Thu 15 Feb 15:00: Adaptive Intrusive Methods for Forward UQ in PDEs
In this talk we discuss a so-called intrusive approach for the forward propagation of uncertainty in PDEs with uncertain coefficients. Specifically, we focus on stochastic Galerkin finite element methods (SGFEMs). Multilevel variants of such methods provide polynomial-based surrogates with spatial coefficients that reside in potentially different finite element spaces. For elliptic PDEs with diffusion coefficients represented as affine functions of countably infinitely many parameters, well established theoretical results state that such methods can achieve rates of convergence independent of the number of input parameters, thereby breaking the curse of dimensionality. Moreover, for nice enough test problems, it is even possible to prove convergence rates afforded to the chosen finite element method for the associated deterministic PDE . However, achieving these rates in practice using automated computational algorithms remains highly challenging, and non-intrusive multilevel sampling methods are often preferred for their ease of use. We discuss an adaptive framework that is driven by a classical hierarchical a posteriori error estimation strategy — modified for the more challenging parametric PDE setting — and present numerical results.
- Speaker: Catherine Powell (University of Manchester)
- Thursday 15 February 2024, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Nicolas Boulle.
Thu 01 Feb 15:00: What happens when you chop an equation?
This talk will discuss a tricky business: truncating a differential equation to produce finite solutions. A truncation scheme is often built directly into the steps needed to create a numerical system. E.g., finite differences replace exact differential operators with more manageable shadows, sweeping the exact approach off the stage.
In contrast, this talk will discuss the “tau method” which adds an explicit parameterised perturbation to an original equation. By design, the correction calls into existence an exact (finite polynomial) solution to the updated analytic system. The hope is that the correction comes out minuscule after comparing it with a hypothetical exact solution. The tau method has worked splendidly in practice, starting with Lanczos’s original 1938 paper outlining the philosophy. However, why the scheme works so well (and when it fails) remains comparably obscure. While addressing the theory behind the Tau method, this talk will answer at least one conceptual question: Where does an infinite amount of spectrum go when transitioning from a continuous differential equation to an exact finite matrix representation?
- Speaker: Geoff Vasil (University of Edinburgh)
- Thursday 01 February 2024, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Nicolas Boulle.
Thu 01 Feb 15:00: What happens when you chop an equation?
This talk will discuss a tricky business: truncating a differential equation to produce finite solutions. A truncation scheme is often built directly into the steps needed to create a numerical system. E.g., finite differences replace exact differential operators with more manageable shadows, sweeping the exact approach off the stage.
In contrast, this talk will discuss the “tau method” which adds an explicit parameterised perturbation to an original equation. By design, the correction calls into existence an exact (finite polynomial) solution to the updated analytic system. The hope is that the correction comes out minuscule after comparing it with a hypothetical exact solution. The tau method has worked splendidly in practice, starting with Lanczos’s original 1938 paper outlining the philosophy. However, why the scheme works so well (and when it fails) remains comparably obscure. While addressing the theory behind the Tau method, this talk will answer at least one conceptual question: Where does an infinite amount of spectrum go when transitioning from a continuous differential equation to an exact finite matrix representation?
- Speaker: Geoff Vasil (University of Edinburgh)
- Thursday 01 February 2024, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Nicolas Boulle.
Thu 25 Jan 15:00: The future of governing equations
A major challenge in the study of dynamical systems is that of model discovery: turning data into reduced order models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SIN Dy to scale efficiently to problems with multiple time scales, noise and parametric dependencies. For systems with incomplete observations, we show that the Hankel alternative view of Koopman (HAVOK) method, based on time-delay embedding coordinates and the dynamic mode decomposition, can be used to obtain a linear models and Koopman invariant measurement systems that nearly perfectly captures the dynamics of nonlinear quasiperiodic systems. Neural networks are used in targeted ways to aid in the model reduction process. Together, these approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.
- Speaker: J. Nathan Kutz (University of Washington)
- Thursday 25 January 2024, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR2.
- Series: Applied and Computational Analysis; organiser: Nicolas Boulle.
Thu 25 Jan 15:00: The future of governing equations
A major challenge in the study of dynamical systems is that of model discovery: turning data into reduced order models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SIN Dy to scale efficiently to problems with multiple time scales, noise and parametric dependencies. For systems with incomplete observations, we show that the Hankel alternative view of Koopman (HAVOK) method, based on time-delay embedding coordinates and the dynamic mode decomposition, can be used to obtain a linear models and Koopman invariant measurement systems that nearly perfectly captures the dynamics of nonlinear quasiperiodic systems. Neural networks are used in targeted ways to aid in the model reduction process. Together, these approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.
- Speaker: J. Nathan Kutz (University of Washington)
- Thursday 25 January 2024, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR2.
- Series: Applied and Computational Analysis; organiser: Nicolas Boulle.
Thu 25 Jan 15:00: The future of governing equations
A major challenge in the study of dynamical systems is that of model discovery: turning data into reduced order models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SIN Dy to scale efficiently to problems with multiple time scales, noise and parametric dependencies. For systems with incomplete observations, we show that the Hankel alternative view of Koopman (HAVOK) method, based on time-delay embedding coordinates and the dynamic mode decomposition, can be used to obtain a linear models and Koopman invariant measurement systems that nearly perfectly captures the dynamics of nonlinear quasiperiodic systems. Neural networks are used in targeted ways to aid in the model reduction process. Together, these approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.
