nt molecules in one pathway. These case studies showcase how computational geometry\, topology and dynamics can provide new insights in the biologic al systems\, specifically how changes at the molecular scale (e.g. molecul ar mutations) result in kinetic differences that are observed as phenotypi c changes (e.g.mutations in fruit fly wings).\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Thomas Surowiec (Philipps-Universität Marburg\, Germany) DTSTART:20201123T210000Z DTEND:20201123T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/11 DESCRIPTION:Title: A Primal-Dual Algorithm for Risk Minimization in PDE-Co nstrained Optimization\nby Thomas Surowiec (Philipps-Universität Marb urg\, Germany) as part of CRM Applied Math Seminar\n\n\nAbstract\nWe prese nt an algorithm for the solution of risk-averse optimization problems. The setting is sufficiently general so as to encompass both finite-dimensiona l and PDE-constrained stochastic optimization problems. Due to a lack of s moothness of many popular risk measures and non-convexity of the objective functions\, both the numerical approximation and numerical solution is a major computational challenge. The proposed algorithm addresses these issu es in part by making use of the favorable dual properties of coherent risk measures. The algorithm itself is motivated by the classical method of mu ltipliers and exploits recent results on epigraphical regularization of ri sk measures. Consequently\, the algorithm requires the solution of a seque nce of smooth problems using derivative-based methods. We prove convergenc e of the algorithm in the fully continuous setting and conclude with sever al numerical examples. The algorithm is seen to outperform a popular bundl e-trust method and a direct smoothing-plus-continuation approach.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Damek Davis (Cornell University\, USA) DTSTART:20201130T210000Z DTEND:20201130T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/12 DESCRIPTION:Title: Nonconvex Optimization for Estimation and Learning: Dyn amics\, Conditioning\, and Nonsmoothness\nby Damek Davis (Cornell Univ ersity\, USA) as part of CRM Applied Math Seminar\n\n\nAbstract\nNonconvex optimization algorithms play a major role in solving statistical estimati on and learning problems. Indeed\, simple nonconvex heuristics\, such as t he stochastic gradient method\, often provide satisfactory solutions in pr actice\, despite such problems being NP hard in the worst case. Key exampl es include deep neural network training and signal estimation from nonline ar measurements. While practical success stories are common\, strong theor etical guarantees\, are rarer. The purpose of this talk is to overview a f ew (highly non exhaustive!) settings where rigorous performance guarantees can be established for nonconvex optimization\, focusing on the interplay of algorithm dynamics\, problem conditioning\, and nonsmoothness.\n\nBio: Damek Davis received his Ph.D. in mathematics from the University of Cali fornia\, Los Angeles in 2015. In July 2016 he joined Cornell University's School of Operations Research and Information Engineering as an Assistant Professor. Damek is broadly interested in the mathematics of data science\ , particularly the interplay of optimization\, signal processing\, statist ics\, and machine learning. He is the recipient of several awards\, includ ing the INFORMS Optimization Society Young Researchers Prize in (2019) and a Sloan Research Fellowship in Mathematics (2020).\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Tim Hoheisel (McGill University) DTSTART:20210111T210000Z DTEND:20210111T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/13 DESCRIPTION:Title: Halting Time is Predictable for Large Models: A Univers ality Property and Average-case Analysis\nby Tim Hoheisel (McGill Univ ersity) as part of CRM Applied Math Seminar\n\n\nAbstract\nAverage-case an alysis computes the complexity of an algorithm averaged over all possible inputs. Compared to worst-case analysis\, it is more representative of the typical behavior of an algorithm\,but remains largely unexplored in optim ization. One difficulty is that the analysis can depend on the probability distribution of the inputs to the model. However\, we show that this is n ot the case for a class of large-scale problems trained with first-order m ethods including random least squares and one-hidden layer neural networks with random weights. In fact\, the halting time exhibits a universality property: it is independent of the probability distribution. With this bar rier for average-case analysis removed\, we provide the first explicit ave rage-case convergence rates showing a tighter complexity not captured by t raditional worst-case analysis. Finally\, numerical simulations suggest th is universality property holds for a more general class of algorithms and problems.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael P. Friedlander (University of British Columbia) DTSTART:20210118T210000Z DTEND:20210118T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/14 DESCRIPTION:Title: Polar deconvolution of mixed signals\nby Michael P. Friedlander (University of British Columbia) as part of CRM Applied Math Seminar\n\n\nAbstract\nThe signal demixing problem seeks to separate the s uperposition of multiple signals into its constituent components. We mode l the superposition process as the polar convolution of atomic sets\, whic h allows us to use the duality of convex cones to develop an efficient two -stage algorithm with sublinear iteration complexity and linear storage. If the signal measurements are random\, the polar deconvolution approach s tably recovers low-complexity and mutually-incoherent signals with high pr obability and with optimal sample complexity. Numerical experiments on bo th real and synthetic data confirm the theory and efficiency of the propos ed approach. Joint work with Zhenan Fan\, Halyun Jeong\, and Babhru Joshi at the University of British Columbia.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Nathan Kutz (University of Washington) DTSTART:20210125T210000Z DTEND:20210125T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/15 DESCRIPTION:Title: Targeted use of deep learning for physics and engineeri ng\nby Nathan Kutz (University of Washington) as part of CRM Applied M ath Seminar\n\n\nAbstract\nMachine learning and artificial intelligence al gorithms are now being used to automate the discovery of governing physica l equations and coordinate systems from measurement data alone. However\, positing a universal physical law from data is challenging: (i) An approp riate coordinate system must also be advocated and (ii) simultaneously pro posing an accompanying discrepancy model to account for the inevitable mis match between theory and measurements must be considered. Using a combina tion of deep learning and sparse regression\, specifically the sparse iden tification of nonlinear dynamics (SINDy) algorithm\, we show how a robust mathematical infrastructure can be formulated for simultaneously learning physics models and their coordinate systems. This can be done with limite d data and sensors. We demonstrate the methods on a diverse number of exa mples\, showing how data can maximally be exploited for scientific and eng ineering applications.