Videos and presentation materials from other INI events are also available.
Event  When  Speaker  Title  Presentation Material 

GCSW01 
8th July 2019 10:00 to 11:00 
Hans MuntheKaas 
Why Bseries, rooted trees, and free algebras?  1
"We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problemoriented work can lead to farreaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's Bseries. The aim of this joined lecture is to answer the question posed in the title by drawing a line from Bseries to those farreaching conceptional results they originated. Unfolding the precise mathematical picture underlying Bseries requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre/postLie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures. In the first lecture we will outline the geometric foundations of Bseries, and their cousins LieButcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes.


GCSW01 
8th July 2019 11:30 to 12:30 
Charlie Elliott  PDEs in Complex and Evolving Domains I  
GCSW01 
8th July 2019 14:00 to 15:00 
Elizabeth Mansfield 
Introduction to Lie groups and algebras  1
In this series of three lectures, I
will give a gentle introduction to Lie groups and their associated Lie
algebras, concentrating on the major examples of importance in applications. I
will next discuss Lie group actions, their invariants, their associated
infinitesimal vector fields, and a little on moving frames. The final topic
will be Noether’s Theorem, which yields conservations laws for variational
problems which are invariant under a Lie group action.
Time
permitting, I will show a little of the finite difference and finite element
versions of Noether’s Theorem. In this first talk, I will consider some of the simplest and most useful Lie groups. I will show how to derive their Lie algebras, and will further discuss the Adjoint action of the group on its Lie algebra, the Lie bracket on the algebra, and the exponential map. 

GCSW01 
8th July 2019 15:00 to 16:00 
Christopher Budd 
Modified error estimates for discrete variational derivative methods
My three talks will be an exploration of geometric integration methods in the context of the numerical solution of PDEs. I will look both at discrete variational methods and their analysis using modified equations, and also of the role of adaptivity in studying and retaining qualitative features of PDEs. 

GCSW01 
8th July 2019 16:30 to 17:30 
Paola Francesca Antonietti 
Highorder Discontinuous Galerkin methods for the numerical modelling of earthquake ground motion
A number of challenging geophysical applications requires a flexible representation of the geometry and an accurate approximation of the solution field. Paradigmatic examples include seismic wave propagation and fractured reservoir simulations. The main challenges are i) the complexity of the physical domain, due to the presence of localized geological irregularities, alluvial basins, faults and fractures; ii) the heterogeneities in the medium, with significant and sharp contrasts; and iii) the coexistence of different physical models. The highorder discontinuous Galerkin FEM possesses the builtin flexibility to naturally accommodate both nonmatching meshes, possibly made of polygonal and polyhedral elements, and highorder approximations in any space dimension. In this talk I will discuss recent advances in the development and analysis of highorder DG methods for the numerical approximation of seismic wave propagation phenomena. I will analyse the stability and the theoretical properties of the scheme and present some simulations of real largescale seismic events in threedimensional complex media.


GCSW01 
9th July 2019 09:00 to 10:00 
Douglas Arnold 
Finite Element Exterior Calculus  1
These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key toolschain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexesand explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications.


GCSW01 
9th July 2019 10:00 to 11:00 
Elizabeth Mansfield 
Introduction to Lie groups and algebras  2
I
will discuss Lie group actions, their invariants, their associated
infinitesimal vector fields, and a little on moving frames.


GCSW01 
9th July 2019 11:30 to 12:30 
Kurusch EbrahimiFard 
Why Bseries, rooted trees, and free algebras?  2
"We regard Butcher’s work on the classification of
numerical integration methods as an impressive example that concrete
problemoriented work can lead to farreaching conceptual results”. This quote
by Alain Connes summarises nicely the mathematical depth and scope of the
theory of Butcher's Bseries.
The aim of this joined lecture is to answer the question
posed in the title by drawing a line from Bseries to those farreaching
conceptional results they originated. Unfolding the precise mathematical
picture underlying Bseries requires a combination of different perspectives
and tools from geometry (connections); analysis (generalisations of Taylor
expansions), algebra (pre/postLie and Hopf algebras) and combinatorics (free
algebras on rooted trees). This summarises also the scope of these lectures.
In the first lecture we will outline the geometric
foundations of Bseries, and their cousins LieButcher series. The latter is
adapted to studying differential equations on manifolds. The theory of
connections and parallel transport will be explained. In the second and third
lectures we discuss the algebraic and combinatorial structures arising from the
study of invariant connections. Rooted trees play a particular role here as
they provide optimal index sets for the terms in Taylor series and
generalisations thereof. The final lecture will discuss various applications of
the theory in the numerical analysis of integration schemes.


GCSW01 
9th July 2019 14:00 to 15:00 
Charlie Elliott  PDEs in Complex and Evolving Domains II  
GCSW01 
9th July 2019 15:00 to 16:00 
Christian Lubich 
Variational Gaussian wave packets revisited
The talk reviews Gaussian wave packets that evolve according to the DiracFrenkel timedependent variational principle for the semiclassically scaled Schr\"odinger equation. Old and new results on the approximation to the wave function are given, in particular an $L^2$ error bound that goes back to Hagedorn (1980) in a nonvariational setting, and a new error bound for averages of observables with a Weyl symbol, which shows the double approximation order in the semiclassical scaling parameter in comparison with the norm estimate. The variational equations of motion in Hagedorn's parametrization of the Gaussian are presented. They show a perfect quantumclassical correspondence and allow us to read off directly that the Ehrenfest time is determined by the Lyapunov exponent of the classical equations of motion. A variational splitting integrator is formulated and its remarkable conservation and approximation properties are discussed. A new result shows that the integrator approximates averages of observables with the full order in the time stepsize, with an error constant that is uniform in the semiclassical parameter. The material presented here for variational Gaussians is part of an Acta Numerica review article on computational methods for quantum dynamics in the semiclassical regime, which is currently in preparation in joint work with Caroline Lasser. 

