# Bath Numerical Analysis Seminar

Fridays at **12.15** either **online** (Zoom) or **hybrid** (Wolfson 4W 1.7 and Zoom).

** Everyone is welcome at these talks. **

Date | Speaker | Title |

11 Feb 2022 | Gianluca Frasca-Caccia (University of Salerno) |
## Bespoke preservation of local conservation laws
The main goal of geometric integration is to reproduce, in a numerical approximation, key geometric properties of a given continuous differential problem. In the numerical treatment of partial differential equations, the benefits of conserving global integral invariants are well-known. Preserving the underlying local conservation laws gives, in general, a stricter constraint. Recently, a new approach has been introduced to develop bespoke finite difference schemes that preserve conservation laws. The schemes obtained in this way typically feature certain free parameters that can be arbitrarily chosen. The first part of the talk illustrates the new approach and its application to the Korteweg-de Vries equation. Numerical tests are presented to compare the obtained schemes and other integrators known in literature. The experiments show that a convenient choice of the parameters yields to very accurate approximations. However, the parameters optimal values are not available a priori and depend heavily on the initial conditions. A new procedure for identifying their optimal values is introduced in the second part of the talk. The effectiveness and efficiency of the new strategy is tested. The last part of the talk shows recent advances on the efficient solution of highly oscillatory problems. |

18 Feb 2022 | Tobias Jawecki (TU Vienna) |
## Krylov subspace approximations to the action of unitary matrix exponentials
Defect-based a posteriori error estimates for the polynomial Krylov approximation to the action of unitary matrix exponentials (i.e. exponentials of skew-Hermitian matrices) are discussed. Furthermore, a priori convergence results such as the near-best approximation property in the class of polynomial approximations are recalled. The performance of approximations to the action of matrix functions depends on the matrix spectrum and spectral coefficients of the initial vector. Based on properties of Gaussian-quadrature formulae, Krylov subspace techniques provide piecewise estimates on these quantities, and can be further used to classify the underlying problems. For the action of the matrix exponential, this approach is applied to discuss problems for which rational Krylov approximations potentially outperform polynomial Krylov approximations. |

25 Feb 2022 | Sheehan Olver (Imperial College London) |
## (hybrid) Computing equilibrium distributions with power law interactions
When particles interact, say by attracting or repulsing, they tend to form nice distributions as the number of particles become large. Examples include both physical (electrons in a potential well, space dust) and biological (flocking birds, bacteria). Naïve simulation via differential equations proves insufficient, with computational cost becoming prohibitively expensive in more than one dimensions. Instead, we will introduce techniques based on a measure minimisation reformulation using expansions in weighted orthogonal polynomials to approximate the measures, whereby incorporating the correct singularities of the distributions we can rapidly and accurately compute many such distributions in arbitrary dimensions. This leads to high accuracy confirmation of open conjectures on gap formation (imagine a flock of birds forming a ring, with no density in the middle). These techniques involve new recurrence relationship of power law kernels applied to weighted orthogonal polynomials. |

4 Mar 2022 | Michele Firmo (University of Bath) |
## Bayesian Inverse Problems for NMR
In petrophysics, information about the internal porosity structure of logging samples is indirectly recovered using the NMR T2 relaxation time distribution P(T2). As the measured magnetisation is noisy, solving the model for P(T2) is a highly ill-posed problem. So far, the petrophysicists have tackled this problem using Tikhonov regularization. Although quick and directly interpretable, this technique lacks information about the uncertainty of the solution. Instead, we proposed a Bayesian approach based on an atomic P(T2), i.e., a weighted sum of Dirac Deltas. Defining a posterior distribution for P(T2), we can sample from it. This allows to recover that distribution as well as the distribution of different quantities of interest, such as the porosity, quantifying their uncertainty. In this talk we will uncover the different issues associated with the exploration of a posterior distribution with many parameters, as well as the different techniques used to overcome them. This leads to a new MCMC algorithm overan unbounded domain, and a convergence result is discussed as well. Finally, we will show some numerical results of the exploration on real NMR dataset, together with a formal comparison between Bayesian and Tikhonov approaches. |

11 Mar 2022 | Margaret Duff (University of Bath) |
## (hybrid) Regularising inverse imaging problems using generative deep learning models
In this talk we consider the use of generative models in a variational regularisation approach to inverse problems. Generative models learn, from observations, approximations to high-dimensional data distributions. The considered regularisers penalise image reconstructions that are far from the range of a generative model that has learned to produce images similar to a training dataset. In contrast to other data driven approaches, this approach does not require paired training data and models are learned independently of the forward model. We will see an introduction to inverse problems and generative deep learning methods before seeing different approaches to generative regularisers. We will consider desired criteria for the generative model, to enable generative regularisers to be successful. Finally, we will discuss how generative models could be improved for better use in inverse problems. |

18 Mar 2022 | Eric Baruch Gutierrez (University of Bath) |
## (hybrid) Random Primal-Dual Method with applications to Parallel MRI
Stochastic Primal-Dual Hybrid Gradient, or SPDHG, is an algorithm proposed by Chambolle et al. to efficiently solve a wide class of nonsmooth large-scale optimization problems. Recently we proved its almost sure convergence for all convex functionals. In addition, to test performance we look into applications of SPDHG to parallel Magnetic Resonance Imaging reconstruction. Numerical results suggest significantly faster convergence for SPDHG than its deterministic counterpart. Furthermore, we will see how the physical properties of MRI models might give us a clue on how to optimize our random process. |