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Zhaojun Bai (UC Davis) DTSTART:20210201T210000Z DTEND:20210201T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/16 DESCRIPTION:Title: Rayleigh quotient optimizations and eigenvalue problems \nby Zhaojun Bai (UC Davis) as part of CRM Applied Math Seminar\n\n\nA bstract\nMany computational science and data analysis techniques lead to o ptimizing Rayleigh quotient (RQ) and RQ-type objective functions\, such as computing excitation states (energies) of electronic structures\, robust classification to handle uncertainty and constrained data clustering to in corporate domain knowledge. We will discuss emerging RQ optimization prob lems\, variational principles\, and reformulations to algebraic linear and nonlinear eigenvalue problems. We will show how to exploit underlying pr operties of these eigenvalue problems for designing fast solvers\, and ill ustrate the efficacy of these solvers in applications.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Marwa El Halabi (MILA) DTSTART:20210208T210000Z DTEND:20210208T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/17 DESCRIPTION:Title: Optimal approximation for unconstrained non-submodular minimization\nby Marwa El Halabi (MILA) as part of CRM Applied Math Se minar\n\n\nAbstract\nSubmodular function minimization is well studied\, an d existing algorithms solve it exactly or up to arbitrary accuracy. Howev er\, in many applications\, such as structured sparse learning or batch Ba yesian optimization\, the objective function is not exactly submodular\, b ut close. In this case\, no theoretical guarantees exist. Indeed\, submo dular minimization algorithms rely on intricate connections between submod ularity and convexity. We show how these relations can be extended to obt ain approximation guarantees for minimizing non-submodular functions\, cha racterized by how close the function is to submodular. We also extend thi s result to noisy function evaluations. Our approximation results are the first for minimizing non-submodular functions\, and are optimal\, as esta blished by our matching lower bound.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Patrick Combettes (NC State) DTSTART:20210215T210000Z DTEND:20210215T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/18 DESCRIPTION:Title: Perspective Functions and Applications\nby Patrick Combettes (NC State) as part of CRM Applied Math Seminar\n\n\nAbstract\nIn this talk I will discuss mathematical and computational issues pertaining to perspective functions\, a powerful concept that permits to extend a co nvex function to a jointly convex one in terms of an additional scale vari able. Applications in inverse problems and statistics will be presented.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Heinz Bauschke (UBC) DTSTART:20210222T210000Z DTEND:20210222T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/19 DESCRIPTION:Title: Compositions of projection mappings: fixed point sets a nd difference vectors\nby Heinz Bauschke (UBC) as part of CRM Applied Math Seminar\n\n\nAbstract\nProjection operators and associated projection algorithms are fundamental building blocks in fixed point theory and opti mization. In this talk\, I will survey recent results on the displacement mapping of the right-shift operator and sketch a new application deepenin g our understanding of the geometry of the fixed point set of the composit ion of projection operators in Hilbert space. Based on joint works with S alha Alwadani\, Julian Revalski\, and Shawn Wang.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Paul E. Hand (Northeastern University) DTSTART:20210308T210000Z DTEND:20210308T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/20 DESCRIPTION:Title: Signal Recovery with Generative Priors\nby Paul E. Hand (Northeastern University) as part of CRM Applied Math Seminar\n\n\nAb stract\nRecovering images from very few measurements is an important task in imaging problems. Doing so requires assuming a model of what makes som e images natural. Such a model is called an image prior. Classical prior s such as sparsity have led to the speedup of Magnetic Resonance Imaging i n certain cases. With the recent developments in machine learning\, neura l networks have been shown to provide efficient and effective priors for i nverse problems arising in imaging. In this talk\, we will discuss the us e of neural network generative models for inverse problems in imaging. We will present a rigorous recovery guarantee at optimal sample complexity f or compressed sensing and other inverse problems under a suitable random m odel. We will see that generative models enable an efficient algorithm fo r phase retrieval from generic measurements with optimal sample complexity . In contrast\, no efficient algorithm is known for this problem in the c ase of sparsity priors. We will discuss strengths\, weaknesses\, and futu re opportunities of neural networks and generative models as image priors. These works are in collaboration with Vladislav Voroninski\, Reinhard He ckel\, Ali Ahmed\, Wen Huang\, Oscar Leong\, Jorio Cocola\, Muhammad Asim\ , and Max Daniels.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Clarice Poon (University of Bath) DTSTART:20210315T200000Z DTEND:20210315T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/21 DESCRIPTION:Title: Off-the-grid sparse estimation\nby Clarice Poon (Un iversity of Bath) as part of CRM Applied Math Seminar\n\n\nAbstract\nThe b ehaviour of sparse regularization using the Lasso method is well understoo d when dealing with discretized linear models. However\, the behaviour o f Lasso is poor when dealing with models with very large parameter spaces and in recent years\, there has been much interest in the use of "off-the -grid" approaches\, using a continuous parameter space in conjunction with convex optimization problem over measures. In my talk\, I will present some recent results which explain the behaviour of this method in arbitrar y dimensions. Some highlights include the use of the Fisher metric to st udy the performance of Blasso over general domains and the application of this for quantitative MRI.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Olga Mula (Paris Dauphine) DTSTART:20210322T200000Z DTEND:20210322T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/22 DESCRIPTION:Title: Depth-Adaptive Neural Networks from the Optimal Control viewpoint\nby Olga Mula (Paris Dauphine) as part of CRM Applied Math Seminar\n\n\nAbstract\nIn recent years\, deep learning has been connected with optimal control as a way to define a notion of a continuous underlyin g learning problem. In this view\, neural networks can be interpreted as a discretization of a parametric Ordinary Differential Equation which\, i n the limit\, defines a continuous-depth neural network. The learning  task then consists in finding the best ODE parameters for the problem unde r consideration\, and their number increases with the accuracy of the tim e discretization. Although important steps have been taken to realize the advantages of such continuous formulations\, most current learning techn iques fix a discretization (i.e.~the number of layers is fixed). In this work\, we propose an iterative adaptive algorithm where we progressively refine the time discretization (i.e.