GCSW01 
10th July 2019 09:00 to 10:00 
Christopher Budd 
Blowup in PDES and how to compute it
My three talks will be an exploration of geometric integration methods in the context of the numerical solution of PDEs. I will look both at discrete variational methods and their analysis using modified equations, and also of the role of adaptivity in studying and retaining qualitative features of PDEs. 

GCSW01 
10th July 2019 10:00 to 11:00 
Douglas Arnold 
Finite Element Exterior Calculus  2
These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key toolschain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexesand explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications.


GCSW01 
10th July 2019 11:30 to 12:30 
Charlie Elliott  PDEs in Complex and Evolving Domains III  
GCSW01 
11th July 2019 09:00 to 10:00 
Charlie Elliott  PDEs in Complex and Evolving Domains IV  
GCSW01 
11th July 2019 10:00 to 11:00 
Beth Wingate 
An introduction to timeparallel methods
I
will give an introduction to timeparallel methods and discuss how they apply
to PDEs, in particular those with where resonance plays an important role. I’ll
give some examples from ODEs before finally discussing PDEs.


GCSW01 
11th July 2019 11:30 to 12:30 
Douglas Arnold 
Finite Element Exterior Calculus  3
These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key toolschain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexesand explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications.


GCSW01 
11th July 2019 14:00 to 15:00 
Kurusch EbrahimiFard 
Why Bseries, rooted trees, and free algebras?  3
"We regard Butcher’s work on the classification of
numerical integration methods as an impressive example that concrete
problemoriented work can lead to farreaching conceptual results”. This quote
by Alain Connes summarises nicely the mathematical depth and scope of the
theory of Butcher's Bseries.
The aim of this joined lecture is to answer the question
posed in the title by drawing a line from Bseries to those farreaching
conceptional results they originated. Unfolding the precise mathematical
picture underlying Bseries requires a combination of different perspectives
and tools from geometry (connections); analysis (generalisations of Taylor
expansions), algebra (pre/postLie and Hopf algebras) and combinatorics (free
algebras on rooted trees). This summarises also the scope of these lectures.
In the first lecture we will outline the geometric
foundations of Bseries, and their cousins LieButcher series. The latter is
adapted to studying differential equations on manifolds. The theory of
connections and parallel transport will be explained. In the second and third
lectures we discuss the algebraic and combinatorial structures arising from the
study of invariant connections. Rooted trees play a particular role here as
they provide optimal index sets for the terms in Taylor series and
generalisations thereof. The final lecture will discuss various applications of
the theory in the numerical analysis of integration schemes.


GCSW01 
11th July 2019 15:00 to 16:00 
Elizabeth Mansfield 
Noether's conservation laws  smooth and discrete
The final topic
will be Noether’s Theorem, which yields conservation laws for variational
problems which are invariant under a Lie group action.
. I will show a little of the finite difference and finite element
versions of Noether’s Theorem.


GCSW01 
11th July 2019 16:30 to 17:30 
Panel  
GCSW01 
12th July 2019 09:00 to 10:00 
Reinout Quispel 
Discrete Darboux polynomials and the preservation of measure and integrals of ordinary differential equations
Preservation of phase space volume (or more generally
measure), first integrals (such as energy), and second integrals have been
important topics in geometric numerical integration for more than a decade, and
methods have been developed to preserve each of these properties separately.
Preserving two or more geometric properties
simultaneously, however, has often been difficult, if not impossible.
Then it was discovered that Kahan’s ‘unconventional’
method seems to perform well in many cases [1]. Kahan himself, however, wrote:
“I have used these unconventional methods for 24 years without quite
understanding why they work so well as they do, when they work.”
The first approximation to such an understanding in
computational terms was:
Kahan’s method works so well because
1. It is very successful at preserving multiple quantities simultaneously, eg modified energy and modified measure. 2. It is linearly implicit 3. It is the restriction of a RungeKutta method However, point 1 above raises a further obvious question: Why does Kahan’s method preserve both certain (modified) first integrals and certain (modified) measures? In this talk we invoke Darboux polynomials to try and answer this question. The method of Darboux polynomials (DPs) for ODEs was introduced by Darboux to detect rational integrals. Very recently we have advocated the use of DPs for discrete systems [2,3]. DPs provide a unified theory for the preservation of polynomial measures and second integrals, as well as rational first integrals. In this new perspective the answer we propose to the above question is: Kahan’s method works so well because it is good at preserving (modified) Darboux polynomials. If time permits we may discuss extensions to polarization methods. [1] Petrera et al, Regular and Chaotic Dynamics 16 (2011), 245–289. [2] Celledoni et al, arxiv:1902.04685. [3] Celledoni et al, arxiv:1902.04715. 

GCSW01 
12th July 2019 10:00 to 10:30 
James Jackaman 
Lie symmetry preserving finite element methods
Through
the mathematical construction of finite element methods, the “standard”
finite element method will often preserve the underlying symmetry of a given
differential equation. However, this is not always the case, and while
historically much attention has been paid to the preserving of conserved
quantities the preservation of Lie symmetries is an open problem. We
introduce a methodology for the design of arbitrary order finite element
methods which preserve the underlying Lie symmetries through an equivariant
moving frames based invariantization procedure.


GCSW01 
12th July 2019 10:30 to 11:00 
Candan Güdücü 
PortHamiltonian Systems
The framework of portHamiltonian systems (PH systems) combines both the Hamiltonian approach and the network approach, by associating with the interconnection structure of the network model a geometric structure given by a Dirac structure. In this talk, I introduce portHamiltonian (pH) systems and their underlying Dirac structures. Then, a coordinatebased representation of PH systems and some properties are shown. A Lanczos method for the solution of linear systems with nonsymmetric coefficient matrices and its application to pH systems are presenred.