- Speaker: J. Nathan Kutz (University of Washington)
- Thursday 25 January 2024, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Nicolas Boulle.
Thu 25 Jan 15:00: The future of governing equations
A major challenge in the study of dynamical systems is that of model discovery: turning data into reduced order models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SIN Dy to scale efficiently to problems with multiple time scales, noise and parametric dependencies. For systems with incomplete observations, we show that the Hankel alternative view of Koopman (HAVOK) method, based on time-delay embedding coordinates and the dynamic mode decomposition, can be used to obtain a linear models and Koopman invariant measurement systems that nearly perfectly captures the dynamics of nonlinear quasiperiodic systems. Neural networks are used in targeted ways to aid in the model reduction process. Together, these approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.
- Speaker: J. Nathan Kutz (University of Washington)
- Thursday 25 January 2024, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Nicolas Boulle.
Mon 12 Feb 18:00: Going beyond emissions reduction – Climate Repair Check website for latest updates and booking information http://www.cambridgephilosophicalsociety.org
The lecture will firstly summarise exactly where we are with climate change and crucially what the scientists are now considering in terms of the future. A future based purely on emissions reductions cannot keep the world below 1.5C.
We discuss some of the exciting ideas for greenhouse gas removal, and importantly going beyond terrestrial-based carbon dioxide removal. We will explore some of the approaches for marine carbon dioxide removal as well as the development of materials to accelerate the rate of oxidation of methane.
We will then spend time discussing what additional options we might have beyond emissions reduction and greenhouse gas removal; whilst these are necessary, even the most optimistic and ambitious scenarios considered by the IPCC indicate that they are not sufficient to keep temperatures below 1.5C. We will therefore review engineering concepts to limit temperature rise or interventions to protect glaciers and sea-ice, and ostensibly buy us time to stave off the worst effects of climate change whilst we get greenhouse gas levels down.
We will explore the different technologies which are being researched at the University of Cambridge in collaboration with multiple partner universities around the world, as well as the issues of public attitudes, governance and ethics associated with such research and potential deployment.
Check website for latest updates and booking information http://www.cambridgephilosophicalsociety.org
- Speaker: Dr Shaun Fitzgerald FREng OBE, Director of Research, Centre for Climate Repair, Department of Engineering
- Monday 12 February 2024, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
Mon 29 Jan 18:00: G I TAYLOR LECTURE - The influence of GI Taylor: granular collapses, viscous gravity currents, explosive eruptions and chemical gardens Check website for latest updates and booking information http://www.cambridgephilosophicalsociety.org
The presentation will start will a short summary of the seminal work of G. I. Taylor and his most famous student, G. K. Batchelor. Evaluations of the propagation of muti-sized granular material under a variety of conditions will then be described, as well as being illustrated with desk top experiments.
The lecture will then discuss the all important flow of viscous gravity currents, again illustrated by desk top experiments and actual photos and explanations of the recent eruption of the Soufriere of St. Vincent. A description of the development of chemical gardens will then be described, initially experimented upon by Johan Glauber, said to be the first chemical engineer, and then by Isaac Newton. It is said by some that chemical gardens are the origin of life, at deep-sea smokers, as will be described.
Check website for latest updates and booking information http://www.cambridgephilosophicalsociety.org
- Speaker: Professor Herbert Huppert FRS FRSN, Professor of Theoretical Geophysics, Faculty of Mathematics
- Monday 29 January 2024, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
Mon 26 Feb 18:00: The quest for the first stars and first black holes with the James Webb Space Telescope Check website for latest updates and booking information http://www.cambridgephilosophicalsociety.org
Finding and understanding the nature of the first stars at cosmic dawn is one of the most important and most ambitious goals for modern astrophysics. The first populations of stars produced the first chemical elements heavier than helium and formed the first, small protogalaxies, which then evolved, across the cosmic epoch, into the large and mature galaxies, such as the Milky Way and those in our local neighbour. Equally important and equally challenging is the search, in the early Universe, of the seeds of the first population of black holes, which later evolved in the supermassive black holes at the centre of galaxies, with masses even exceeding a billion times the mass of the Sun. When matter accretes on such supermassive black holes it can become so luminous to vastly outshine the light emitted by all stars in their host galaxy.
Since its launch, about two years ago, the James Webb Space Telescope has been revolutionizing this area of research. Its sensitivity in detecting infrared light from the remotest parts of the Universe is orders of magnitude higher than any previous observatory, an historical leap in astronomy and, more broadly, in science. I will presents some of the first, extraordinary discoveries from the Webb telescope, which have resulted in several unexpected findings. I will also discuss the new puzzles and areas of investigation that have been opened by Webb’s observations, how these challenge theoretical models, and the prospects of further progress in the coming years.
Check website for latest updates and booking information http://www.cambridgephilosophicalsociety.org
- Speaker: Professor Roberto Maiolino FRS, Professor of Experimental Astrophysics, Kavli Institute for Cosmology
- Monday 26 February 2024, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.