~we increase the number of layers). P rovided that certain tolerances are met across the iterations\, we prove that the strategy converges to the underlying continuous problem. One sal ient advantage of such a shallow-to-deep approach is that it helps to ben efit in practice from the higher approximation properties of deep networks by mitigating over-parametrization issues. The performance of the approa ch is illustrated in several numerical examples.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Sasha Aravkin (University of Washington) DTSTART:20210412T200000Z DTEND:20210412T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/23 DESCRIPTION:Title: A tale of two models for Covid-19 scenarios\nby Sas ha Aravkin (University of Washington) as part of CRM Applied Math Seminar\ n\n\nAbstract\nCovid-19 Pandemic is a defining global health event in the 21st century. Forecasting the evolution of the pandemic is a key problem for anyone trying to plan ahead. Since March 2020\, IHME has been generat ing Covid-19 scenarios\, first for US states and then for all Admin-1 loca tions around the world. These scenarios have been intensively used\; resu lts are uploaded weekly to an interactive website: https://covid19.healthd ata.org/ \nIn this talk\, we describe two core mathematical models underly ing the IHME scenarios. The first model\, dubbed CurveFit\, used strong a ssumptions to get useful predictions using extremely limited data\, and wa s used during March and April of 2020. The second model\, a data-driven S EIIR model\, was put in play in June 2020\, and provides a flexible way to incorporate relationships with key drivers such as mobility\, mask use\, and pneumonia seasonality. We describe the mathematics underlying both mo dels\, and discuss the interplay between stability\, scalability\, and com plexity in mathematical modeling.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/23/ END:VEVENT BEGIN:VEVENT SUMMARY:test DTSTART:20210614T200000Z DTEND:20210614T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/24 DESCRIPTION:Title: test\nby test as part of CRM Applied Math Seminar\n \n\nAbstract\nWe show that intertwining operators for the discrete Fourier transform form a cubic algebra $C_q$ with $q$ a root of unity. This algeb ra is intimately related to the two other well-known\nrealizations of the cubic algebra: the Askey-Wilson algebra and the Askey-Wilson-Heun algebra. \nThis is joint work with Mesuma Atakishiyeva (Universidad Autónoma del E stado de Morelos\,\nCentro de Investigación en Ciencias\, Cuernavaca\, 62 250\, Morelos\, México) and Alexei Zhedanov (School of Mathematics\, Renm in University of China\, Beijing 100872\, China) curious theorem on S-inte grables D$\\Delta$Es and its consequences\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Kuhn (EPFL) DTSTART:20210913T183000Z DTEND:20210913T193000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/26 DESCRIPTION:Title: Mathematical Foundations of Robust and Distributionally Robust Optimization\nby Daniel Kuhn (EPFL) as part of CRM Applied Mat h Seminar\n\n\nAbstract\nRobust and distributionally robust optimization a re modeling paradigms for decision-making under uncertainty where the unce rtain parameters are only known to reside in an uncertainty set or are gov erned by any probability distribution from within an ambiguity set\, respe ctively\, and a decision is sought that minimizes a cost function under th e most adverse outcome of the uncertainty. In this paper\, we develop a r igorous and general theory of robust and distributionally robust nonlinear optimization using the language of convex analysis. Our framework is bas ed on a generalized `primal-worst-equals-dual-best' principle that establi shes strong duality between a semi-infinite primal worst and a non-convex dual best formulation\, both of which admit finite convex reformulations. This principle offers an alternative formulation for robust optimization problems that may be computationally advantageous\, and it obviates the ne ed to mobilize the machinery of abstract semi-infinite duality theory to p rove strong duality in distributionally robust optimization. We illustrat e the modeling power of our approach through convex reformulations for dis tributionally robust optimization problems whose ambiguity sets are define d through general optimal transport distances\, which generalize earlier r esults for Wasserstein ambiguity sets.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Diane Guignard (University of Ottawa) DTSTART:20210920T200000Z DTEND:20210920T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/27 DESCRIPTION:Title: Nonlinear reduced models for parametric PDEs\nby Di ane Guignard (University of Ottawa) as part of CRM Applied Math Seminar\n\ n\nAbstract\nWe consider model reduction methods for parametric partial di fferential equations. The usual approach to model reduction is to constru ct a low dimensional linear space which accurately approximates the parame ter-to-solution map\, and use it to build an efficient forward solver. Ho wever\, the construction of a suitable linear space is not always feasible numerically. It is well-known that nonlinear methods may provide improve d efficiency. In a so-called library approximation\, the idea is to repla ce the linear space by a collection of linear (or affine) spaces of smalle r dimension. In this talk\, we first review standard linear methods for m odel reduction. Then\, we present a strategy which can be used to generat e a nonlinear reduced model\, namely a library based on piecewise (Taylor) polynomials. We provide an analysis of the method\, in particular the de rivation of an upper bound on the size of the library\, and illustrate its performance through several numerical experiments.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Gitta Kutyniok (LMU Munich) DTSTART:20210927T183000Z DTEND:20210927T193000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/28 DESCRIPTION:Title: The Modern Mathematics of Deep Learning\nby Gitta K utyniok (LMU Munich) as part of CRM Applied Math Seminar\n\n\nAbstract\nDe spite the outstanding success of deep neural networks in real-world applic ations\, ranging from science to public life\, most of the related researc h is empirically driven and a comprehensive mathematical foundation is sti ll missing. At the same time\, these methods have already shown their imp ressive potential in mathematical research areas such as imaging sciences\ , inverse problems\, or numerical analysis of partial differential equatio ns\, sometimes by far outperforming classical mathematical approaches for particular problem classes. The goal of this lecture is to first provide an introduction into this new vibrant research area. We will then survey recent advances in two directions\, namely the development of a mathematic al foundation of deep learning and the introduction of novel deep learning -based approaches to mathematical problem settings.