GCSW01 
12th July 2019 11:30 to 12:30 
Christopher Budd 
Adaptivity and optimal transport
My three talks will be an exploration of geometric integration methods in the context of the numerical solution of PDEs. I will look both at discrete variational methods and their analysis using modified equations, and also of the role of adaptivity in studying and retaining qualitative features of PDEs. 

GCSW01 
12th July 2019 14:00 to 15:00 
Douglas Arnold 
Finite element exterior calculus  4
These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key toolschain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexesand explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications.


GCSW01 
12th July 2019 15:00 to 16:00 
Hans MuntheKaas 
Why Bseries, rooted trees, and free algebras?  2
"We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problemoriented work can lead to farreaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's Bseries. The aim of this joined lecture is to answer the question posed in the title by drawing a line from Bseries to those farreaching conceptional results they originated. Unfolding the precise mathematical picture underlying Bseries requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre/postLie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures. In the first lecture we will outline the geometric foundations of Bseries, and their cousins LieButcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes.


GCS 
17th July 2019 11:00 to 12:00 
Christian Offen 
Detection of high codimensional bifurcations in variational PDEs
We derive bifurcation test equations for Aseries singularities of nonlinear functionals and, based on these equations, we propose a numerical method for detecting high codimensional bifurcations in parameterdependent PDEs such as parameterdependent semilinear Poisson equations. As an example, we consider a Bratutype problem and show how high codimensional bifurcations such as the swallowtail bifurcation can be found numerically. Lisa Maria Kreusser, Robert I McLachlan, Christian Offen 

GCS 
22nd July 2019 14:00 to 15:00 
Ralf Hiptmair  Exterior Shape Calculus  
GCS 
23rd July 2019 14:00 to 15:00 
Annalisa Buffa  Dual complexes and mortaring for regular approximations of electromagnetic fields  
GCS 
24th July 2019 15:00 to 16:00 
Onno Bokhove 
A new wavetowire waveenergy model: from variational principle to compatible spacetime discretisation
Amplification phenomena in a socalled
boresolitonsplash have led us to develop a novel waveenergy device with wave
amplification in a contraction used to enhance waveactivated buoy motion and
magneticallyinduced energy generation.
An experimental proofofprinciple shows that our
waveenergy device works.
Most importantly, we develop a novel wavetowire
mathematical model of the combined wave hydrodynamics, waveactivated buoy
motion and electric power generation by magnetic induction, from first principles,
satisfying one grand variational principle in its conservative limit.
Wave and buoy dynamics are coupled via a Lagrange
multiplier, which boundary value at the waterline is subtly solved explicitly
by imposing incompressibility in a weak sense.
Dissipative features, such as electrical wire resistance
and nonlinear LEDloads, are added a posteriori.
New is also the intricate and compatible (finiteelement)
spacetime discretisation of the linearised dynamics, guaranteeing numerical
stability and the correct energy transfer between the three subsystems.
Preliminary simulations of our simplified and linearised
waveenergy model are encouraging, yet suboptimal, and involve a first study of
the resonant behaviour and parameter dependence of the device.


GCS 
30th July 2019 14:00 to 15:00 
Melvin Leok  The Connections Between Discrete Geometric Mechanics, Information Geometry and Machine Learning  
GCS 
30th July 2019 15:00 to 16:00 
Benjamin Tapley  Computational methods for simulating inertial particles in discrete incompressible flows.  
GCS 
31st July 2019 15:00 to 16:00 
Anthony Bloch 
Optimal control and the geometry of integrable systems In this talk we discuss a geometric approach to certain optimal control problems and discuss the relationship of the solutions of these problem to some classical integrable dynamical systems and their generalizations. We consider the socalled Clebsch optimal control problem and its relationship to Lie group actions on manifolds. The integrable systems discussed include the rigid body equations, geodesic flows on the ellipsoid, flows on Stiefel manifolds, and the Toda lattice flows. We discuss the Hamiltonian structure of these systems and relate our work to some work of Moser. We also discuss the link to discrete dynamics and symplectic integration. 

GCS 
7th August 2019 14:00 to 15:00 
Arieh Iserles 
Fast approximation on the real line
Abstract: While approximation theory in an
interval is thoroughly understood, the real line represents something of a
mystery. In this talk we review the state of the art in this area, commencing
from the familiar Hermite functions and moving to recent results
characterising all orthonormal sets on $L_2(\infty,\infty)$ that have a
skewsymmetric (or skewHermitian) tridiagonal differentiation matrix and such
that their first $n$ expansion coefficients can be calculated in $O(n \log n)$
operations. In particular, we describe the generalised Malmquist–Takenaka
system. The talk concludes with a (too!) long list of open problems and
challenges. 

GCS 
7th August 2019 15:00 to 16:00 
Robert McLachlan 
The Lie algebra of classical mechanics
Classical
mechanical systems are defined by their kinetic and potential energies.
They
generate a Lie algebra under the canonical Poisson bracket which is
useful
in geometric integration.
But
because the kinetic energy is quadratic in the
momenta,
the Lie algebra obeys identities beyond those implied by skew symmetry
and
the Jacobi identity. Some Poisson brackets, or combinations of brackets, are
zero
for all choices of kinetic and potential energy.
Therefore,
we study and give a complete description of
the
universal object in this setting, the ‘Lie algebra of
classical
mechanics’ modelled on the Lie algebra generated by kinetic and potential
energy of a simple mechanical system with respect to
the canonical Poisson bracket. Joint work with Ander Murua. 

