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Thu 23 Nov 15:00: Pseudospectra: Two Different Ways

Talks - Mon, 20/11/2023 - 14:01
Pseudospectra: Two Different Ways

I will talk about how pseudospectra comes up in two different ways in my research. Firstly, I will talk the certification and computation of pseudospectra for a non-self-adjoint linear operator. Efficient calculation of the pseudospectrum for unbounded operators in infinite dimensional spaces is a relatively unexplored territory. Often finite-dimensional subspaces are used in place of the infinite dimensional space. However, the use of finite dimensional subspaces changes the qualitative nature of pseudospectral contours from unbounded curves with asymptotic behaviour at infinity to closed loops. The overlap between these curves forms the well-resolved of the pseudospectrum. It is useful to have a certification process for these calculated points. Secondly, I will talk about more recent work in using pseudospectra to model the HPA axis. Pseudospectra can be obtained via creating models or via data driven approaches. I will discuss the pros and cons of these different methods, which include interpretability and matching to experimental data.

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Thu 23 Nov 15:00: Pseudospectra: Two Different Ways

Other events - Mon, 20/11/2023 - 14:01
Pseudospectra: Two Different Ways

I will talk about how pseudospectra comes up in two different ways in my research. Firstly, I will talk the certification and computation of pseudospectra for a non-self-adjoint linear operator. Efficient calculation of the pseudospectrum for unbounded operators in infinite dimensional spaces is a relatively unexplored territory. Often finite-dimensional subspaces are used in place of the infinite dimensional space. However, the use of finite dimensional subspaces changes the qualitative nature of pseudospectral contours from unbounded curves with asymptotic behaviour at infinity to closed loops. The overlap between these curves forms the well-resolved of the pseudospectrum. It is useful to have a certification process for these calculated points. Secondly, I will talk about more recent work in using pseudospectra to model the HPA axis. Pseudospectra can be obtained via creating models or via data driven approaches. I will discuss the pros and cons of these different methods, which include interpretability and matching to experimental data.

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Mon 20 Nov 18:00: The quiet AI revolution in weather forecasting Check website for latest updates and booking information http://www.cambridgephilosophicalsociety.org

Talks - Thu, 16/11/2023 - 17:54
The quiet AI revolution in weather forecasting

Abstract: Over the last 18 months a quiet AI revolution has begun in the field of numerical weather prediction. Medium-term weather prediction involves forecasting several days to a couple of weeks in the future and these forecasts are critical for making many social and economic decisions. The standard approach to this problem is to run detailed global simulations of the earth’s atmosphere using a supercomputer, so-called numerical weather prediction (NWP). As little as one year ago, researchers in this field had thought it unlikely that machine learning approaches would be competitive with numerical weather prediction any time soon. However, over the last year, the same advances that underpin large language models, like ChatGPT, have been applied to weather prediction. Surprisingly, these models achieve a performance which is already competitive with standard NWP , but with a computational cost that is 1000s of times cheaper. The deep learning based forecasts have also been shown to be surprisingly robust, performing reasonably even when faced with rare or extreme events. Consequently, weather prediction centres like the World Meteorological Organisation and the European Centre for Medium-Range Weather Forecasts (ECMWF) are now racing to build machine learning teams and publicly testing AI forecasts. This talk will describe this quieter AI revolution and it will end with a discussion of the opportunities for AI and machine learning in weather and climate, and speak a little more widely about the balancing act that must be struck between regulation and adoption of AI technology.

Check website for latest updates and booking information http://www.cambridgephilosophicalsociety.org

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Thu 30 Nov 15:00: Randomized methods for joint eigenvalue problems

Talks - Mon, 13/11/2023 - 17:13
Randomized methods for joint eigenvalue problems

By a basic linear algebra result, a family of two or more commuting symmetric matrices has a common eigenvector basis and can thus be jointly diagonalized. Such joint eigenvalue problems come in several flavors and they play an important role in a variety of applications, including independent component analysis in signal processing, multivariate polynomial systems, tensor decompositions, and computational quantum chemistry. Perhaps surprisingly, the development of robust numerical algorithms for solving such problems is by no means trivial. To start with, roundoff error or other forms of error will inevitably destroy commutativity assumptions. In turn, one can at best hope to find approximate solutions to joint eigenvalue problems and, in turn, most existing approaches are based on optimization techniques, which may or may not recover the approximate solution. In this talk, we propose randomized methods that address joint eigenvalue problems via the solution of one or a few standard eigenvalue problems. The methods are simple but surprisingly effective. We provide a theoretical explanation for their success by establishing probabilistic guarantees for robust recovery. Through numerical experiments on synthetic and real-world data, we show that our algorithms reach or outperform state-of-the-art optimization-based methods. This talk is based on joint work with Haoze He.