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Jason Bramburger (George Mason University) DTSTART:20211004T200000Z DTEND:20211004T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/29 DESCRIPTION:Title: Deep learning of conjugate mappings\nby Jason Bramb urger (George Mason University) as part of CRM Applied Math Seminar\n\n\nA bstract\nDespite many of the most common chaotic dynamical systems being c ontinuous in time\, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincaré first made this connecti on by tracking consecutive iterations of the continuous flow with a lower- dimensional\, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now ref erred to as a Poincaré map\, and it is the primary method available for i nterpreting and classifying chaotic dynamics. Unfortunately\, in all but t he simplest systems\, an explicit form for such a mapping remains outstand ing. In this talk I present a method of discovering explicit Poincaré map pings using deep learning to construct an invertible coordinate transforma tion into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is ba sed on an autoencoder\, which allows for dimensionality reduction\, and ha s the advantage of classifying chaotic systems using the equivalence relat ion of topological conjugacies. We illustrate with low-dimensional systems such as the Rössler and Lorenz systems\, while also demonstrating the ut ility of the method on the infinite-dimensional Kuramoto--Sivashinsky equa tion.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Terry Rockafellar (University of Washington) DTSTART:20211018T200000Z DTEND:20211018T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/30 DESCRIPTION:Title: Hidden convexity in nonconvex optimization\nby Terr y Rockafellar (University of Washington) as part of CRM Applied Math Semin ar\n\n\nAbstract\nIn nonconvex optimization\, not only the objective but e ven the feasible set may lack convexity. It may seem therefore that the c oncepts and methodology of convex optimization can no longer have a fundam ental role\, but this is actually wrong. Standard sufficient conditions f or local optimality in nonlinear programming and its extensions turn out t o correspond to characterizing optimality in terms of a local convex-conca ve-type saddle point of an augmented Lagrangian function. Algorithms that effectively in both primal and dual elements are thereby revealed as work ing just as they would in the convex case.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Aaron Berk (University of British Columbia) DTSTART:20211025T200000Z DTEND:20211025T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/31 DESCRIPTION:Title: On LASSO parameter sensitivity\nby Aaron Berk (Univ ersity of British Columbia) as part of CRM Applied Math Seminar\n\n\nAbstr act\nCompressed sensing theory explains why LASSO programs recover structu red high-dimensional signals with minimax order-optimal error. Yet\, the optimal choice of the program's governing parameter is often unknown in pr actice. It is still unclear how variation of the governing parameter impa cts recovery error in compressed sensing\, which is otherwise provably sta ble and robust. We provide an overview of parameter sensitivity in LASSO programs in the setting of proximal denoising\; and of compressed sensing with subgaussian measurement matrices and gaussian noise. We demonstrate how two popular ell-1 minimization programs exhibit sensitivity with respe ct to their parameter choice and illustrate the theory with numerical simu lations. For example\, a 1% error in the estimate of a parameter can caus e the error to increase by a factor of 10^9\, while choosing a different L ASSO program avoids such sensitivity issues. We hope that revealing param eter sensitivity regimes of LASSO programs helps to inform a practitioner' s choice.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Frank E. Curtis (Lehigh University) DTSTART:20211101T200000Z DTEND:20211101T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/32 DESCRIPTION:Title: Algorithms for Deterministically Constrained Stochast ic Optimization\nby Frank E. Curtis (Lehigh University) as part of CR M Applied Math Seminar\n\n\nAbstract\nI will present the recent work by my research group on the design\, analysis\, and implementation of algorithm s for solving nonlinear optimization problems that involve a stochastic  objective function and deterministic constraints.  The talk will focus o n our sequential quadratic optimization (commonly known as SQP) methods fo r cases when the constraints are defined by nonlinear systems of equatio ns\, which arise in various applications including optimal control\, PDE-c onstrained optimization\, and network optimization problems.  One might also consider our techniques for training machine learning (e.g.\, deep le arning) models with constraints.  I will also discuss the various extens ions that my group is exploring along with other related open questions.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Rainer Groh (Bristol Composites Institute (ACCIS)) DTSTART:20211108T210000Z DTEND:20211108T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/33 DESCRIPTION:Title: Experimental continuation of nonlinear load-bearing str uctures\nby Rainer Groh (Bristol Composites Institute (ACCIS)) as part of CRM Applied Math Seminar\n\n\nAbstract\nThe drive for lightweighting i n structural engineering leads to ever thinner structures that deform in n onlinear ways and that are prone to sudden instabilities. Simultaneously\ , a renewed interest in structural instability revolves around purposefull y embedding instabilities in structures to add functionality beyond struct ural load-carrying capability (e.g. dynamic shape adaptivity). To date\, the design of nonlinear structures is guided almost entirely by computati onal modelling\, in particular the use of numerical continuation tools. A dvances in experimental testing of nonlinear structures\, on the other han d\, are significantly lagging behind numerical methods. While numerical c ontinuation principles such as path-following\, calculation of bifurcation s\, branch-switching\, and bifurcation tracking are now well established\, nonlinear experimental methods of structures have not advanced beyond sim ple displacement and force control. This means that the nonlinear respons e of even simple nonlinear structures cannot be fully characterised\, as e stablished techniques induce dynamic snaps at limit points and subcritical bifurcations. There is thus huge potential for devising novel and non-de structive ways of testing nonlinear structures by applying concepts from t he field of continuation to experimental mechanics. At the University of Bristol\, we have developed a testing method based on adding control point s with auxiliary sensors and actuators to a structure to: (i) stabilise ot herwise unstable equilibria\; (ii) control the shape of the structure to t ransition between different stable equilibria\; and (iii) compute an exper imental “tangential” stiffness matrix (the Jacobian)\, which u ltimately allows Newton's root-finding algorithm to be implemented experim entally. With this approach all the features of the numerical techniques mentioned above can (theoretically) be replicated. The testing method has been applied to laboratory scale test specimens such as the snap-through of a shallow arch\, and this seminar will provide an overview of the mathe matical background to experimental continuation\, its application\, and ou tlook to future experiments.