GCS 
14th August 2019 14:00 to 15:00 
David Garfinkle 
Numerical General Relativity
This
talk will cover the basic properties of the equations of General Relativity, and issues involved in performing numerical simulations of those equations. Particular emphasis will be placed on three issues: (1) hyperbolicity of the equations. (2) preserving constraints. (3) dealing with black holes and spacetime singularities.


GCS 
14th August 2019 15:00 to 16:00 
Ari Stern 
Finite element methods for Hamiltonian PDEs
Hamiltonian
ODEs satisfy a symplectic conservation law, and there are many advantages to
using numerical integrators that preserves this structure. This talk will
discuss how the canonical Hamiltonian structure, and its preservation by a
numerical method, can be generalized to PDEs. I will also provide a basic
introduction to the finite element method and, time permitting, discuss how
some classic symplectic integrators can be understood from this point of view.


GCS 
21st August 2019 14:00 to 15:00 
Fernando Casas 
More on composition methods: error estimation and pseudosymmetry
In this talk I will review composition methods
for the time integration of differential equations, paying special attention to two recent contributions in this area. The first one is the construction of a new local error estimator so that the additional computational effort required is almost insignificant. The second one is related to a new family of highorder methods obtained from a basic symmetric (symplectic) scheme in such a way that they are timesymmetric (symplectic) only up to a certain order. 

GCS 
21st August 2019 15:00 to 16:00 
Yajuan Sun 
Contact Hamiltonian system and its application in solving VlasovPoisson FokkerPlanck system
The VlasovPoisson FokkerPlanck system
is a kinetic description of Brownian motion for
a large number of particles in a surrounding bath. In solving
this system, the contact structure is investigated. This
motivates the study for the contact system and the construction of
corresponding numerical methods. This talk will introduce the contact system
and the construction of numerical methods based on generating
function and variational principle. 

GCS 
28th August 2019 14:00 to 15:00 
Gerard Awanou 
Computational geometric optics: MongeAmpere
I will review recent developments in the numerical resolution of the second boundary value problem for MongeAmpere type equations and their applications to the design of reflectors and refractors. 

GCS 
28th August 2019 15:00 to 16:00 
Milo Viviani 
LiePoisson methods for isospectral flows and their application to longtime simulation of spherical ideal hydrodynamics
The theory of isospectral flows comprises a
large class of continuous dynamical systems, particularly integrable
systems and Lie–Poisson systems. Their discretization is a classical
problem in numerical analysis. Preserving the spectra in the discrete flow
requires the conservation of high order polynomials, which is hard to
come by. Existing methods achieving this are complicated and usually fail to
preserve the underlying LiePoisson structure. Here we present a class of
numerical methods of arbitrary order for Hamiltonian and nonHamiltonian
isospectral flows, which preserve both the spectra and the LiePoisson
structure. The methods are surprisingly simple, and avoid the use of
constraints or exponential maps. Furthermore, due to preservation
of the Lie–Poisson structure, they exhibit near conservation of the
Hamiltonian function. As an illustration, we apply the methods to
longtime simulation of the Euler equations on a sphere. Our findings
suggest that our structurepreserving algorithms, on the one hand, perform
at least as well as other popular methods (i.e. CLAM) without adding
spurious hyperviscosity terms, on the other hand, show that the
conservation of the Casimir functions can be actually used to predict
the final state of the fluid 

GCS 
4th September 2019 14:00 to 15:00 
Sina OberBlöbaum 
Variational formulations for dissipative systems
Variational
principles are powerful tools for the modelling and simulation of conservative
mechanical and electrical systems. As it is wellknown, the fulfilment of a
variational principle leads to the EulerLagrange equations of motion
describing the dynamics of such systems. Furthermore, a variational
discretisation directly yields unified numerical schemes with powerful
structurepreserving properties. Since many years there have been several
attempts to provide a variational description also for dissipative mechanical
systems, a task that is addressed in the talk in order to construct both
Lagrangian and Hamiltonian pictures of their dynamics.
One
way doing this is to use fractional terms in the Lagrangian or Hamiltonian
function which allows for a purely variational derivation of dissipative
systems. Another approach followed in this talk is to embed the
nonconservative systems in larger conservative systems. These concepts are
used to develop variational integrators for which superior qualitative numerical
properties such as the correct energy dissipation rate are demonstrated.


GCS 
4th September 2019 15:00 to 16:00 
Sigrid Leyendecker 
Mixed order and multirate variational integrators for the simulation of dynamics on different time scales
Mechanical
systems with dynamics on varying time scales, e.g. including highly oscillatory
motion, impose challenging questions for numerical integration schemes. High
resolution is required to guarantee a stable integration of the fast
frequencies. However, for the simulation of the slow dynamics, integration with
a lower resolution is accurate enough  and computationally cheaper, especially
for costly function evaluations. Two approaches are presented, a mixed order
Galerkin variational integrator and a multirate variational integrator, and
analysed with respect to the preservation of invariants, computational costs,
accuracy and linear stability.


GCS 
5th September 2019 14:15 to 15:15 
Chus SanzSerna  Numerical Integrators for the Hamiltonian Monte Carlo Method  
GCS 
11th September 2019 14:00 to 15:00 
Peter Hydon 
Conservation laws and Euler operators
A (local) conservation law of a given system of
differential or difference equations is a divergence expression that is zero on
all solutions. The Euler operator is a powerful tool in the formal theory of
conservation laws that enables key results to be proved simply, including
several generalizations of Noether's theorems.
This talk begins with a short survey of the main ideas and results.
The current method for inverting the divergence operator
generates many unnecessary terms by integrating in all directions
simultaneously. As a result, symbolic algebra packages create overcomplicated
representations of conservation laws, making it difficult to obtain efficient
conservative finite difference approximations symbolically. A new approach
resolves this problem by using partial Euler operators to construct
nearoptimal representations. The talk explains this approach, which was
developed during the GCS programme.