25 Mar 2022 | Shi Jin (Shanghai Jiao Tong University) |
## Random Batch Methods for interacting particle systems and molecular dynamics
We first develop random batch methods for classical interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N^2) per time step to O(N), for a system with N particles with binary interactions. For one of the methods, we give a particle number independent error estimate under some special interactions. This method is also extended to molecular dynamics with Coulomb interactions, in the framework of Ewald summation. We will show its superior performance compared to the current state-of-the-art methods (for example PPPM) for the corresponding problems, in the computational efficiency and parallelizability. |

1 Apr 2022 | Georg Maierhofer (Sorbonne University) |
## Highly oscillatory quadrature and low-regularity integrators for nonlinear dispersive equations
Even after decades of thorough research, the efficient approximation of highly oscillatory integrals remains a challenging and fascinating topic. Amongst the most promising modern techniques are so-called Filon methods which aim to provide accurate approximations of integrals over rapidly varying functions at frequency-independent cost. However, for general oscillatory kernels, one of the central difficulties encountered in Filon methods is the fast computation of the quadrature moments. In this talk, we will describe a novel approach to the solution of this ‘moment-problem’ by providing a unified framework for the construction of moment recursions that can achieve this desired task for a wide range of oscillators. We will then exploit the deep connection of oscillatory phenomena and the regularity of functions to provide a bridge between Filon methods and low-regularity integrators for nonlinear dispersive partial differential equations. Based on recent advances in resonance-based integrators, this connection allows us to design innovative numerical schemes, which can efficiently approximate low-regularity solutions to nonlinear systems, even when classical methods (such as exponential integrators) fail. Most importantly, for the first time, we are able to describe and analyse low-regularity integrators with structure preservation properties, thus providing initial results on geometric numerical integration in low-regularity regimes. This is joint work with Arieh Iserles, Nigel Peake and Katharina Schratz. |

8 Apr 2022 | Tatiana Bubba (University of Bath) |
## (hybrid) Limited angle tomography, wavelets and convolutional neural networks
Sparsity promotion is a popular regularization technique for inverse problems, reflecting the prior knowledge that the exact solution is expected to have few non-vanishing components, e.g. with respect to a suitable wavelet basis. In this talk, I will present a convolutional neural network designed for sparsity-promoting regularization for linear inverse problems. The architecture of the network is deduced by unrolling the well-known Iterative Soft Thresholding Algorithm (ISTA), together with a novel convolutional structure of each layer, motivated by the wavelet representation of the involved operator. As a result, the proposed network is able to replicate the application of ISTA and outperform it, by learning a suitable pseudodifferential operator. By a combination of techniques and tools from regularization theory of inverse problems, harmonic wavelet analysis and microlocal analysis, we are able to theoretically analyze the network and to prove approximation error estimates. Our case study is limited-angle computed tomography - we test two different implementations of our network on simulated data from limited-angle geometry, achieving promising results. |

29 Apr 2022 | Catherine Higham (Glasgow University) |
## (hybrid) Quantum Deep Learning
Deep learning is being applied and obtaining impressive results in many novel emerging quantum technologies. Commercially available quantum annealing promises to complement these activities. In this talk, I will introduce some machine learning approaches, in quantum optics imaging applications (single-pixel camera/video/LiDAR), where neural networks including convolutional autoencoders and generative adversarial networks are being used to solve experimental optimization, inverse and classification/regression problems. I will also briefly discuss quantum machine learning approaches using D-Wave’s quantum annealer. |

6 May 2022 | Théophile Chaumont-Frelet (INRIA Sophia-Antipolis) |
## (hybrid) A posteriori error estimates for high-frequency Maxwell's equations
Maxwell’s equations play a central role in a vast number of applications involving electromagnetic fields. The numerical approximation of the solution is often required is practice and is particularly challenging when the frequency is high. In this talk, I will consider finite element discretizations and focus on a posteriori error estimates. These estimates not only allow for a robust control of the error, but they also enable iterative mesh refinements that are locally adapted to the features of the solution. In this context, Maxwell’s equations exhibit two specific difficulties: (a) in the high-frequency regime, their variational formulation is not coercive and (b) the functional framework associated with the ‘curl’ operator is substantially more involved than the one used for scalar wave propagation problems. The goal of this talk is to explain, as simply as possible, how these two issues manifest and how they can be handled. To do so, I will proceed in three steps. (i) I will first describe how a posteriori error estimators are constructed in the simple case of the (coercive) Poisson problem. (ii) I will then examine the high-frequency aspect by considering the Helmholtz equation modelling high-frequency scalar wave propagation problems. Finally (iii), I will present the necessary adjustments to correctly handle Maxwell’s equations. Numerical examples will be presented to illustrate the key theoretical results. |

## Subscribe to seminar calendar

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## How to get to Bath

See here for instructions how to get to Bath. Please email Pranav Singh (ps2106@bath.ac.uk) if you intend to come by car and require a parking permit for Bath University Campus for the day.## Tips for giving talks

#### Tips for new students on giving talks

Since the audience of the NA seminar contains both PhD students and staff with quite wide interests and backgrounds, the following are some guidelines/hints to make sure people don't give you evil looks at lunch afterwards.

Before too much time passes in your talk, ideally the audience should know the answers to the following 4 questions:

- What is the problem you're considering?
- Why do you find this interesting?
- What has been done before on this problem/what's the background?
- What is your approach/what are you going to talk about?

There are lots of different ways to communicate this information. One way, if you're doing a slide show, could be for the first 4 slides to cover these 4 questions; although in this case you may want to revisit these points later on in the talk (e.g. to give more detail).

Remember:

- "vertebrate style" (structure hidden inside - like the skeleton of a vertebrate) = good for detective stories, bad for maths talks.
- "crustacean style" (structure visible from outside - like the skeleton of a crustacean) = bad for detective stories, good for maths talks.