Numerical Analysis and Scientific Computing Seminars

Forthcoming Seminars

Past Seminars

Fri 19/05/2023 12:00 – 13:00
Frank Adams 1

Marcus Webb (UoM)

Title: Are Sketch-and-Precondition Least Squares Solvers Numerically Stable?

Over the last 15 years there has been an explosion in interest in randomised methods in numerical linear algebra, many of which involve a process called “sketching”. Randomised sketching can be used to precondition a large least squares problem, and can outperform classical approaches in terms of flop counts and number of passes over the data, making them attractive in big data settings. In this talk I will give an introduction to a sketch-and-precondition techniques and discuss their numerical stability properties. In recent work, I showed with collaborators that while computation of the random preconditioner itself is extremely stable even for very ill-conditioned matrices, the application of the preconditioner to solve the least squares system by an iterative solver such as LSQR does not produce a backward stable numerical result without certain modifications. This is joint work with Maike Meier (Oxford), Yuji Nakatsukasa (Oxford) and Alex Townsend (Cornell). https://arxiv.org/abs/2302.07202

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Fri 12/05/2023 12:00 – 13:00
Frank Adams 1

Chris Hickey (Arup): “Finding the Useful Frequencies for Structural Analysis Using Shift and Invert Lanczos”

Nick Higham (UoM): “Perfectly Conditioned Matrices”

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Fri 21/04/2023 12:00 – 13:00
Frank Adams 1

Beth Wingate (Exeter)

Title: On the way to the limit: time-parallel algorithms for oscillatory, multiscale PDEs
Abstract:

Motivated by using exascale computers for high resolution simulations of time-evolution problems, in this talk I will discuss time-parallelism in the context of oscillatory PDEs. I will give an introduction to time-parallel time-stepping methods and then go onto discuss work on understanding and using time-parallel methods for multiscale oscillatory PDEs (fast singular limits) like those found in the atmosphere, ocean, magnetic fields and plasmas. I will give concrete examples from ODES, such as the swinging spring, and PDEs, such as the rotating shallow water equations. All these problems share the common structure of having a parameter, epsilon, associated with purely oscillatory fast frequencies. I will show results for superlinear convergence in the limit as epsilon goes to zero, and sketch the proof of convergence for the more important case, when epsilon is finite. Time permitting, I will also discuss new directions for this work, including new work on multilevel parareal method with Juliane Rosemeier and Terry Haut (https://arxiv.org/abs/2211.17239).

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Fri 24/03/2023 12:00 – 13:00
Frank Adams 1

Speaker: Ramaseshan Kannan (Arup)

Title: How can surrogates and uncertainty quantification be useful for sustainable design and reuse within the built environment?

Abstract: The Algorithms and Numerical Analysis team at Arup works on computational science problems within the built environment. In this talk I will introduce who we are and present on two ongoing applied research projects. The first concerns the use of sensor data and inverse problems to enable the condition assessment and prognosis of structures such as buildings and bridges. The second is about how machine learning-based surrogates, if trained to be “uncertainty aware”, can be useful for design prototyping without losing confidence in the accuracy of the process. I will conclude with some open questions encountered in both areas.

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Fri 10/03/2023 12:00 – 13:00
Frank Adams Room 2

Jitse Niesen (University of Leeds)

Plasma simulations using Vlasov equations and particle-in-cell methods

A plasma is an ionized gas. The charged particles in a plasma are influenced by electromagnetic fields, but they also generate their own electromagnetic fields. This talk will outline various approaches to modeling plasmas and describe the research of collaborators and myself in numerical methods for simulating plasmas. The simplified definition above of what a plasma is naturally leads to the particle-in-cell method. We investigated the use of a spectral deferred correction method to compute the motion of the charged particles to a higher order of accuracy. A continuum approximation for the charged particles leads to a Vlasov PDE. We studied numerical methods that respect the Hamiltonian structure of the PDE. Unfortunately, the Vlasov PDE is an equation in six dimensions and thus extremely expensive to solve numerically. The third modeling approach is magnetohydrodynamics (MHD), which describes the plasma as a fluid. We have used this to simulate the core of the Earth.