GCS 
11th September 2019 15:00 to 16:00 
Gianluca FrascaCaccia 
Numerical preservation of local conservation laws
In the numerical treatment of partial differential
equations (PDEs), the benefits of preserving global integral invariants are
wellknown. Preserving the underlying local conservation law gives, in general,
a stricter constraint than conserving the global invariant obtained by
integrating it in space. Conservation laws, in fact, hold throughout the domain
and are satisfied by all solutions, independently of initial and boundary conditions.
A new approach that uses symbolic algebra to develop
bespoke finite difference schemes that preserve multiple local conservation
laws has been recently applied to PDEs with polynomial nonlinearity.
The talk illustrates this new strategy using some
wellknown equations as benchmark examples and shows comparisons between the
obtained schemes and other integrators known in literature.


GCS 
18th September 2019 14:05 to 14:50 
Blanca Ayuso De Dios 
Constructing Discontinuous Galerkin methods for Vlasovtype systems
The
VlasovPoisson and the VlasovMaxwell systems are two classical models in
collisionless kinetic theory. They are both derived as meanfield limit
description of a large ensemble of interacting particles by electrostatic
and electromagnetic forces, respectively. In this talk we describe how to design (semidiscrete!) discontinuous Galerkin finite element methods for approximating such Vlasovtype systems. We outline the error analysis of the schemes and discuss further properties of the proposed schemes, as well as their shortcomings. If time allows, we discuss further endeavours in alleviating the drawbacks of the schemes. 

GCS 
18th September 2019 15:05 to 15:50 
Linyu Peng 
Variational systems on the variational bicomplex
It is well know that
symplecticity plays a fundamentally important role in Lagrangian and
Hamiltonian systems. Numerical methods preserving symplecticity (or
multisymplecticity for PDEs) have been greatly developed and applied
during last decades. In this talk, we will show how the
variational bicomplex, a double cochain complex on jet manifolds, provides
a natural framework for understanding multisymplectic systems. The
discrete counterpart, discrete multisymplectic systems on the
difference variational bicomplex will briefly be introduced if time
permits.


GCS 
19th September 2019 16:00 to 17:00 
Marcus Webb  Energy preserving spectral methods on the real line whose analysis strays into the complex plane (copy)  
GCS 
25th September 2019 14:05 to 14:50 
Martin Licht 
Newest Results in Newest Vertex Bisection
The algorithmic refinement of triangular meshes
is an important component in numerical simulation codes. Newest vertex
bisection is one of the most popular methods for geometrically stable local
refinement. Its complexity analysis, however, is a fairly intricate recent
result and many combinatorial aspects of this method are not yet fully
understood. In this talk, we access newest vertex bisection from the
perspective of theoretical computer science. We outline the amortized
complexity analysis over generalized triangulations. An immediate application
is the convergence and complexity analysis of adaptive finite element methods
over embedded surfaces and singular surfaces. Moreover, we
"combinatorialize" the complexity estimate and remove any
geometrydependent constants, which is only natural for this purely
combinatorial algorithm and improves upon prior results. This is joint work with
Michael Holst and Zhao Lyu. 

GCSW02 
30th September 2019 09:30 to 10:30 
Deirdre Shoemaker 
Numerical Relativity in the Era of Gravitational Wave Observations
The birth and future of gravitational wave astronomy offers new opportunities and challenges for numerical methods in general relativity. Numerical relativity in particular provides critical support to detect and interpret gravitational wave measurements. In this talk, I’ll discuss the role numerical relativity is playing in the observed black hole binaries by LIGO and Virgo and its future potential for unveiling strongfield gravity in future detections with an emphasis on the computational challenges. I'll frame a discussion about what demands will be placed on this field to maximize the science output of the new era. 

GCSW02 
30th September 2019 11:00 to 12:00 
Michael Holst 
Some Research Problems in Mathematical and Numerical General Relativity
The 2017 Nobel Prize in Physics was awarded to three of the key scientists involved in the development of LIGO and its eventual successful first detections of gravitational waves. How do LIGO (and other gravitational wave detector) scientists know what they are detecting? The answer is that the signals detected by the devices are shown, after extensive data analysis and numerical simulations of the Einstein equations, to be a very close match to computer simulations of wave emission from very particular types of binary collisions. In this lecture, we begin with a brief overview of the mathematical formulation of Einstein (evolution and constraint) equations, and then focus on some fundamental mathematics research questions involving the Einstein constraint equations. We begin with a look at the most useful mathematical formulation of the constraint equations, and then summarize the known existence, uniqueness, and multiplicity results through 2009. We then present a number of new existence and multiplicity results developed since 2009 that substantially change the solution theory for the constraint equations. In the second part of the talk, we consider approaches for developing "provably good" numerical methods for solving these types of geometric PDE systems on 2 and 3manifolds. We examine how one proves rigorous error estimates for particular classes of numerical methods, including both classical finite element methods and newer methods from the finite element exterior calculus. This lecture will touch on several joint projects that span more than a decade, involving a number of collaborators. The lecture is intended both for mathematicians interested in potential research problems in mathematical and numerical general relativity, as well as physicists interested in relevant new developments in mathematical and numerical methods for nonlinear geometric PDE. 

GCSW02 
30th September 2019 13:30 to 14:30 
Pau Figueras 
numerical relativity beyond astrophysics: new challenges and new dynamics
Motivated by more fundamental theories of gravity such as string theory, in recent years there has been a growing interesting in solving the Einstein equations numerically beyond the traditional astrophysical set up. For instance in spacetime dimensions higher than the four that we have observed, or in exotic spaces such as antide Sitter spaces. In this talk I will give an overview of the challenges that are often encountered when solving the Einstein equations in these new settings. In the second part of the talk I will provide some examples, such as the dynamics of unstable black holes in higher dimensions and gravitational collapse in antide Sitter spaces.