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Thu 30 Nov 15:00: Randomized methods for joint eigenvalue problems

Other events - Mon, 13/11/2023 - 17:13
Randomized methods for joint eigenvalue problems

By a basic linear algebra result, a family of two or more commuting symmetric matrices has a common eigenvector basis and can thus be jointly diagonalized. Such joint eigenvalue problems come in several flavors and they play an important role in a variety of applications, including independent component analysis in signal processing, multivariate polynomial systems, tensor decompositions, and computational quantum chemistry. Perhaps surprisingly, the development of robust numerical algorithms for solving such problems is by no means trivial. To start with, roundoff error or other forms of error will inevitably destroy commutativity assumptions. In turn, one can at best hope to find approximate solutions to joint eigenvalue problems and, in turn, most existing approaches are based on optimization techniques, which may or may not recover the approximate solution. In this talk, we propose randomized methods that address joint eigenvalue problems via the solution of one or a few standard eigenvalue problems. The methods are simple but surprisingly effective. We provide a theoretical explanation for their success by establishing probabilistic guarantees for robust recovery. Through numerical experiments on synthetic and real-world data, we show that our algorithms reach or outperform state-of-the-art optimization-based methods. This talk is based on joint work with Haoze He.

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Fri 10 Nov 15:00: Causal Discovery from Observations: Introduction and Some Recent Advances

Talks - Tue, 07/11/2023 - 10:02
Causal Discovery from Observations: Introduction and Some Recent Advances

In causal discovery, the aim is to uncover the underlying causal mechanisms that drive the relationships between a collection of observed variables. It is a research topic with applications in many areas, including medicine, biology, economics, and social sciences. In principle, identifying causal relationships requires interventions (a.k.a., experiments). However, this is often impossible, impractical, or unethical, which has stimulated much research on causal discovery from purely observational data or mixed observational-interventional data. In this talk, after surveying the causal discovery field, I will discuss some recent advances, namely on causal discovery from data with latent interventions and on the quintessential causal discovery problem: distinguishing cause from effect on a pair of dependent variables.

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Fri 10 Nov 15:00: Causal Discovery from Observations: Introduction and Some Recent Advances

Other events - Tue, 07/11/2023 - 10:02
Causal Discovery from Observations: Introduction and Some Recent Advances

In causal discovery, the aim is to uncover the underlying causal mechanisms that drive the relationships between a collection of observed variables. It is a research topic with applications in many areas, including medicine, biology, economics, and social sciences. In principle, identifying causal relationships requires interventions (a.k.a., experiments). However, this is often impossible, impractical, or unethical, which has stimulated much research on causal discovery from purely observational data or mixed observational-interventional data. In this talk, after surveying the causal discovery field, I will discuss some recent advances, namely on causal discovery from data with latent interventions and on the quintessential causal discovery problem: distinguishing cause from effect on a pair of dependent variables.

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Thu 09 Nov 15:00: Solving partial differential equations exactly over polynomials

Talks - Mon, 06/11/2023 - 19:43
Solving partial differential equations exactly over polynomials

Numerical simulations of partial differential equations (PDEs) are indispensable across science and engineering. For simple geometries, spectral methods are a powerful class of techniques that produce exceptionally accurate solutions for wide ranges of equations. But many variations of these methods exist, each with distinct properties and performance, and developing the best method for a complex nonlinear problem is often quite challenging.

In this context, we present a framework that unifies all polynomial and trigonometric spectral methods, from classical “collocation” to the more recent “ultraspherical” schemes. In particular, we examine the exact discrete equations solved by each method and characterize their deviation from the original PDE in terms of perturbations called “tau corrections”. By analyzing these corrections, we can precisely categorize existing methods and design new solvers that robustly accommodate new boundary conditions, eliminate spurious numerical modes, and satisfy exact conservation laws.

This approach conceptually separates what discrete model a spectral scheme solves from how it solves it. This separation provides much more freedom when building and optimizing new numerical models. We will illustrate these advantages with some examples from fluid dynamics using Dedalus, an open-source package for solving PDEs with modern spectral methods.