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Fri 17/02/2023 12:00 – 13:00
Frank Adams 1

Vanni Noferini (Aalto University)
Perturbation theory for simple eigenvalues of rational matrices.

Rational eigenvalue problems (REPs) are often solved via an associated (minimal) polynomial system matrix. In this talk, we define and study a structured condition number for simple zeros of a locally minimal polynomial system matrix. In particular, we consider structured perturbations that preserve the degree of each block in the system matrix. We thus develop a perturbation theory for any simple eigenvalue of a rational matrix expressed as a transfer function, regardless of whether the eigenalue is also a pole or not: the case of eigenpoles can be dealt with using the concept of root vectors. We compare the newly defined structured condition number with Tisseur’s unstructured conditional number for polynomial matrices (that is, perturbations that preserve the global degree but not the degrees of each block), and we show that the latter can be unboundedly larger than the former. In fact, numerical experiments highlight the fact that this also happens in practice for some real-world problems. Our results are relevant for the analysis of numerical methods for the REP because Dopico, Quintana and Van Dooren (2023) showed that, in some cases, structure-preserving backward stable methods are available.

The talk is based on joint work with L. Nyman (Aalto), M.C. Quintana (Aalto) and J. Pérez (Montana).

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Thu 16/02/2023 12:00 – 13:00
Frank Adams 2

Speaker: Oleg Balabanov (Sorbonne University)
Title: Randomized Krylov methods for solution of linear systems and eigenvalue problems

Abstract: In recent years, randomization techniques have gained popularity in the field of numerical linear algebra due to their ability to greatly reduce the computational complexity of algorithms and to fully leverage modern computational architectures. In this talk, we integrate the random sketching technique into Krylov methods for the efficient solution of sparse linear systems and eigenvalue problems. The core ingredient of our methodology is the estimation of inner products of high-dimensional vectors by inner products of their small, efficiently computable random images, called sketches. This enables the imposition of orthogonality conditions inherent in Krylov methods by operating on sketches rather than high-dimensional vectors, resulting in significant computational savings. We will present a randomized version of the Arnoldi iteration for constructing a Krylov basis and randomized versions of GMRES, FOM, and Rayleigh-Ritz methods for obtaining a solution in this basis. Rigorous guarantees of accuracy and stability of the methods will be established based on the fact that the sketching embedding is, with high probability, an approximate isometry for the Krylov space. The talk will conclude with a discussion of the methodology’s potential for modern computational architectures, with a particular focus on s-step Krylov methods.

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Fri 10/02/2023 12:00 – 13:00
Frank Adams 1

Speaker: Igor Simunec (SNS Pisa)
Title: Computation of the von Neumann entropy of large matrices via trace estimation and Krylov subspace methods

In this talk I will consider the computation of the von Neumann entropy of a large density matrix $A$ (i.e. a symmetric positive semidefinite matrix with unit trace), which is defined as $\trace(f(A))$, where $f(z) = -z \log(z)$. The computation or approximation of the von Neumann entropy is an important task in several fields, such as quantum mechanics, quantum information theory, and network science.
The straightforward computation of the entropy via diagonalization of $A$ has a cubic cost in the matrix size $n$, which quickly becomes too expensive for large values of $n$.
As an alternative to diagonalization, the approximation of $\trace(f(A))$ can be reduced to the computation of several quadratic forms $x^T f(A) x$ and matrix vector products $f(A) x$, using either a probing method [1] or a stochastic trace estimator [2]; in turn, these can be efficiently approximated using polynomial and rational Krylov subspace methods [3].
After introducing these techniques, I will present some error bounds and heuristics for the special case of the von Neumann entropy, exploiting an integral expression of the entropy function. The analysis leads to algorithms that, given an input tolerance $\epsilon$, attempt to compute an approximation of the entropy with relative accuracy $\epsilon$, using either theoretical bounds or heuristic estimates as stopping criteria. The speed of the algorithms is dependent on certain properties of the matrix $A$, such as its sparsity pattern, but they are often significantly faster than diagonalization and they have much better scaling for large values of $n$. The methods are tested on several density matrices from network theory to demonstrate their effectiveness.