GCSW02 
30th September 2019 14:30 to 15:30 
Ari Stern 
Structurepreserving time discretization: lessons for numerical relativity?
In numerical ODEs, there is a rich literature on methods that preserve certain geometric structures arising in physical systems, such as Hamiltonian/symplectic structure, symmetries, and conservation laws. I will give an introduction to these methods and discuss recent work extending some of these ideas to numerical PDEs in classical field theory.


GCSW02 
30th September 2019 16:00 to 17:00 
Frans Pretorius 
Computational Challenges in Numerical Relativity
I will give a brief overview of the some of the challenges in computational solution of the Einstein field equations.I will then describe the background error subtraction technique, designed to allow for more computationally efficient solution of scenarios where a significant portion of the domain is close to a know, exact solution. To demonstrate, I will discuss application to tidal disruption of a star by a supermassive black hole, and studies of black hole superradiance.


GCSW02 
1st October 2019 09:30 to 10:30 
Lee Lindblom  Solving PDEs Numerically on Manifolds with Arbitrary Spatial Topologies  
GCSW02 
1st October 2019 11:00 to 12:00 
Melvin Leok 
Variational discretizations of gauge field theories using groupequivariant interpolation spaces
Variational integrators are geometric structurepreserving numerical methods that preserve the symplectic structure, satisfy a discrete Noether's theorem, and exhibit exhibit excellent longtime energy stability properties. An exact discrete Lagrangian arises from Jacobi's solution of the HamiltonJacobi equation, and it generates the exact flow of a Lagrangian system. By approximating the exact discrete Lagrangian using an appropriate choice of interpolation space and quadrature rule, we obtain a systematic approach for constructing variational integrators. The convergence rates of such variational integrators are related to the best approximation properties of the interpolation space. Many gauge field theories can be formulated variationally using a multisymplectic Lagrangian formulation, and we will present a characterization of the exact generating functionals that generate the multisymplectic relation. By discretizing these using groupequivariant spacetime finite element spaces, we obtain methods that exhibit a discrete multimomentum conservation law. We will then briefly describe an approach for constructing groupequivariant interpolation spaces that take values in the space of Lorentzian metrics that can be efficiently computed using a generalized polar decomposition. The goal is to eventually apply this to the construction of variational discretizations of general relativity, which is a secondorder gauge field theory whose configuration manifold is the space of Lorentzian metrics. 

GCSW02 
1st October 2019 13:30 to 14:30 
Oscar Reula 
Hyperbolicity and boundary conditions.
Abstract: (In collaboration with Fernando Abalos.) Very often in physics, the evolution systems we have to deal with are not purely hyperbolic, but contain also constraints and gauge freedoms. After fixing these gauge freedoms we obtain a new system with constraints which we want to solve subject to initial and boundary values. In particular, these values have to imply the correct propagation of constraints. In general, after fixing some reduction to a purely evolutionary system, this is asserting by computing by hand what is called the constraint subsidiary system, namely a system which is satisfied by the constraints quantities when the fields satisfy the reduced evolution system. If the subsidiary system is also hyperbolic then for the initial data case the situation is clear: we need to impose the constraints on the initial data and then they will correctly propagate along evolution. For the boundary data, we need to impose the constraint for all incoming constraint modes. These must be done by fixing some of the otherwise free boundary data, that is the incoming modes. Thus, there must be a relation between some of the incoming modes of the evolution system and all the incoming modes of the constraint subsidiary system. Under certain conditions on the constraints, this relation is known and understood, but those conditions are very restrictive. In this talk, we shall review the known results and discuss what is known so far for the general case and what are the open questions that still remain. 

GCSW02 
1st October 2019 14:30 to 15:30 
Snorre Christiansen 
Compatible finite element spaces for metrics with curvature
I will present some new finite element spaces for metrics with integrable curvature. These were obtained in the framework of finite element systems, developed for constructing differential complexes with adequate gluing conditions between the cells of a mesh. The new spaces have a higher regularity than those of Regge calculus, for which the scalar curvature contains measures supported on lower dimensional simplices (Dirac deltas). This is joint work with Kaibo Hu.


GCSW02 
1st October 2019 16:00 to 17:00 
Second Chances
The formalism of discrete differential forms has been used very successfully in computational electrodynamics. It is based on the idea that only the observables (i.e., the electromagnetic field) should be discretised and that coordinates should not possess any relevance in the numerical method. This is reflected in the fact that Maxwell's theory can be written entirely in geometric terms using differential forms. Einstein's theory is entirely geometric as well and can also be written in terms of differential forms. In this talk I will describe an attempt to discretise Einstein's theory in a way similar to Maxwell's theory. I will describe the advantages and point out disadvantages. I will conclude with some remarks about more general discrete structures on manifolds.


GCSW02 
2nd October 2019 09:30 to 10:30 
Anil Hirani 
Discrete Vector Bundles with Connection and the First Chern Class
The use of differential forms in general relativity requires ingredients like the covariant exterior derivative and curvature. One potential approach to numerical relativity would require discretizations of these ingredients. I will describe a discrete combinatorial theory of vector bundles with connections. The main operator we develop is a discrete covariant exterior derivative that generalizes the coboundary operator and yields a discrete curvature and a discrete Bianchi identity. We test this theory by defining a discrete first Chern class, a topological invariant of vector bundles. This discrete theory is built by generalizing discrete exterior calculus (DEC) which is a discretization of exterior calculus on manifolds for realvalued differential forms. In the first part of the talk I will describe DEC and its applications to the HodgeLaplace problem and NavierStokes equations on surfaces, and then I will develop the discrete covariant exterior derivative and its implications. This is joint work with Daniel BerwickEvans and Mark Schubel.