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Thu 09 Nov 15:00: Solving partial differential equations exactly over polynomials

Other events - Mon, 06/11/2023 - 19:43
Solving partial differential equations exactly over polynomials

Numerical simulations of partial differential equations (PDEs) are indispensable across science and engineering. For simple geometries, spectral methods are a powerful class of techniques that produce exceptionally accurate solutions for wide ranges of equations. But many variations of these methods exist, each with distinct properties and performance, and developing the best method for a complex nonlinear problem is often quite challenging.

In this context, we present a framework that unifies all polynomial and trigonometric spectral methods, from classical “collocation” to the more recent “ultraspherical” schemes. In particular, we examine the exact discrete equations solved by each method and characterize their deviation from the original PDE in terms of perturbations called “tau corrections”. By analyzing these corrections, we can precisely categorize existing methods and design new solvers that robustly accommodate new boundary conditions, eliminate spurious numerical modes, and satisfy exact conservation laws.

This approach conceptually separates what discrete model a spectral scheme solves from how it solves it. This separation provides much more freedom when building and optimizing new numerical models. We will illustrate these advantages with some examples from fluid dynamics using Dedalus, an open-source package for solving PDEs with modern spectral methods.

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Thu 30 Nov 15:00: Title to be confirmed

Talks - Mon, 23/10/2023 - 12:06
Title to be confirmed

Abstract not available

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Thu 30 Nov 15:00: Title to be confirmed

Other events - Mon, 23/10/2023 - 12:06
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Thu 23 Nov 15:00: Title to be confirmed

Talks - Mon, 23/10/2023 - 12:06
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Thu 23 Nov 15:00: Title to be confirmed

Other events - Mon, 23/10/2023 - 12:06
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Thu 09 Nov 15:00: Title to be confirmed

Talks - Mon, 23/10/2023 - 12:03
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Thu 09 Nov 15:00: Title to be confirmed

Other events - Mon, 23/10/2023 - 12:03
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Thu 26 Oct 15:00: Essential spectra and spectral pollution for Maxwell and Drude-Lorentz systems

Talks - Mon, 23/10/2023 - 12:01
Essential spectra and spectral pollution for Maxwell and Drude-Lorentz systems

In order to discuss uniqueness of solutions to inverse problems for inhomogeneous Maxwell systems with conductivity, it is useful to have good a-priori estimates of where the spectra of such systems lies; in order to solve the spectral problems numerically, one needs to understand how much of a problem spectral pollution may be. A priori one expects these problems to be greatly complicated by features such as the well known lack of coercivity, and consequent low regularity of solutions, for Maxwell and Drude-Lorentz systems. The fact that Drude-Lorentz systems are also nonlinear in the spectral parameter adds a further layer of interest. The investigation of these problems reveals many more interesting questions and phenomena. Plasmon-type quasi-modes concentrated around discontinuity interfaces (black hole quasi-modes) may characterise one of the components of the essential spectrum. Spectral pollution turns out to be confined to a much smaller set than anyone had dared expect. Resolvent estimates are possible using an abstract Morawetz-type trick.

This talk discusses work with co-authors including Giovanni Alberti (Genova), Sabine Boegli (Durham), Malcolm Brown (Cardiff; deceased) Francesco Ferraresso (Sassari), Christiane Tretter (Bern) and Ian Wood (Kent).

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Thu 26 Oct 15:00: Essential spectra and spectral pollution for Maxwell and Drude-Lorentz systems

Other events - Mon, 23/10/2023 - 12:01
Essential spectra and spectral pollution for Maxwell and Drude-Lorentz systems

In order to discuss uniqueness of solutions to inverse problems for inhomogeneous Maxwell systems with conductivity, it is useful to have good a-priori estimates of where the spectra of such systems lies; in order to solve the spectral problems numerically, one needs to understand how much of a problem spectral pollution may be. A priori one expects these problems to be greatly complicated by features such as the well known lack of coercivity, and consequent low regularity of solutions, for Maxwell and Drude-Lorentz systems. The fact that Drude-Lorentz systems are also nonlinear in the spectral parameter adds a further layer of interest. The investigation of these problems reveals many more interesting questions and phenomena. Plasmon-type quasi-modes concentrated around discontinuity interfaces (black hole quasi-modes) may characterise one of the components of the essential spectrum. Spectral pollution turns out to be confined to a much smaller set than anyone had dared expect. Resolvent estimates are possible using an abstract Morawetz-type trick.

