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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
Updated: 1 hour 2 min ago

Thu 07 Mar 15:00: Hamiltonian simulation and optimal control

Tue, 27/02/2024 - 09:51
Hamiltonian simulation and optimal control

Hamiltonian simulation on quantum computers is one of the primary candidates for demonstration of quantum advantage. A central tool in Hamiltonian simulation is the matrix exponential. While uniform polynomial approximations (Chebyshev), best polynomial approximations, and unitary but asymptotic rational approximations (Padé) are well known and are extensively used in computational quantum mechanics, there was an important gap which has now been filled by the development of the theory and algorithms for unitary rational best approximations. This class of approximants leads to geometric numerical integrators with excellent approximation properties. In the second part of the talk I will talk about time-dependent Hamiltonians for many-body two-level systems, including a quantum algorithm for their simulation and some (classical) optimal control algorithms for quantum gate design.

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Thu 23 May 15:00: TBA

Thu, 22/02/2024 - 15:06
TBA

Abstract not available

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Thu 29 Feb 15:00: Efficient frequency-dependent numerical simulation of wave scattering problems

Wed, 21/02/2024 - 14:26
Efficient frequency-dependent numerical simulation of wave scattering problems

Wave propagation in homogeneous media is often modelled using integral equation methods. The boundary element method (BEM) is for integral equations what the finite element method is for partial differential equations. One difference is that BEM typically leads to dense discretization matrices. A major focus in the field has been the development of fast solvers for linear systems involving such dense matrices. Developments include the fast multipole method (FMM) and more algebraic methods based on the so-called H-matrix format. Yet, for time-harmonic wave propagation, these methods solve the original problem only for a single frequency. In this talk we focus on the frequency-sweeping problem: we aim to solve the scattering problem for a range of frequencies. We exploit the wavenumber-dependence of the dense discretization matrix for the 3D Helmholtz equation and demonstrate a memory-compact representation of all integral operators involved which is valid for a continuous range of frequencies, yet comes with a cost of a only small number of single frequency simulations. This is joined work at KU Leuven with Simon Dirckx, Kobe Bruyninckx and Karl Meerbergen.

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Mon 19 Feb 14:00: SINDy-RL: Interpretable and Efficient Model-Based Reinforcement Learning

Mon, 19/02/2024 - 21:12
SINDy-RL: Interpretable and Efficient Model-Based Reinforcement Learning

Deep Reinforcement Learning (DRL) has shown significant promise for uncovering sophisticated control policies that interact in environments with complicated dynamics, such as stabilizing the magnetohydrodynamics of a tokamak reactor and minimizing the drag force exerted on an object in a fluid flow. However, these algorithms require many training examples and can become prohibitively expensive for many applications. In addition, the reliance on deep neural networks results in an uninterpretable, black-box policy that may be too computationally challenging to use with certain embedded systems. Recent advances in sparse dictionary learning, such as the Sparse Identification of Nonlinear Dynamics (SINDy), have shown to be a promising method for creating efficient and interpretable data-driven models in the low-data regime. In this work, we extend ideas from the SIN Dy literature to introduce a unifying framework for combining sparse dictionary learning and DRL to create efficient, interpretable, and trustworthy representations of the dynamics model, reward function, and control policy. We demonstrate the effectiveness of our approaches on benchmark control environments and challenging fluids problems, achieving comparable performance to state-of-the-art DRL algorithms using significantly fewer interactions in the environment and an interpretable control policy orders of magnitude smaller than a deep neural network policy.

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Wed 14 Feb 14:00: SINDy-RL: Interpretable and Efficient Model-Based Reinforcement Learning

Wed, 14/02/2024 - 13:35
SINDy-RL: Interpretable and Efficient Model-Based Reinforcement Learning

Deep Reinforcement Learning (DRL) has shown significant promise for uncovering sophisticated control policies that interact in environments with complicated dynamics, such as stabilizing the magnetohydrodynamics of a tokamak reactor and minimizing the drag force exerted on an object in a fluid flow. However, these algorithms require many training examples and can become prohibitively expensive for many applications. In addition, the reliance on deep neural networks results in an uninterpretable, black-box policy that may be too computationally challenging to use with certain embedded systems. Recent advances in sparse dictionary learning, such as the Sparse Identification of Nonlinear Dynamics (SINDy), have shown to be a promising method for creating efficient and interpretable data-driven models in the low-data regime. In this work, we extend ideas from the SIN Dy literature to introduce a unifying framework for combining sparse dictionary learning and DRL to create efficient, interpretable, and trustworthy representations of the dynamics model, reward function, and control policy. We demonstrate the effectiveness of our approaches on benchmark control environments and challenging fluids problems, achieving comparable performance to state-of-the-art DRL algorithms using significantly fewer interactions in the environment and an interpretable control policy orders of magnitude smaller than a deep neural network policy.