This is a joint work with Michele Benzi* and Michele Rinelli*. A preprint is available on arXiv [4].

* Scuola Normale Superiore, Pisa, Italy

References
[1] A. Frommer, C. Schimmel, and M. Schweitzer, Analysis of probing techniques for sparse approximation and trace estimation of decaying matrix functions, SIAM Journal on Matrix Analysis and Applications, 2021 42:3, 1290-1318.
[2] R. A. Meyer, C. Musco, C. Musco, and D. P. Woodruff, Hutch++: optimal stochastic trace estimation, Symposium on Simplicity in Algorithms (SOSA), 2021, 142-155.
[3] S. Güttel, Rational Krylov approximation of matrix functions: numerical methods and optimal pole selection, GAMM-Mitteilungen, 2013, 36: 8-31.
[4] M. Benzi, M. Rinelli, I. Simunec, Computation of the von Neumann entropy of large matrices via trace estimation and Krylov subspace methods, arXiv preprint arXiv:2212.09642 (2022).

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Dec 9th, 2022 – Ludmil Zikatanov (Penn State)
2pm in Frank Adams 2

Multilevel methods for nearly-singular problems in mixed dimensions.

We consider nearly singular problems, that is, problems
with operators that are small, but nonsingular perturbations of
singular operators. Discretizations of such problems lead to matrices
with condition numbers of the system growing rapidly with mesh size
and model parameters. This results in slow convergence even when using
preconditioners which are optimal when the nonsingular perturbation
dominates.

To design efficient preconditioners we follow the theory of the method
of subspace corrections and construct block Schwarz smoothers for the
underlying multilevel solution method. The blocks are chosen
specifically to cover the supports of the vectors/functions spanning
the kernel of the singular part of the operator. We demonstrate key
features of such solvers on a mixed-dimensional model of
electrodiffusion in brain tissue. This is a joint work with Ana
Budisa, Miroslav Kuchta, Kent-Andre Mardal (from University of Oslo
and Simula) and James Adler and Xiaozhe Hu (Tufts University).

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Nov 18th, 2022 – Yunan Yang (ETH Zurich)
2pm in Frank Adams 1

Title: Optimal Transport for Learning Chaotic Dynamics via Invariant Measures

Abstract: Standard data-driven techniques for learning dynamical systems struggle when observational data has been sampled slowly, and derivatives cannot be accurately estimated. To address this challenge, we assume that the available measurements reliably describe the asymptotic statistics of the dynamical process in question, and we instead treat invariant measures as inference data. We reformulate the inversion as a PDE-constrained optimization problem by viewing invariant measures as stationary distributional solutions to the Fokker-Planck equation, which is discretized via an upwind finite volume scheme. The velocity is parameterized by fully-connected neural networks, and we use the adjoint-state method along with backpropagation to efficiently perform model identification. Numerical results for the Van der Pol Oscillator and Lorenz-63 system, as well as real-world applications to temperature prediction, are presented to demonstrate the effectiveness of the proposed approach.

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Nov 2nd, 2022 – Daan Huybrechs (KU Leuven)
Wednesday at 10am in Frank Adams 1

Title: Approximating functions with overcomplete representations

The representation of continuous functions is a foundational aspect of scientific computing. A case that often arises is the following. An expert practitioner has accumulated knowledge on the properties of the solution to a mathematical model of the problem at hand. Perhaps the solution has singularities at known locations – corners, edges, interfaces… Perhaps the solution oscillates in a certain way, has certain frequency content, it is transient, small in some parts of the domains and large in others,… How does one use that knowledge in a discretization? How can one exploit expert knowledge for computational gains? Most methods in scientific computing are based on piecewise polynomial approximations. Piecewise polynomials are powerful, but highly suboptimal in the presence of additional information. In this talk we discuss a number of ground rules that apply when we go beyond that setting. We identify a mathematical setting that has provable guarantees for stability and convergence, and we explore some novel algorithms which, at least in some settings, allow to pursue creative discretizations along with computational efficiency.