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Wolfgang Dahmen (University of South Carolina) DTSTART:20211115T210000Z DTEND:20211115T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/34 DESCRIPTION:Title: Some Thoughts on Physics Informed Neural Networks\n by Wolfgang Dahmen (University of South Carolina) as part of CRM Applied M ath Seminar\n\n\nAbstract\nEmploying Deep Learning concepts to "learn" phy sical laws\, has been recently attracting significant attention. In partic ular\, so called "Physics Informed Neural Networks” (PINN) refers to a p aradigm where the training of model surrogates is based on empirical risks that require only point-wise evaluation of residuals. This avoids the exp ensive computation of a sufficiently large number of training data\, typic ally given in terms of high fidelity approximations of model states. \n\nT he core issue addressed in this talk is the prediction capability of such methods for models given in terms of parameter-dependent families of parti al differential equations. Related specific questions concern\, for instan ce\, the choice of "variationally correct" training risks\, that convey ce rtifiable information about the achieved accuracy in problem relevant metr ics\, the role of a priori versus a posteriori error bounds\, connections with Generative Adversarial Networks\, as well as related implications on training strategies and network adaptation.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Soizic Terrien (CNRS - Université du Mans) DTSTART:20211122T210000Z DTEND:20211122T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/35 DESCRIPTION:Title: Equidistant and non equidistant pulsing patterns in an excitable microlaser with delayed feedback\nby Soizic Terrien (CNRS - Université du Mans) as part of CRM Applied Math Seminar\n\n\nAbstract\nEx citability is observed in many natural and artificial systems\, from spiki ng neurons to cardiac cells and semiconductor lasers. It corresponds to t he all-or-none pulse-shaped response of a system to an external perturbati on\, depending whether or not the perturbation amplitude exceeds the so-ca lled excitable threshold. When subject to delayed feedback\, an excitable system can regenerate its own excitable response when it is reinjected af ter a delay time τ. As the process repeats\, this results in sustained p ulsing regimes\, which can be of interest for many applications\, from dat a transmission to all-optical signal processing or neuromorphic photonic n etworks. \n\nHere we investigate the short-term and long-term dynamics of an excitable microlaser subject to delayed optical feedback. This is don e both experimentally and numerically through a bifurcation analysis of a suitable model written in the form of three delay-differential equations ( DDEs) with one fast and two slow variables. We show that almost any pulse sequence can be excited and regenerated by the system over short periods of time. In the long-term\, on the other hand\, the system settles down t o one of the coexisting\, slowly-attracting periodic orbits\, which corres pond to different numbers of pulses in the feedback cavity. We show that\ , depending on the internal timescales of the excitable system\, these pul ses appear to be either equidistant (i.e. with equalized pulse intervals) or non-equidistant in the feedback cavity. A bifurcation analysis demons trates that non-equidistant pulsing patterns originate in resonance phenom ena. The mechanism for the emergence of very large locking regions in the parameter space in investigated. \n\nJoint work with Bernd Krauskopf (Un iversity of Auckland)\, Neil Broderick (University of Auckland) and Sylvai n Barbay (C2N\, CNRS / Univ. Paris Saclay)\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrus Giraldo (The University of Auckland) DTSTART:20211129T210000Z DTEND:20211129T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/36 DESCRIPTION:Title: Degenerate singular cycles and chaotic switching in the two-site open Bose--Hubbard model\nby Andrus Giraldo (The University of Auckland) as part of CRM Applied Math Seminar\n\n\nAbstract\nThe two-si te open Bose--Hubbard dimer model is a celebrated fundamental quantum opti cal model that accounts for the dynamics of bosons at two lossy interactin g sites. Recently\, two coupled\, driven\, and lossy photonic crystal nano cavities ---which are optical devices that operate with only a few hundred photons due to their extremely small size--- have been shown to realise t his model experimentally. Thus\, there is much interest in understanding t he different behaviours that such model exhibits for theoretical and pract ical purposes.\nThis talk will show the different dynamics in the semiclas sical approximation of this quantum optical system by presenting a compreh ensive bifurcation analysis. We characterised different transitions of cha otic attractors in parameter plane by numerically computing tangency bifur cations between stable and unstable manifolds of saddle equilibria and per iodic orbits. By doing so\, we identify codimension-two degenerate singula r cycles\, and their generalisations\, as responsible for the organisation s of different tangency and heteroclinic bifurcations between saddle equil ibria periodic orbits in parameter plane. Thus\, we provide a roadmap for observable chaotic dynamics in the semiclassical approximation of the two- site Bose--Hubbard dimer model\, which connects novel results in bifurcati on theory with novel applications through numerical continuation technique s.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Haesun Park (Georgia Institute of Technology) DTSTART:20211206T210000Z DTEND:20211206T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/37 DESCRIPTION:Title: Multi-view Unsupervised and Semi-Supervised Clustering based on Content and Connection Information\nby Haesun Park (Georgia I nstitute of Technology) as part of CRM Applied Math Seminar\n\n\nAbstract\ nConstrained Low Rank Approximation (CLRA) is a powerful framework for a v ariety of important tasks in large scale data analytics such as topic disc overy in text data and community detection in social network data. In this talk\, a hybrid method called Joint Nonnegative Matrix Factorization (Joi ntNMF) is introduced for latent information discovery from multi-view data sets that contain both text content and connection structure information. The method jointly optimizes an integrated objective function\, which is a combination of the Nonnegative Matrix Factorization (NMF) objective func tion for handling text content/attribute information and the Symmetric NMF (SymNMF) objective function for handling relation/connection information. An effective algorithm for the joint NMF objective function is proposed u tilizing the block coordinate descent (BCD) method.\nThe proposed hybrid m ethod simultaneously discovers content associations and related latent con nections without any need for post-processing or additional clustering. In addition\, known partial label information can be incorporated into a Joi ntNMF for semi-supervised clustering framework. The experimental results f rom several real-life application problems illustrate the advantages of th e proposed approaches.