GCSW02 
2nd October 2019 11:00 to 12:00 
Soeren Bartels 
Approximation of Harmonic Maps and Wave Maps
Partial differential equations with a nonlinear pointwise constraint defined by a manifold occur in a variety of applications: the magnetization of a ferromagnet can be described by a unit length vector field and the orientation of the rodlike molecules that constitute a liquid crystal is often modeled by a restricted vector field. Other applications arise in geometric modeling, nonlinear bending of solids, and quantum mechanics. Nodal finite element methods have to appropriately relax the pointwise constraint leading to a variational crime. Since exact solutions are typically nonunique and do not admit higher regularity properties, the correctness of discretizations has to be established by weaker means avoiding unrealistic conditions. The iterative solution of the nonlinear systems of equations can be based on linearizations of the constraint or by using appropriate constraintpreserving reformulations. The talk focusses on the approximation of harmonic maps and wave maps. The latter arise as a model problem in general relativity.


GCSW02 
2nd October 2019 13:30 to 14:30 
Helvi Witek 
New prospects in numerical relativity
Both observations and deeply theoretical considerations indicate that general relativity, our elegant standard model of gravity, requires modifications at high curvatures scales. Candidate theories of quantum gravity, in their lowenergy limit, typically predict couplings to additional fields or extensions that involve higher curvature terms. At the same time, the breakthrough discovery of gravitational waves has provided a new channel to probe gravity in its most extreme, strongfield regime. Modelling the expected gravitational radiation in these extensions of general relativity enables us to search for  or place novel observational bounds on  deviations from our standard model. In this talk I will give an overview of the recent progress on simulating binary collisions in these situations and address renewed mathematical challenges such as wellposedness of the underlying initial value formulation. 

GCSW02 
3rd October 2019 09:30 to 10:30 
Fernando Abalos 
On necessary and sufficient conditions for strong hyperbolicity in systems with differential constraints
In many physical applications, due to the presence of constraints, the number of equations in the partial differential equation systems is larger than the number of unknowns, thus the standard Kreiss conditions can not be directly applied to check whether the system admits a well posed initial value formulation. In this work we show necessary and sufficient conditions such that there exists a reduced set of equations, of the same dimensionality as the set of unknowns, which satisfy Kreiss conditions and so are well defined and properly behaved evolution equations. We do that by decomposing the systems using the Kronecker decomposition of matrix pencils and, once the conditions are met, we look for specific families of reductions. We show the power of the theory in the densitized, pseudodifferential ADM equations.


GCSW02 
3rd October 2019 11:00 to 12:00 
David Hilditch 
Putting Infinity on the Grid
I will talk about an ongoing research program relying on a dual frame approach to treat numerically the field equations of GR (in generalized harmonic gauge) on compactified hyperboloidal slices. These slices terminate at futurenull infinity, and the hope is to eventually extract gravitational waves from simulations there. The main obstacle to their use is the presence of 'infinities' coming from the compactified coordinates, which have to somehow interact well with the assumption of asymptotic flatness so that we may arrive at regular equations for regular unknowns. I will present a new 'subtract the logs' regularization strategy for a toy nonlinear wave equation that achieves this goal.


GCSW02 
3rd October 2019 13:30 to 14:30 
Warner Miller 
General Relativity: One Block at a Time
This talk will provide an overview and motivation for Regge calculus (RC). We will highlight our insights into unique features of building GR on a discrete geometry in regards to structure preservation, and highlight some relative strengths and weaknesses of RC. We will review some numerical applications of RC, including our more recent work on discrete Ricci flow.


GCSW02 
3rd October 2019 14:30 to 15:30 
Ragnar Winther 
Finite element exterior calculus as a tool for compatible discretizations
The purpose of this talk is to review the basic results of finite element exterior calculus (FEEC) and to illustrate how the set up gives rise to to compatible discretizations of various problems. In particular, we will recall how FEEC, combined with the BernsteinGelfandGelfand framework, gave new insight into the construction of stable schemes for elasticity methods based on the HellingerReissner variational principle. 

GCSW02 
3rd October 2019 16:00 to 17:00 
Douglas Arnold 
FEEC 4 GR?
The finite element exterior calculus (FEEC) has proven to be a powerful tool for the design and understanding of numerical methods for solving PDEs from many branches of physics: solid mechanics, fluid flow, electromagnetics, etc. Based on preserving crucial geometric and topological structures underlying the equations, it is a prime example of a structurepreserving numerical method. It has organized many known finite element methods resulting in the periodic table of finite elements. For elasticity, which is not covered by the table, it led to new methods with long soughtafter properties. Might the FEEC approach lead to better numerical solutions of the Einstein equations as well? This talk will explore this question through two examples: the EinsteinBianchi formulation of the Einstein equations based on Bel decomposition of the Weyl tensor, and the Regge elements, a family of finite elements inspired by Regge calculus. Our goal in the talk is to raise questions and inspire future work; we do not purport to provide anything near definitive answers. 

GCSW02 
4th October 2019 11:00 to 12:00 
Pablo Laguna 
Inside the Final Black Hole from Black Hole Collisions
Modeling black hole singularities as punctures in spacetime is common in binary black hole simulations. As the punctures approach each other, a common apparent horizon forms, signaling the coalescence of the black holes and the formation of the final black hole. I will present results from a study that investigates the fate of the punctures and in particular the dynamics of the trapped surfaces on each puncture. Coauthors: Christopher Evans, Deborah Ferguson, Bhavesh Khamesra and Deirdre Shoemaker 

GCSW02 
4th October 2019 13:30 to 14:30 
Charalampos Markakis 
On numerical conservation of the PoincaréCartan integral invariant in relativistic fluid dynamics
The motion of strongly gravitating fluid bodies is described by the EulerEinstein system of partial differential equations. We report progress on formulating wellposed, acoustical and canonical hydrodynamic schemes, suitable for binary inspiral simulations and gravitationalwave source modelling. The schemes use a variational principle by CarterLichnerowicz stating that barotropic fluid motions are conformally geodesic, a corollary to Kelvin's theorem stating that initially irrotational flows remain irrotational, and Christodoulou's acoustic metric approach adopted to numerical relativity, in order to evolve the canonical momentum of a fluid element via Hamilton or HamiltonJacobi equations. These mathematical theorems leave their imprints on inspiral waveforms from binary neutron stars observed by the LIGOVirgo detectors. We describe a constraint damping scheme for preserving circulation in numerical general relativity, in accordance with Helmholtz's third theorem.