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Oct 28th, 2022 – Stefan Güttel (UoM)
2pm in Frank Adams 1

Randomized sketching for Krylov approximations of large-scale matrix functions

The computation of f(A)b, the action of a matrix function on a vector, is a task arising in many areas of scientific computing. In many applications, the matrix A is very large and hence computing f(A) explicitly is unfeasible. Here we discuss a new approach to overcome memory limitations of Krylov approximation methods for this task by combining randomized sketching with an integral representation of f(A)b. Two different approximations are introduced, one based on sketched FOM and another based on sketched GMRES. The convergence of the latter method is analyzed for Stieltjes functions of positive real matrices. We also derive a closed form expression for the sketched FOM approximant and bound its distance to the full FOM approximant. Numerical experiments demonstrate the potential of the sketching approaches. This is joint work with Marcel Schweitzer (Bergische Universität Wuppertal).

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Oct 7th, 2022 – Ian Mcinerney (UoM)
2pm in Frank Adams 1

Tuning Numerical Methods for Linear Predictive Control using Toeplitz Operator Theory

First-order optimization solvers, such as the Fast Gradient Method (FGM), are increasingly being used to solve Model Predictive Control (MPC) problems in resource-constrained environments where the computation time and available power are limited. Unfortunately, these implementations face two major issues: a poor convergence rate with illconditioned problem data, and instability/non-convergence when implemented using reduced precision data-types. In this work, we exploit the block Toeplitz structure of the MPC problem’s matrices to derive two results that are independent of the length of the prediction horizon: a preconditioner and data-type sizing rules that ensure stability. Horizon-independence allows one to use only the predicted system and cost matrices to compute the preconditioner and data-type sizes, instead of the full Hessian. The proposed preconditioner has equivalent performance to an optimal preconditioner in numerical examples, producing speed-ups between 2x and 9x for the Fast Gradient Method, but is computable in closed-form and without the need to solve large semi-definite optimization problems. We propose a metric for measuring the amount of round-off error the FGM iteration can tolerate before becoming unstable, and then combine that metric with a round-off error model derived from the block Toeplitz Hessian structure to derive a method for computing the minimum number of fractional bits needed for an implementation of the FGM using a fixed-point data type. This new design rule nearly halves the number of fractional bits needed to implement an example problem, and results in significant decreases in resource usage, computational energy and execution time for an implementation on a Field Programmable Gate Array. Finally, we derive horizon-independent spectral bounds for the Hessian in terms of the transfer function of the predicted system, and show how these can be used to compute a novel horizon-independent bound on the condition number for the preconditioned Hessian.

6 Jun 2022

Leonardo Robol

Senior Assistant Professor, University of Pisa
Title : Sampling the eigenvalues of random matrices
Abstract : Random matrices (that is, matrices with random variables as entries) appear in several areas of mathematics and physics. Often, one is interested in sampling their eigenvalues (which are again random variables); we briefly recap the main tools that can be used to sample the joint eigenvalue distribution for Hermitian matrices from the Gaussian unitary Ensemble (GUE) with O(n^2) flops, and discuss how similar ideas can be used to efficiently perform the analogous operation for Haar-distributed orthogonal or unitary matrices.

The latter operation requires to find a suitable compact canonical form for upper Hessenberg unitary matrices obtained by reducing random orthogonal/unitary matrices, and then efficiently operate on this compact representation to compute the eigenvalues.
Time : 2:00 PM to 3:00 PM.
Venue : Frank Adams 1, Alan Turing Building.