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Alex Bihlo (Memorial University) DTSTART:20220110T210000Z DTEND:20220110T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/38 DESCRIPTION:Title: Deep neural networks for solving differential equations on general orientable surfaces\nby Alex Bihlo (Memorial University) a s part of CRM Applied Math Seminar\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Matus Benko (Johannes Kepler University Linz) DTSTART:20220117T210000Z DTEND:20220117T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/39 DESCRIPTION:Title: Variational Analysis: Basics\, Calculus\, and Semismoot hness*\nby Matus Benko (Johannes Kepler University Linz) as part of CR M Applied Math Seminar\n\n\nAbstract\nThe purpose of this talk is to offer a brief introduction into set-valued and variational analysis and to try to motivate the study of this area. To this end\, we first discuss some ba sic notions and ideas. Namely\, we try to explain why set-valued mappings should be analyzed\, what properties of such mappings seems to be useful a nd are typically studied\, as well as how one can analyze them\, i.e.\, wh at are the available tools. It should not be very surprising that\, just l ike in the standard analysis of functions\, derivatives play a crucial rol e. Thus\, we clarify how to differentiate set-valued mappings using the ma chinery of variational geometry (tangent and normal cones). Then we discus s in more depth the topic of calculus rules that enable one to properly ma nipulate with generalized derivatives and apply them to practically releva nt problems. We conclude with some remarks about the new property of semis moothness* for set-valued mappings and the related Newton method for solvi ng generalized equations (inclusions).\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/39/ END:VEVENT BEGIN:VEVENT SUMMARY:David Rolnick (McGill University) DTSTART:20220124T210000Z DTEND:20220124T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/40 DESCRIPTION:Title: Expressivity and learnability in deep neural networks\nby David Rolnick (McGill University) as part of CRM Applied Math Semin ar\n\n\nAbstract\nIn this talk\, we show that there is a large gap between the maximum complexity of the functions that a neural network can express and the expected complexity of the functions that it learns in practice. Deep ReLU networks are piecewise linear functions\, and the number of dis tinct linear regions is a natural measure of their expressivity. It is we ll-known that the maximum number of linear regions grows exponentially wit h the depth of the network\, and this has often been used to explain the s uccess of deeper networks. We show that the expected number of linear reg ions in fact grows polynomially in the size of the network\, far below the exponential upper bound and independent of the depth of the network. Thi s statement holds true both at initialization and after training\, under n atural assumptions for gradient-based learning algorithms. We also show t hat the linear regions of a ReLU network reveal information about the netw ork's parameters. In particular\, it is possible to reverse-engineer the weights and architecture of an unknown deep ReLU network merely by queryin g it.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Sebastien Le Digabel (Polytechnique Montreal) DTSTART:20220214T210000Z DTEND:20220214T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/41 DESCRIPTION:by Sebastien Le Digabel (Polytechnique Montreal) as part of CR M Applied Math Seminar\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Tom Trogdon (U of Washington) DTSTART:20220221T210000Z DTEND:20220221T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/42 DESCRIPTION:by Tom Trogdon (U of Washington) as part of CRM Applied Math S eminar\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Guy Wolf (UdeM) DTSTART:20220307T210000Z DTEND:20220307T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/43 DESCRIPTION:by Guy Wolf (UdeM) as part of CRM Applied Math Seminar\n\nAbst ract: TBA\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Margarida Carvalho (UdeM) DTSTART:20220321T200000Z DTEND:20220321T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/44 DESCRIPTION:Title: Mathematical Programming Games: pushing the limits of e quilibria computation\nby Margarida Carvalho (UdeM) as part of CRM App lied Math Seminar\n\n\nAbstract\nMathematical programming games (MPGs) enc ompass flexible problem modeling when decision makers interact. Through t hem\, we can reflect each player's goal in the game through a parametric o ptimization problem. In this talk\, we will first provide examples of suc h games and their MPG formulation. Then\, we will focus on MPGs where dec isions can take integer values\, the so-called integer programming games. We will also discuss Nash games among Stackelberg leaders. The theoretic al intractability of these games will be presented as well as algorithmic schemes to solve them in practice.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Soledad Villar (Johns Hopkins) DTSTART:20220404T200000Z DTEND:20220404T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/45 DESCRIPTION:Title: Units-equivariant machine learning\nby Soledad Vill ar (Johns Hopkins) as part of CRM Applied Math Seminar\n\n\nAbstract\nWe c ombine ideas from dimensional analysis and from equivariant machine learni ng to provide an approach for units-equivariant machine learning. Units eq uivariance is the exact symmetry that follows from the requirement that re lationships among measured quantities must obey self-consistent dimensiona l scalings. Our approach is to construct a dimensionless version of the le arning task\, using classic results from dimensional analysis\, and then p erform the learning task in the dimensionless space. This approach can be used to impose units equivariance on almost any contemporary machine-learn ing methods\, including those that are equivariant to rotations and other groups. Units equivariance is expected to be particularly valuable in the contexts of symbolic regression and emulation. We discuss the in-sample an d out-of-sample prediction accuracy gains one can obtain if exact units eq uivariance is imposed\; the symmetry is extremely powerful in some context s. We illustrate these methods with simple numerical examples involving dy namical systems in physics and ecology.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Giang Tran (University of Waterloo) DTSTART:20220411T200000Z DTEND:20220411T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/46 DESCRIPTION:Title: Sparse Random Feature Models: Theoretical Guarantees an d Applications\nby Giang Tran (University of Waterloo) as part of CRM Applied Math Seminar\n\n\nAbstract\nRandom feature methods have been succe ssful in various machine learning tasks\, are easy to compute\, and come w ith theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent similar function spaces without a costly training phase. However\, for accuracy\, random feature methods require more measurements than trainable parameters\, limiting th eir use for data-scarce applications or problems in scientific machine lea rning. In this talk\, we will introduce the sparse random feature expansi on to obtain parsimonious random feature models. Specifically\, we levera ge ideas from compressive sensing to generate random feature expansions wi th theoretical guarantees even in the data-scarce setting. We also presen t a random feature model for approximating high-dimensional sparse additiv e functions and a sparse random mode decomposition to extract intrinsic mo des from challenging time-series data. Comparisons show that our proposed approaches perform better or are comparable to other state-of-the-art or popular methods. Applications of our methods on identifying important var iables in high-dimensional settings as well as on decomposing music pieces and visualizing black-hole mergers will be addressed.