This talk discusses work with co-authors including Giovanni Alberti (Genova), Sabine Boegli (Durham), Malcolm Brown (Cardiff; deceased) Francesco Ferraresso (Sassari), Christiane Tretter (Bern) and Ian Wood (Kent).

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Thu 02 Nov 15:00: Sparsistency for inverse optimal transport

Talks - Mon, 23/10/2023 - 09:44
Sparsistency for inverse optimal transport

Optimal Transport is a useful metric to compare probability distributions and to compute a pairing given a ground cost. Its entropic regularization variant (eOT) is crucial to have fast algorithms and reflect fuzzy/noisy matchings. This work focuses on Inverse Optimal Transport (iOT), the problem of inferring the ground cost from samples drawn from a coupling that solves an eOT problem. It is a relevant problem that can be used to infer unobserved/missing links, and to obtain meaningful information about the structure of the ground cost yielding the pairing. On one side, iOT benefits from convexity, but on the other side, being ill-posed, it requires regularization to handle the sampling noise. This work presents a study of l1 regularization to model for instance Euclidean costs with sparse interactions between features. Specifically, we derive a sufficient condition for the robust recovery of the sparsity of the ground cost that can be seen as a generalization of the Lasso’s celebrated ``Irrepresentability Condition’’. To provide additional insight into this condition, we consider the Gaussian case. We show that as the entropic penalty varies, the iOT problem interpolates between a graphical Lasso and a classical Lasso, thereby establishing a connection between iOT and graph estimation. This is joint work with Francisco Andrade and Gabriel Peyré.

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Thu 02 Nov 15:00: Sparsistency for inverse optimal transport

Other events - Mon, 23/10/2023 - 09:44
Sparsistency for inverse optimal transport

Optimal Transport is a useful metric to compare probability distributions and to compute a pairing given a ground cost. Its entropic regularization variant (eOT) is crucial to have fast algorithms and reflect fuzzy/noisy matchings. This work focuses on Inverse Optimal Transport (iOT), the problem of inferring the ground cost from samples drawn from a coupling that solves an eOT problem. It is a relevant problem that can be used to infer unobserved/missing links, and to obtain meaningful information about the structure of the ground cost yielding the pairing. On one side, iOT benefits from convexity, but on the other side, being ill-posed, it requires regularization to handle the sampling noise. This work presents a study of l1 regularization to model for instance Euclidean costs with sparse interactions between features. Specifically, we derive a sufficient condition for the robust recovery of the sparsity of the ground cost that can be seen as a generalization of the Lasso’s celebrated ``Irrepresentability Condition’’. To provide additional insight into this condition, we consider the Gaussian case. We show that as the entropic penalty varies, the iOT problem interpolates between a graphical Lasso and a classical Lasso, thereby establishing a connection between iOT and graph estimation. This is joint work with Francisco Andrade and Gabriel Peyré.

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Thu 26 Oct 15:00: Essential spectra and spectral pollution for Maxwell and Drude-Lorentz systems

Talks - Thu, 19/10/2023 - 17:02
Essential spectra and spectral pollution for Maxwell and Drude-Lorentz systems

In order to discuss uniqueness of solutions to inverse problems for inhomogeneous Maxwell systems with conductivity, it is useful to have good a-priori estimates of where the spectra of such systems lies; in order to solve the spectral problems numerically, one needs to understand how much of a problem spectral pollution may be. A priori one expects these problems to be greatly complicated by features such as the well known lack of coercivity, and consequent low regularity of solutions, for Maxwell and Drude-Lorentz systems. The fact that Drude-Lorentz systems are also nonlinear in the spectral parameter adds a further layer of interest. The investigation of these problems reveals many more interesting questions and phenomena. Plasmon-type quasi-modes concentrated around discontinuity interfaces (black hole quasi-modes) may characterise one of the components of the essential spectrum. Spectral pollution turns out to be confined to a much smaller set than anyone had dared expect. Resolvent estimates are possible using an abstract Morawetz-type trick.

This talk discusses work with co-authors including Giovanni Alberti (Genova), Sabine Boegli (Durham), Malcolm Brown (Cardiff; deceased) Francesco Ferraresso (Sassari), Christiane Tretter (Bern) and Ian Wood (Kent).

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