13 May 2022	Alex Townsend Associate Professor, Cornell University Title : Building a bridge between numerical linear algebra and theoretical computer science Abstract : The numerical linear algebra (NLA) and theoretical computer science (TCS) research communities work on similar linear algebra problems from strikingly different viewpoints. During the pandemic, we organized an online fortnightly seminar to explore the connections between these two communities. In the first half of the talk, I will discuss some observations in how the two communities formulate problems, design and analyse algorithms, and publicise their findings. In the second half, I will give a personal account of how TCS ideas has impacted my research. The aim of the seminar was to foster future collaborations between NLA and TCS and generally bring about a greater appreciation for each other’s work. The organizing committee for the seminar was Ilse Ipsen (NCSU), Mike Mahoney (Berkeley), Yuji Nakatsukasa (University of Oxford), Nikhil Srivastava (Berkeley), Alex Townsend (Cornell), and Joel Tropp (Caltech), along with founding members Daniel Kressner (EPFL) and Cleve Moler (MathWorks). Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
6 May 2022	Tim Sullivan Associate Professor in Predictive Modelling, University of Warwick and Alan Turing Institute Title : Γ-convergence of Onsager–Machlup functionals and MAP estimation in non-parametric Bayesian inverse problems Abstract : The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e.\ a MAP estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager–Machlup functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Γ-convergence of Onsager–Machlup functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
29 Apr 2022	Oliver Rhodes Lecturer in Bio-Inspired Computing, University of Manchester Title : Neuromorphic Computing Abstract : This talk will introduce the field of neuromorphic computing: building machines to explore brain function, and how we can use our enhanced understanding of the brain to build better computer hardware and algorithms. Specifically it will discuss spiking neural networks, including how they can be used to model neural circuits, and how these models can be employed to develop low-power bio-inspired AI systems. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
1 Apr 2022	Matthew Thorpe Lecturer in Applied Mathematics, University of Manchester Title : A Random Walk from Semi Supervised Learning to Neural Networks Abstract : Semi-supervised learning is the problem of finding missing labels; more precisely one has a data set of feature vectors of which a (often small) subset are labelled. The semi-supervised learning assumption is that similar feature vectors should have similar labels which implies one needs a geometry on the set of feature vectors. A typical way to represent this geometry is via a graph where the nodes are the feature vectors and the edges are weighted by some measure of similarity. Laplace learning is a popular graph-based method for solving the semi-supervised learning problem which essentially requires one to minimise a Dirichlet energy defined on the graph (hence the Euler-Lagrange equation is Laplace’s equation). However, at low labelling rates Laplace learning typically performs poorly. This is due to the lack of regularity, or the ill-posedness, of solutions to Laplace’s equation in any dimension higher (or equal to) two. The random walk interpretation allows one to characterise how close one is to entering the ill-posed regime. In particular, it allows one to give a lower bound on the number of labels required and even provides a route for correcting the bias. Correcting the bias leads to a new method, called Poisson learning. Finally, the ideas behind correcting the bias in Laplace learning have motivated a new graph neural network architecture which do not suffer from the over-smoothing phenomena. In particular, this type of neural network, which we call GRAND++ (GRAph Neural Diffusion with a source term) enables one to employ deep architectures. This is joint work with Jeff Calder, Dejan Slepčev, Brendan Cook, Tan Nguyen, Hedi Xia, Thomas Strohmer, Andrea Bertozzi, Stanley Osher and Bao Wang. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