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Fabian Pedregosa (Google) DTSTART:20220207T210000Z DTEND:20220207T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/47 DESCRIPTION:Title: Efficient and Modular Implicit Differentiation\nby Fabian Pedregosa (Google) as part of CRM Applied Math Seminar\n\n\nAbstrac t\nAutomatic differentiation (autodiff) has revolutionized machine learnin g. It allows expressing complex computations by composing elementary ones in creative ways and removes the burden of computing their derivatives by hand. More recently\, differentiation of optimization problem solutions has attracted widespread attention with applications such as optimization layers\, and in bi-level problems such as hyper-parameter optimization and meta-learning. However\, so far\, implicit differentiation remained diff icult to use for practitioners\, as it often required case-by-case tedious mathematical derivations and implementations. In this paper\, we propose a unified\, efficient and modular approach for implicit differentiation o f optimization problems. In our approach\, the user defines directly in P ython a function F capturing the optimality conditions of the problem to b e differentiated. Once this is done\, we leverage autodiff of F and impli cit differentiation to automatically differentiate the optimization proble m. Our approach thus combines the benefits of implicit differentiation an d autodiff. It is efficient as it can be added on top of any state-of-the -art solver and modular as the optimality condition specification is decou pled from the implicit differentiation mechanism. We show that seemingly simple principles allow to recover many exiting implicit differentiation m ethods and create new ones easily. We demonstrate the ease of formulating and solving bi-level optimization problems using our framework. We also showcase an application to the sensitivity analysis of molecular dynamics. \n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Ma (University of California\, Irvine) DTSTART:20220314T200000Z DTEND:20220314T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/48 DESCRIPTION:Title: The Kaczmarz Algorithm: Greed\, Randomness\, and Tensor s\nby Anna Ma (University of California\, Irvine) as part of CRM Appli ed Math Seminar\n\n\nAbstract\nIn settings where data sets become extremel y large-scale\, stochastic iterative methods such as the Kaczmarz algorith m and Randomized Coordinate Descent become advantageous due to their low m emory footprint. The Randomized Kaczmarz algorithm in particular has garn ered attention owing to its applicability in large-scale settings and its elegant geometric interpretation. In this talk\, we will discuss the Rand omized Kaczmarz algorithm\, it's connection to the popular Stochastic Grad ient Descent algorithm and it's greedy counter-part: Motzkin's Method. Th is presentation contains joint work with Jamie Haddock and Denali Molitor. \n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Tom Trogdon (U of Washington) DTSTART:20220328T200000Z DTEND:20220328T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/49 DESCRIPTION:Title: Perturbations of orthogonal polynomials: Riemann-Hilber t problems\, random matrices and numerical linear algebra\nby Tom Trog don (U of Washington) as part of CRM Applied Math Seminar\n\n\nAbstract\nW e consider the perturbation of orthogonal polynomials (OPs) with respect t o changes in the orthogonality measure. While the transformation from a m easure to its orthogonal polynomials is typically ill-conditioned as the d egree of the polynomial grows\, using the Fokas-Its-Kitaev Riemann--Hilber t problem\, we show that in certain settings this mapping is well-conditio ned. A usable perturbation theory can then be obtained. The results are strengthened when the asymptotics of the OPs with respect to a limiting me asure are known. Our main applications are to random matrices and to numer ical algorithms and dynamical systems applied to these random matrices. T his is joint work with Percy Deift\, Xuicai Ding and Elliot Paquette.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Bruno Després (Jacques-Louis Lions Laboratory) DTSTART:20220519T200000Z DTEND:20220519T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/50 DESCRIPTION:Title: Neural Networks from the viewpoint of Numerical Analysi s\nby Bruno Després (Jacques-Louis Lions Laboratory) as part of CRM A pplied Math Seminar\n\n\nAbstract\nThe presentation will focus on the inte rplay between\, on the one hand Neural Networks and Machine Learning which are emerging hot topics\, and on the other hand Numerical Analysis which is now a classical topic. The Yarotsky theorem will be discussed together with a recent alternative to the polarisation formula (D.-Ancellin 19'). T he stability of the Adam algorithm will be shown with a particular Lyapuno v function. Discretization of transport equations for CFD will serve as an applicative illustration.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/50/ END:VEVENT BEGIN:VEVENT SUMMARY:James Forbes (McGill) DTSTART:20220912T200000Z DTEND:20220912T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/51 DESCRIPTION:Title: Regularization Techniques in Koopman-based System Ident ification\nby James Forbes (McGill) as part of CRM Applied Math Semina r\n\n\nAbstract\nUsing the Koopman operator\, nonlinear systems can be exp ressed as infinite-dimensional linear systems. Data-driven methods can the n be used to approximate a finite-dimensional Koopman operator\, which is particularly useful for system identification\, control\, and state estima tion tasks. However\, approximating large Koopman operators is numerically challenging\, leading to unstable Koopman operators being identified for otherwise stable systems.\nThis talk will present a selection of technique s to regularize the Koopman regression problem\, including a novel H-infin ity norm regularizer. In particular\, how to re-pose the system identifica tion problem in order to leverage numerically efficient optimization tools \, such as linear matrix inequalities\, will be presented. This talk is ba sed on the pre-print arxiv\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/51/ END:VEVENT BEGIN:VEVENT SUMMARY:Quentin Bertrand (Mila) DTSTART:20220919T200000Z DTEND:20220919T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/52 DESCRIPTION:Title: Implicit Differentiation in Non-Smooth Convex Learning< /a>\nby Quentin Bertrand (Mila) as part of CRM Applied Math Seminar\n\n\nA bstract\nFinding the optimal hyperparameters of a model can be cast as a b ilevel optimization problem\, typically solved zero-order techniques. In t his work we study first-order methods when the inner optimization problem is convex but non-smooth. We show that the forward-mode differentiation of proximal gradient descent and proximal coordinate descent yield sequences of Jacobians converging toward the exact Jacobian. Using implicit differe ntiation\, we show it is possible to leverage the non-smoothness of the in ner problem to speed up the computation. Finally\, we provide a bound on t he error made on the hypergradient when the inner optimization problem is solved approximately. Results on regression and classification problems re veal computational benefits for hyperparameter optimization\, especially w hen multiple hyperparameters are required.