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A weekly seminar seires with talks in all areas of applied and computational analysis, of broad interest to the mathematical community. For questions about this series or if you have suggestions, please contact: m.colbrook@damtp.cam.ac.uk
URL: http://talks.cam.ac.uk/show/index/9811
Updated: 46 min 14 sec ago

Thu 16 Mar 15:00: Variational Convergence of Liquid Crystal Energies to Line and Surface Energies

Mon, 13/03/2023 - 20:20
Variational Convergence of Liquid Crystal Energies to Line and Surface Energies

We study a model of an inclusion inside a liquid crystal, modelled by a Landau-De Gennes energy, in a regime where the line singularities (hedgehog) of the director field and the surface singularities (flip of direction) have energies of the same order. In the limit, we find an energy of a minimal 2D flat chain with minimal 1D boundary, attached to the inclusion. We will rapidly mention possible directions for numerical experiments. This is a joint work with Dominik Stantejsky (now at McMaster) and François Alouges (now at ENS Saclay).

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Thu 09 Mar 15:00: The challenge of AI errors and the limitations of training accurate and verifiably stable data-driven AI

Mon, 06/03/2023 - 14:13
The challenge of AI errors and the limitations of training accurate and verifiably stable data-driven AI

Since the seminal work by Szegedy et al. revealing the apparent sensitivity of deep learning classifiers to small adversarial perturbations of their input data, the robustness of modern data-driven AI systems has been a widely discussed and broadly debated issue. Among these adversarial perturbations, there can exist even universal perturbations which trigger the instability of the network for seemingly any input. The presence of such instabilities in a tool which is so widely used in applications gives rise to a fundamental question: are these instabilities typical, and to be expected in modern large-scale AI and deep learning models? Moreover, is it even possible to compute a data-driven AI model which is both accurate and verifiably stable at the same time?

In the talk, we will present and discuss a list of scenarios enabling the formulation of high-level verifiable criteria for the detection of instabilities in a broad class of trained models. However, as we will show too, major limitations on the pathway to compute accurate and verifiably stable AI from data remain. These limitations constitute a fundamental issue around the possibility of building data-driven systems which are indeed accurate and verifiably stable. We will discuss potential approaches to alleviate the problem by accepting the inevitability of errors and finding computationally efficient ways to correct them “on-the-job” with given performance guarantees and without re-training.

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Thu 08 Jun 15:00: Title to be confirmed

Sun, 05/03/2023 - 18:27
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Abstract not available

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Thu 01 Jun 15:00: Title to be confirmed

Sun, 05/03/2023 - 18:26
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Thu 18 May 15:00: Title to be confirmed

Sun, 05/03/2023 - 18:25
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Thu 11 May 15:00: Title to be confirmed

Sun, 05/03/2023 - 18:25
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Thu 04 May 15:00: Title to be confirmed

Sun, 05/03/2023 - 18:24
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Thu 27 Apr 15:00: Title to be confirmed

Sun, 05/03/2023 - 18:24
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Thu 16 Mar 15:00: Title to be confirmed

Sun, 05/03/2023 - 18:23
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Thu 09 Mar 15:00: Title to be confirmed

Sun, 05/03/2023 - 18:22
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Tue 07 Mar 15:00: Multiply accelerated value iteration for affine fixed point problems and application to Markov decision processes

Thu, 02/03/2023 - 20:41
Multiply accelerated value iteration for affine fixed point problems and application to Markov decision processes

We analyze a modified version of Nesterov accelerated gradient algorithm, which applies to affine fixed point problems with non self-adjoint matrices, such as the ones appearing in the theory of Markov decision processes with discounted or mean payoff criteria. We characterize the spectra of matrices for which this algorithm does converge with an accelerated asymptotic rate. We also introduce a dth-order algorithm, and show that it yields a multiply accelerated rate under more demanding conditions on the spectrum. We subsequently apply these methods to develop accelerated schemes for non-linear fixed point problems arising from Markov decision processes.

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Tue 07 Mar 15:00: Multiply accelerated value iteration for affine fixed point problems and application to Markov decision processes

Thu, 02/03/2023 - 20:38
Multiply accelerated value iteration for affine fixed point problems and application to Markov decision processes

We analyze a modified version of Nesterov accelerated gradient algorithm, which applies to affine fixed point problems with non self-adjoint matrices, such as the ones appearing in the theory of Markov decision processes with discounted or mean payoff criteria. We characterize the spectra of matrices for which this algorithm does converge with an accelerated asymptotic rate. We also introduce a dth-order algorithm, and show that it yields a multiply accelerated rate under more demanding conditions on the spectrum. We subsequently apply these methods to develop accelerated schemes for non-linear fixed point problems arising from Markov decision processes.

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Tue 21 Feb 15:00: Data-driven schemes for high-dimensional Hamilton-Jacobi-Bellman PDEs

Tue, 21/02/2023 - 18:19
Data-driven schemes for high-dimensional Hamilton-Jacobi-Bellman PDEs

Optimal feedback synthesis for nonlinear dynamics a fundamental problem in optimal control is enabled by solving fully nonlinear Hamilton-Jacobi-Bellman type PDEs arising in dynamic programming. While our theoretical understanding of dynamic programming and HJB PD Es has seen a remarkable development over the last decades, the numerical approximation of HJB -based feedback laws has remained largely an open problem due to the curse of dimensionality. More precisely, the associated HJB PDE must be solved over the state space of the dynamics, which is extremely high-dimensional in applications such as distributed parameter systems or agent-based models.

In this talk we will review recent approaches regarding the effective numerical approximation of very high-dimensional HJB PD Es via data-driven schemes in supervised and semi-supervised learning environments. We will discuss the use of representation formulas as synthetic data generators, and different architectures for the value function, such a polynomial approximation, tensor decompositions, and deep neural networks.

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Thu 02 Feb 15:00: Gaussian Wave Packet Dynamics in Quantum Dynamics

Mon, 30/01/2023 - 11:23
Gaussian Wave Packet Dynamics in Quantum Dynamics

The development of numerical methods for the efficient simulation of quantum dynamics is essential for the theoretical analysis of quantum systems. In this talk, the Time-Sliced Thawed Gaussian (TSTG) propagation method is presented, including a rigorous error analysis of the algorithm and numerical experiments. In the last part of the talk we will focus on the tensor-train format, a special variant of low-rank tensor decompositions that can overcome the curse of dimensionality and is currently at the forefront of research in many different fields. Practical numerical experiments will illustrate how we can use this tool for multi-dimensional quantum dynamics simulations.

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Thu 26 Jan 15:00: Convergence analysis of non-stationary and deep Gaussian process regression

Fri, 20/01/2023 - 11:42
Convergence analysis of non-stationary and deep Gaussian process regression

We are interested in the task of estimating an unknown function from data, given as a set of point evaluations. In this context, Gaussian process regression is often used as a Bayesian inference procedure, and we are interested in the convergence as the number of data points goes to infinity. Using results from scattered data approximation, we provide a convergence analysis of the method applied to a given, unknown function of interest. We are particularly interested in the case of non-stationary covariance kernels, and the extension of the results to deep Gaussian processes.

This is joint work with Conor Osborne (University of Edinburgh).

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Thu 25 May 15:00: Title to be confirmed

Sat, 03/12/2022 - 14:21
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Abstract not available

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