24 March 2022	Matt Colbrook Junior Research Fellow, Trinity College, Cambridge Title : Koopman operators and the computation of spectral properties in infinite dimensions Abstract : Koopman operators are infinite-dimensional operators that globally linearise nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. Their increasing popularity, dubbed “Koopmania”, includes 10,000s of articles over the last decade. However, Koopman operators can have continuous spectra and lack finite-dimensional invariant subspaces, making computing their spectral properties a considerable challenge. This talk describes data-driven algorithms with rigorous convergence guarantees for computing spectral properties of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), the first scheme for computing the spectra and pseudospectra of general Koopman operators from trajectory data without spectral pollution. By combining ResDMD and the resolvent, we compute smoothed approximations of spectral measures associated with measure-preserving dynamical systems. When computing the continuous and discrete spectrum, explicit convergence theorems provide high-order convergence, even for chaotic systems. Kernelized variants of our algorithms allow for dynamical systems with a high-dimensional state-space. These infinite-dimensional numerical linear algebra algorithms are placed within a broader programme on the foundations of infinite-dimensional spectral computations. We end with computing the spectral measures of a protein molecule (20,046-dimensional state-space) and computing nonlinear Koopman modes with error bounds for chaotic turbulent flow past aerofoils with Reynolds number greater than 100,000 (295,122-dimensional state-space). Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
18 Mar 2022	Jemima Tabeart Postdoctoral Associate, University of Edinburgh Title : Novel preconditioners for saddle point weak-constraint 4D-Var Abstract : The saddle point formulation of weak-constraint four-dimensional data assimilation offers the possibility of exploiting modern computer architectures and algorithms due to itsunderlying block structure. Developing good preconditioners which retain the highly-structured nature of the saddle point system has been an area of recent research interest, especially for applications to numerical weather prediction. In this talk I will present two new approaches for preconditioners for the model terms within the saddle point system. Firstly, we consider including a small number of model terms within the preconditioner, which preserves a user-determined level of parallelisability. In contrast our second approach exploits inherent Kronecker structure within a matrixGMRES implementation. I will present theoretical results comparing our new preconditioners to existing standard choices of preconditioners. Finally I will present two numerical experiments for the heat equation and Lorenz 96 problem and show that our new approaches are competitive compared to current state-of-the-art preconditioners. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
4 Mar 2022	Mantas Mikaitis Research Associate, University of Manchester Title : Numerical Behavior of GPU Matrix Multiply-Accumulate Hardware Abstract : Tensor cores and matrix engines are hardware units on the latest GPUs that perform dot product or matrix multiply accumulate (MMA) operations. 127 of the TOP500 supercomputers contain these units and a lot of the numerical libraries begin to utilize them in various algorithms in scientific computing. Tensor cores and similar arithmetic units are targeted at low precision machine learning algorithms and therefore are not necessarily compliant with the IEEE 754 standard. The features such as rounding, normalization, order of operations, subnormal number support and others can differ from a standard software implementation of the matrix multiplication. In this talk I will discuss our recent work on determining various numerical features of MMAs, using NVIDIA tensor cores as an example test case. We determined the features of the three generations of the tensor core with the carefully constructed numerical test cases on the V100, T4 and the A100 NVIDIA GPUs and have explored the effects those features have on matrix multiplication algorithms, comparing real runs on the GPUs with the theoretical rounding error bounds that we have derived. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
18 Feb 2022	Catherine Powell Professor in Applied Mathematics, University of Manchester Title : Stochastic Galerkin Finite Element Approximation for Linear Elasticity and Poroelasticity Problems with Uncertain Inputs Abstract : The study of theoretical and computational aspects of stochastic Galerkin (SG) methods, also known as intrusive polynomial chaos methods, is now mature for several classes of model problems, such as scalar elliptic PDEs with parameter-dependent diffusion coefficients. For such problems, state-of-the-art multilevel adaptive SG methods have been developed and are competitive with more widely used non-intrusive approximation schemes. Since intrusive methods require the solution of large highly complex linear systems, in contrast to sampling methods, much work remains to be done to determine whether SG methods can provide a computationally feasible framework for facilitating forward UQ in more complex PDE models with uncertain inputs that arise in engineering applications. In this talk, we will discuss recent work on developing stochastic Galerkin mixed finite element schemes and solvers for linear elasticity and poroelasticity models, with parameter-dependent Young’s modulus and hydraulic conductivity fields. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.