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Akitoshi Takayasu (University of Tsukuba) DTSTART:20220926T200000Z DTEND:20220926T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/53 DESCRIPTION:Title: A general approach for rigorously integrating PDEs usin g semigroup theory\nby Akitoshi Takayasu (University of Tsukuba) as pa rt of CRM Applied Math Seminar\n\n\nAbstract\nIn this talk we introduce a general rigorous PDE integrator that proves the existence of a solution to the Cauchy problem of time-dependent PDEs. We derive a fixed-point formul ation to prove the existence of a solution locally in time\, which is base d on the solution map of a linearized problem called evolution operator. U sing rigorous numerics we validate the contraction of the fixed-point form on a neighborhood of a numerically computed approximate solution. Then we extend the time interval to exist the solution via time stepping. The mai n advantage of our approach is that the rigorous integrator can be applied to a general class of PDEs\, even performed for higher spatial dimensiona l PDEs.\nThis is joint work with Jean-Philippe Lessard and Gabriel Duchesn e.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/53/ END:VEVENT BEGIN:VEVENT SUMMARY:David Goluskin (U. of Victoria) DTSTART:20221003T200000Z DTEND:20221003T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/54 DESCRIPTION:Title: Verifying global stability of fluid flows despite trans ient growth of energy\nby David Goluskin (U. of Victoria) as part of C RM Applied Math Seminar\n\n\nAbstract\nVerifying nonlinear stability of a laminar fluid flow against all perturbations is a classic challenge in flu id dynamics. All past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. This "energy method" ca nnot show global stability of any flow in which perturbation energy may gr ow transiently. For the many flows that allow transient energy growth but seem to be globally stable (e.g. pipe flow and other parallel shear flows at certain Reynolds numbers) there has been no way to mathematically verif y global stability. After explaining why the energy method was the only wa y to verify global stability of fluid flows for over 100 years\, I will de scribe a different approach that is broadly applicable but more technical. This approach\, proposed in 2012 by Goulart and Chernyshenko\, uses sum-o f-squares polynomials to computationally construct non-quadratic Lyapunov functions that decrease monotonically for all flow perturbations. I will p resent a computational implementation of this approach for the example of 2D plane Couette flow\, where we have verified global stability at Reynold s numbers above the energy stability threshold. This energy stability resu lt for 2D Couette flow had not been improved upon since being found by Orr in 1907. The results I will present are the first verification of global stability – for any fluid flow – that surpasses the energy m ethod. This is joint work with Federico Fuentes (Universidad Católica d e Chile) and Sergei Chernyshenko (Imperial College London).\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/54/ END:VEVENT BEGIN:VEVENT SUMMARY:Nicola Guglielmi (Gran Sasso Science Institute) DTSTART:20221017T200000Z DTEND:20221017T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/55 DESCRIPTION:by Nicola Guglielmi (Gran Sasso Science Institute) as part of CRM Applied Math Seminar\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/55/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Baraldi (Sandia National Labs) DTSTART:20221024T200000Z DTEND:20221024T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/56 DESCRIPTION:by Robert Baraldi (Sandia National Labs) as part of CRM Applie d Math Seminar\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/56/ END:VEVENT BEGIN:VEVENT SUMMARY:Serge Prudhomme (Polytechnique Montreal) DTSTART:20221107T210000Z DTEND:20221107T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/57 DESCRIPTION:by Serge Prudhomme (Polytechnique Montreal) as part of CRM App lied Math Seminar\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/57/ END:VEVENT BEGIN:VEVENT SUMMARY:Kimon Fountoulakis (U. of Waterloo) DTSTART:20221128T210000Z DTEND:20221128T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/58 DESCRIPTION:by Kimon Fountoulakis (U. of Waterloo) as part of CRM Applied Math Seminar\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/58/ END:VEVENT BEGIN:VEVENT SUMMARY:Stephanie Dodson (Colby College) DTSTART:20221205T210000Z DTEND:20221205T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/59 DESCRIPTION:by Stephanie Dodson (Colby College) as part of CRM Applied Mat h Seminar\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/59/ END:VEVENT BEGIN:VEVENT SUMMARY:Tatiana Bubba (University of Bath) DTSTART:20221114T200000Z DTEND:20221114T210000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/60 DESCRIPTION:by Tatiana Bubba (University of Bath) as part of CRM Applied M ath Seminar\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/60/ END:VEVENT BEGIN:VEVENT SUMMARY:Ashwin Pananjady (Georgia Tech) DTSTART:20221121T210000Z DTEND:20221121T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/61 DESCRIPTION:by Ashwin Pananjady (Georgia Tech) as part of CRM Applied Math Seminar\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/61/ END:VEVENT BEGIN:VEVENT SUMMARY:Test (test) DTSTART:20221212T210000Z DTEND:20221212T220000Z DTSTAMP:20260404T110654Z UID:AppliedMathematics/62 DESCRIPTION:Title: test\nby Test (test) as part of CRM Applied Math Se minar\n\n\nAbstract\nA tensor T(x_1\, ...\, x_n) is a multilinear function of the input vectors x_j in F_q^n\, F_q a finite field. T has a small an alytic rank if its output distribution is far from uniform. It has partiti on rank `r' if we can write T = f_1 * g_1 + ... + f_r * g_r\, where f_r an d g_r are tensors in fewer variables. Analytic rank measures the amount of randomness\, and partition rank measures the amount of structure. It is k nown that if `f' has small partition rank\, it must have small analytic ra nk. Green and Tao proved an inverse theorem stating that if `f' has small analytic rank then it has small partition rank. Their bound was qualitativ e\, however\, and several authors gave quantitative improvements. Janzer a nd Milicevic independently proved a polynomial dependence. We prove an opt imal inverse theorem: the analytic rank and partition rank are equivalent up to constant factors over large enough fields. Our techniques are very d ifferent from the usual methods in this area\, we rely on algebraic geomet ry rather than additive combinatorics. This is joint work with Guy Moshkov itz.\n LOCATION:https://stable.researchseminars.org/talk/AppliedMathematics/62/ END:VEVENT END:VCALENDAR