Bath Numerical Analysis Seminar
2019/20 - Semester 1
Fridays at 12.15 in 4W1.7 (also known as the Wolfson Lecture Theatre). Campus maps can be found here.
Everyone is welcome at these talks and don't forget to join us for lunch after the seminar.
|13 Sep 2019||
Alessandro Perelli (DTU)
Randomized second-order algorithm for MAP estimation and fast MMSE estimator applied to Computed Tomography
Two families of algorithms for MSE and MMSE estimation respectively will be presented and applied to a monoenergetic X-ray Computed Tomography acquisition model. For the MAP problem, a randomized second order solver for model-based reconstruction is analysed; to reduce the computational complexity, a partial randomized Hessian sketch is used only for the convex likelihood function and the regularization is designed using the regularization by denoising which retains the complex prior structure. Finally, a fist order iterative method, called approximate message passing, will be presented for performing MMSE estimation efficiently.
|4 Oct 2019||Clarice Poon (Bath)||
The geometry of first order methods and adaptive acceleration
I will discuss the acceleration of first order methods for non smooth optimisation (examples include Forward-Backward, ADMM, Douglas Rachford and Primal-Dual). In the current literature, the most widely used form of acceleration is the ‘inertial technique’, and while there are improved performance guarantees for methods such as gradient descent and the Forward-Backward algorithm, in general, inertial can lead to a slow down in performance. We introduce a general framework for the acceleration of such methods motivated by the trajectory/structural properties of the sequences generated by such methods.
|11 Oct 2019||
Martin Gander (Geneva)
Seven things I would have liked to know when starting to work on Domain Decompositions
|18 Oct 2019||Samuel Groth (Cambridge)||
Hybrid numerical-asymptotic boundary element methods for high-frequency transmission scattering problems
High-frequency wave scattering is challenging for conventional numerical methods based on piecewise polynomial approximation spaces, because of the large number of degrees of freedom required to capture the oscillatory solution. Hybrid numerical-asymptotic (HNA) boundary element methods (BEM) aim to significantly reduce the dimension of the approximation space by enriching it with oscillatory functions, carefully chosen to capture the high-frequency asymptotic behaviour of the wave solution. In this talk I will discuss recent developments in the HNA approach for transmission problems (involving penetrable, or dielectric, scatterers). In our earlier work, we developed an algorithm achieving a fixed accuracy with a frequency-independent number of BEM degrees of freedom, associated with oscillatory basis functions capturing corner-diffracted waves. Our recent investigations suggest that to obtain good performance uniformly across all incident wave angles it is necessary to include basis functions capturing so-called “lateral” or “head” waves, which in the high-frequency asymptotic theory correct for the phase mismatch between the internal and external diffracted waves.
|25 Oct 2019||Patrick Farrell (Oxford)||
A Reynolds-robust preconditioner for the 3D stationary Navier-Stokes
When approximating PDEs with the finite element method, large sparse linear systems must be solved. The ideal preconditioner yields convergence that is algorithmically optimal and parameter robust, i.e. the number of Krylov iterations required to solve the linear system to a given accuracy does not grow substantially as the mesh or problem parameters are changed. Achieving this for the stationary Navier-Stokes has proven challenging: LU factorisation is Reynolds-robust but scales poorly with degree of freedom count, while Schur complement approximations such as PCD and LSC degrade as the Reynolds number is increased. Building on the ideas of Schöberl, Benzi & Olshanskii, in this talk we present the first preconditioner for the Newton linearisation of the stationary Navier–Stokes equations in three dimensions that achieves both optimal complexity and Reynolds-robustness. The scheme combines augmented Lagrangian stabilisation to control the Schur complement, a divergence-capturing additive Schwarz relaxation method on each level, and a specialised prolongation operator involving non-overlapping local Stokes solves on coarse cells. The properties of the preconditioner are tailored to the divergence-free Scott–Vogelius discretisation. We present 3D simulations with over one billion degrees of freedom with robust performance from Reynolds numbers 10 to 5000.
|8 Nov 2019||Audrey Repetti (Heriot-Watt, Edinburgh)||
8W2.27 A forward-backward algorithm for reweighted procedures: Application to astro-imaging
During the last decades, reweighted procedures have shown high efficiency in computational imaging. They aim to handle non-convex composite penalization functions by iteratively solving multiple approximated subproblems. Although the asymptotic behaviour of these methods has recently been investigated in several works, they all necessitate the sub-problems to be solved accurately, which can be sub-optimal in practice. In this work we present a reweighted forward-backward algorithm designed to handle non-convex composite functions. Unlike existing convergence studies in the literature, the weighting procedure is directly included within the iterations, avoiding the need for solving any sub-problem. We show that the obtained reweighted forward-backward algorithm converges to a critical point of the initial objective function. We illustrate the good behaviour of the proposed approach on a Fourier imaging example borrowed to radio-astronomical imaging.
|15 Nov 2019||Pierre Marchand (Bath)||
Robust coarse spaces for the boundary element method
Boundary integral equations are reformulations of partial differential equations with non-local integral operators. Widely used in acoustics, electromagnetism and mechanics, they have the advantage to reduce the dimension of the geometric domain and they naturally satisfy conditions at infinity for problems on unbounded domains. But the matrices obtained after discretisation have the disadvantage to be dense, so that iterative linear solvers, such as conjugated gradient or GMRes, are usually preferred compared to direct solvers. To stabilise the number of iterations of these solvers with respect to the mesh size, a classical technique is to use a preconditioner. In this talk, I will first introduce domain decomposition methods and boundary integral equation, and then I will present how to precondition the matrices stemming from the boundary element method using domain decomposition methods, and more precisely, Schwarz Methods with a particular coarse space named GenEO whose construction is based on Generalized Eigenproblems in the Overlaps. This is a joint work with Xavier Claeys and Frédéric Nataf.
|22 Nov 2019||
Teo Deveney (Bath)
Deep surrogate models for Bayesian inverse problems involving PDEs and integral equations
This talk will discuss a deep learning approach to efficiently perform Bayesian inference in PDE and integral equation models over potentially high dimensional parameter spaces. We will review a deep learning algorithm which can be applied to solve high dimensional PDEs, and describe an extension to approximating the solutions of Fredholm and Volterra integral equations. Then we will then make use of these to describe a deep surrogate approach to efficient sampling from a Bayesian posterior distribution in which the likelihood function depends on the solution of a PDE or integral equation. Our approach allows for accurate surrogates to be approximated in significantly higher dimensions than is possible using classical techniques. These surrogates are cheap to evaluate, making Bayesian inference over large parameter spaces tractable using sampling based methods. Numerical examples using real world problems will be given to assess the accuracy of the integral equation solver, and demonstrate the effectiveness of this approach for Bayesian inverse problems.
|29 Nov 2019||
Shaunagh Downing (Bath)
Sensor Optimisation in Seismic Imaging
As part of the seismic exploration process, waves are emitted from a source into the earth and sensors record the resulting signal. These measurements are used to create an image of the subsurface in a process known as Full Waveform Inversion (FWI). FWI requires the solution to a PDE to generate predicted data, and an optimisation step in which the difference between the predicted and measured data is minimised to obtain a model of the subsurface. A problem of great practical interest, which is not considered in the standard approach to FWI, is the optimal positioning of the sensors in order to obtain the best return from the seismic exploration process. This talk will provide an overview of FWI and current work being done on the sensor optimisation problem.
|6 Dec 2019||
MMath Project Student Talks
How to get to BathSee here for instructions how to get to Bath. Please email Matthias Ehrhardt (email@example.com) 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).
- "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.