6 Jun 2022

Leonardo Robol

13 May 2022	Alex Townsend Associate Professor, Cornell University Title : Building a bridge between numerical linear algebra and theoretical computer science Abstract : The numerical linear algebra (NLA) and theoretical computer science (TCS) research communities work on similar linear algebra problems from strikingly different viewpoints. During the pandemic, we organized an online fortnightly seminar to explore the connections between these two communities. In the first half of the talk, I will discuss some observations in how the two communities formulate problems, design and analyse algorithms, and publicise their findings. In the second half, I will give a personal account of how TCS ideas has impacted my research. The aim of the seminar was to foster future collaborations between NLA and TCS and generally bring about a greater appreciation for each other’s work. The organizing committee for the seminar was Ilse Ipsen (NCSU), Mike Mahoney (Berkeley), Yuji Nakatsukasa (University of Oxford), Nikhil Srivastava (Berkeley), Alex Townsend (Cornell), and Joel Tropp (Caltech), along with founding members Daniel Kressner (EPFL) and Cleve Moler (MathWorks). Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
6 May 2022	Tim Sullivan Associate Professor in Predictive Modelling, University of Warwick and Alan Turing Institute Title : Γ-convergence of Onsager–Machlup functionals and MAP estimation in non-parametric Bayesian inverse problems Abstract : The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e.\ a MAP estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager–Machlup functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Γ-convergence of Onsager–Machlup functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
29 Apr 2022	Oliver Rhodes Lecturer in Bio-Inspired Computing, University of Manchester Title : Neuromorphic Computing Abstract : This talk will introduce the field of neuromorphic computing: building machines to explore brain function, and how we can use our enhanced understanding of the brain to build better computer hardware and algorithms. Specifically it will discuss spiking neural networks, including how they can be used to model neural circuits, and how these models can be employed to develop low-power bio-inspired AI systems. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
1 Apr 2022	Matthew Thorpe Lecturer in Applied Mathematics, University of Manchester Title : A Random Walk from Semi Supervised Learning to Neural Networks Abstract : Semi-supervised learning is the problem of finding missing labels; more precisely one has a data set of feature vectors of which a (often small) subset are labelled. The semi-supervised learning assumption is that similar feature vectors should have similar labels which implies one needs a geometry on the set of feature vectors. A typical way to represent this geometry is via a graph where the nodes are the feature vectors and the edges are weighted by some measure of similarity. Laplace learning is a popular graph-based method for solving the semi-supervised learning problem which essentially requires one to minimise a Dirichlet energy defined on the graph (hence the Euler-Lagrange equation is Laplace’s equation). However, at low labelling rates Laplace learning typically performs poorly. This is due to the lack of regularity, or the ill-posedness, of solutions to Laplace’s equation in any dimension higher (or equal to) two. The random walk interpretation allows one to characterise how close one is to entering the ill-posed regime. In particular, it allows one to give a lower bound on the number of labels required and even provides a route for correcting the bias. Correcting the bias leads to a new method, called Poisson learning. Finally, the ideas behind correcting the bias in Laplace learning have motivated a new graph neural network architecture which do not suffer from the over-smoothing phenomena. In particular, this type of neural network, which we call GRAND++ (GRAph Neural Diffusion with a source term) enables one to employ deep architectures. This is joint work with Jeff Calder, Dejan Slepčev, Brendan Cook, Tan Nguyen, Hedi Xia, Thomas Strohmer, Andrea Bertozzi, Stanley Osher and Bao Wang. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