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Thu 22 Feb 15:00: Computing lower eigenvalues on rough domains

Tue, 06/02/2024 - 22:54
Computing lower eigenvalues on rough domains

In this talk I will describe a strategy for finding sharp upper and lower numerical bounds of the Poincare constant on a class of planar domains with piecewise self-similar boundary. The approach is developed in [A] and it consists of four main blocks: 1) tight inner-outer shape interpolation, 2) conformal mapping of the approximate polygonal regions, 3) grad-div system formulation of the spectral problem and 4) computation of the eigenvalue bounds. After describing the method, justifying its validity and reporting on general convergence estimates, I will show concrete evidence of its effectiveness on the Koch snowflake. I will conclude the talk by discussing potential applications to other linear operators on rough regions. This research has been conducted jointly with Lehel Banjai (Heriot-Watt University).

[A] J. Fractal Geometry 8 (2021) No. 2, pp. 153-188

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Thu 15 Feb 15:00: Adaptive Intrusive Methods for Forward UQ in PDEs

Sun, 28/01/2024 - 17:54
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.

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Thu 01 Feb 15:00: What happens when you chop an equation?

Sat, 27/01/2024 - 11:05
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?

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Thu 25 Jan 15:00: The future of governing equations

Thu, 25/01/2024 - 14:11
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.

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Thu 25 Jan 15:00: The future of governing equations

Sun, 21/01/2024 - 12:28
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.

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Thu 18 Jan 15:00: Computing the Spectra and Pseudospectra of Band-Dominated and Random Operators

Thu, 11/01/2024 - 09:08
Computing the Spectra and Pseudospectra of Band-Dominated and Random Operators

I will give an overview of my work over the last 15 years, with collaborators including Marko Lindner (TU Hamburg), Ratchanikorn Chonchaiya (King Mongkut’s University of Technology, Bangkok), Raffael Hagger (Kiel), and Brian Davies (KCL), on computing the spectra and pseudospectra of banded and band-dominated operators. This will include describing algorithms that, given appropriate inputs, can produce a convergent sequence of approximations to the spectrum of an arbitrary band-dominated operator, with the property that each member of the sequence can be computed in finitely many arithmetical operations. We give a concrete implementation of the algorithm for operators that are pseudoergodic in the sense of Davies (Commun. Math. Phys. 2001) and illustrate this algorithm by spectral computations for the beautiful Feinberg-Zee random hopping matrix. Details can be found at https://arxiv.org/abs/2401.03984

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Thu 14 Mar 15:00: TBA

Mon, 08/01/2024 - 11:01
TBA

Abstract not available

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Thu 07 Mar 15:00: TBA

Mon, 08/01/2024 - 10:58
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Thu 29 Feb 15:00: TBA

Mon, 08/01/2024 - 10:55
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Thu 22 Feb 15:00: TBA

Mon, 08/01/2024 - 10:53
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Thu 15 Feb 15:00: TBA

Mon, 08/01/2024 - 10:52
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Abstract not available

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Thu 01 Feb 15:00: TBA

Mon, 08/01/2024 - 10:51
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Thu 25 Jan 15:00: TBA

Mon, 08/01/2024 - 10:50
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Thu 18 Jan 15:00: TBA

Mon, 08/01/2024 - 10:49
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Abstract not available

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Thu 08 Feb 15:00: Towards Finite Element Tensor Calculus

Mon, 08/01/2024 - 10:46
Towards Finite Element Tensor Calculus

Finite Element Exterior Calculus (FEEC) provides a cohomology framework for structure-preserving discretisation of a large class of PDEs. There has been a relatively mature FEEC theory with de Rham complexes for problems involving differential forms (skew-symmetric tensors) and vector fields. A canonical discretisation exists, which has a discrete topological interpretation and can be generalized to other discrete structures, e.g., graph cohomology.

In recent years, there has been significant interest in extending FEEC to tensor-valued problems with applications in continuum mechanics, differential geometry and general relativity etc. In this talk, we first review the de Rham sequences and their canonical discretisation with Whitney forms. Then we provide an overview of some efforts towards Finite Element Tensor Calculus (FETC). On the continuous level, we characterise tensors and differential structures using the Bernstein-Gelfand-Gelfand (BGG) machinery and incorporate analysis. On the discrete level in 2D and 3D, we discuss analogies of the Whitney forms and establish their cohomology. A special case is Christiansen’s finite element interpretation of Regge calculus, a discrete geometric scheme for metric and curvature. Moreover, we present a correspondence between BGG sequences, continuum mechanics with microstructure and Riemann-Cartan geometry. These efforts are in the direction of establishing a tensor calculus on triangulation and potentially on other discrete structures.

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