24 March 2022	Matt Colbrook Junior Research Fellow, Trinity College, Cambridge Title : Koopman operators and the computation of spectral properties in infinite dimensions Abstract : Koopman operators are infinite-dimensional operators that globally linearise nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. Their increasing popularity, dubbed “Koopmania”, includes 10,000s of articles over the last decade. However, Koopman operators can have continuous spectra and lack finite-dimensional invariant subspaces, making computing their spectral properties a considerable challenge. This talk describes data-driven algorithms with rigorous convergence guarantees for computing spectral properties of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), the first scheme for computing the spectra and pseudospectra of general Koopman operators from trajectory data without spectral pollution. By combining ResDMD and the resolvent, we compute smoothed approximations of spectral measures associated with measure-preserving dynamical systems. When computing the continuous and discrete spectrum, explicit convergence theorems provide high-order convergence, even for chaotic systems. Kernelized variants of our algorithms allow for dynamical systems with a high-dimensional state-space. These infinite-dimensional numerical linear algebra algorithms are placed within a broader programme on the foundations of infinite-dimensional spectral computations. We end with computing the spectral measures of a protein molecule (20,046-dimensional state-space) and computing nonlinear Koopman modes with error bounds for chaotic turbulent flow past aerofoils with Reynolds number greater than 100,000 (295,122-dimensional state-space). Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
18 Mar 2022	Jemima Tabeart Postdoctoral Associate, University of Edinburgh Title : Novel preconditioners for saddle point weak-constraint 4D-Var Abstract : The saddle point formulation of weak-constraint four-dimensional data assimilation offers the possibility of exploiting modern computer architectures and algorithms due to itsunderlying block structure. Developing good preconditioners which retain the highly-structured nature of the saddle point system has been an area of recent research interest, especially for applications to numerical weather prediction. In this talk I will present two new approaches for preconditioners for the model terms within the saddle point system. Firstly, we consider including a small number of model terms within the preconditioner, which preserves a user-determined level of parallelisability. In contrast our second approach exploits inherent Kronecker structure within a matrixGMRES implementation. I will present theoretical results comparing our new preconditioners to existing standard choices of preconditioners. Finally I will present two numerical experiments for the heat equation and Lorenz 96 problem and show that our new approaches are competitive compared to current state-of-the-art preconditioners. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
4 Mar 2022	Mantas Mikaitis Research Associate, University of Manchester Title : Numerical Behavior of GPU Matrix Multiply-Accumulate Hardware Abstract : Tensor cores and matrix engines are hardware units on the latest GPUs that perform dot product or matrix multiply accumulate (MMA) operations. 127 of the TOP500 supercomputers contain these units and a lot of the numerical libraries begin to utilize them in various algorithms in scientific computing. Tensor cores and similar arithmetic units are targeted at low precision machine learning algorithms and therefore are not necessarily compliant with the IEEE 754 standard. The features such as rounding, normalization, order of operations, subnormal number support and others can differ from a standard software implementation of the matrix multiplication. In this talk I will discuss our recent work on determining various numerical features of MMAs, using NVIDIA tensor cores as an example test case. We determined the features of the three generations of the tensor core with the carefully constructed numerical test cases on the V100, T4 and the A100 NVIDIA GPUs and have explored the effects those features have on matrix multiplication algorithms, comparing real runs on the GPUs with the theoretical rounding error bounds that we have derived. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.
18 Feb 2022	Catherine Powell Professor in Applied Mathematics, University of Manchester Title : Stochastic Galerkin Finite Element Approximation for Linear Elasticity and Poroelasticity Problems with Uncertain Inputs Abstract : The study of theoretical and computational aspects of stochastic Galerkin (SG) methods, also known as intrusive polynomial chaos methods, is now mature for several classes of model problems, such as scalar elliptic PDEs with parameter-dependent diffusion coefficients. For such problems, state-of-the-art multilevel adaptive SG methods have been developed and are competitive with more widely used non-intrusive approximation schemes. Since intrusive methods require the solution of large highly complex linear systems, in contrast to sampling methods, much work remains to be done to determine whether SG methods can provide a computationally feasible framework for facilitating forward UQ in more complex PDE models with uncertain inputs that arise in engineering applications. In this talk, we will discuss recent work on developing stochastic Galerkin mixed finite element schemes and solvers for linear elasticity and poroelasticity models, with parameter-dependent Young’s modulus and hydraulic conductivity fields. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.