GCSW02 
4th October 2019 14:30 to 15:30 
Luis Lehner  tba  
GCSW02 
4th October 2019 16:00 to 17:00 
David Garfinkle 
Tetrad methods in numerical relativity
Most numerical relativity simulations use the usual coordinate methods to put the Einstein field equations in the form of partial differential equations (PDE), which are then handled using more or less standard numerical PDE methods, such as finite differences. However, there are some advantages to instead using a tetrad (orthonormal) basis rather than the usual coordinate basis. I will present the tetrad method and its numerical uses, particularly for simulating the approach to a spacetime singularity. I will end with open questions about which tetrad systems are suitable for numerical simulations.


GCS 
9th October 2019 14:05 to 14:50 
Richard Falk 
Numerical Computation of Hausdorff Dimension
We
show how finite element approximation theory can be combined with theoretical results about the properties of the eigenvectors of a class of linear PerronFrobenius operators to obtain accurate approximations of the Hausdorff dimension of some invariant sets arising from iterated function systems. The theory produces rigorous upper and lower bounds on the Hausdorff dimension. Applications to the computation of the Hausdorff dimension of some Cantor sets arising from real and complex continued fraction expansions are described. 

GCS 
9th October 2019 15:05 to 15:50 
Daniele Boffi 
Approximation of eigenvalue problems arising from partial differential equations: examples and counterexamples
We discuss the finite element approximation of
eigenvalue problems arising from elliptic partial differential equations. We present various examples of nonstandard schemes, including mixed finite elements, approximation of operators related to the leastsquares finite element method, parameter dependent formulations such as those produced by the virtual element method. Each example is studied theoretically; advantages and disadvantages of each approach are pointed out. 

GCS 
21st October 2019 16:00 to 17:00 
Donatella Marini 
Kirk Lecture: A recent technology for Scientific Computing: the Virtual Element Method
The Virtual Element Method (VEM) is a recent technology for the numerical solution of boundary value problems for Partial Differential Equations. It could be seen as a generalization of the Finite Element Method (FEM). With FEM the computational domain is typically split in triangles/quads (tetrahedra/hexahedra). VEM responds to the recent interest in using decompositions into polygons/polyhedra of very general shape, whenever more convenient for the approximation of problems of practical interest. Indeed,the possibility of using general polytopal meshes opens up a new range of opportunities in terms of accuracy, efficiency and flexibility. This is for instance reflected by the fact that various (commercial and free) codes recently included and keep developing polytopal meshes, showing in selected applications an improved computational efficiency with respect to tetrahedral or hexahedral grids. In this talk, after a general description of the use and potential of Scientific Computing, basic ideas of conforming VEM will be described on a simple model problem. Numerical results on more general problems in two and three dimension will be shown. Hints on Serendipity versions will be given at the end. These procedures allow to decrease significantly the number of degrees of freedom, that is, to reduce the dimension of the final linear system.


GCS 
22nd October 2019 09:05 to 09:50 
Franco Brezzi 
Serendipity Virtual Elements
Aftera brief reminder of classical ("plain vanilla") Virtual Elements we will see the generalphilosophy of "enhanced Virtual Elements" and the various types of Serendipity spaces as particular cases. The construction will always be conceptually simple (and extremely powerful, in particular for polygons with many edges), but a codeexploitingthe full advantage of havingmany edges might become difficult in the presence of non convex polygons, and in particular for complicated shapes. We shall also discuss differentchoices ensuring various advantages for different amounts of work. 

GCS 
22nd October 2019 09:55 to 10:40 
Andrea Cangiani 
A posteriori error estimation for discontinuous Galerkin methods on general meshes and adaptivity
The application and a priori error analysis of discontinuous Galerkin (dG) methods for general classes of PDEs under general mesh assumptions is by now well developed. dG methods naturally permits local mesh and order adaptivity. However, deriving robust error estimators allowing for curved/degenerating mesh interfaces as well as developing adaptive algorithms able to exploit such flexibility is nontrivial. In this talk we present recent work on a posteriori error estimates for dG methods of interior penalty type which hold on general mesh settings, including elements with degenerating and curved boundaries. The exploitation of general meshes within mesh adaptation algorithms applied to a few challenging problems will also be discussed.


GCS 
22nd October 2019 11:10 to 11:55 
Emmanuil Georgoulis 
Discontinuous Galerkin methods on arbitrarily shaped elements.
We
extend the applicability of the popular interiorpenalty discontinuous Galerkin
(dG) method discretizing advectiondiffusionreaction problems to meshes
comprising extremely general, essentially arbitrarilyshaped element shapes. In
particular, our analysis allows for curved element shapes, without the use of
(iso)parametric elemental maps. The feasibility of the method relies on the
definition of a suitable choice of the discontinuitypenalization parameter,
which turns out to be essentially independent on the particular element shape.
A priori error bounds for the resulting method are given under very mild
structural assumptions restricting the magnitude of the local curvature of
element boundaries. Numerical experiments are also presented, indicating the
practicality of the proposed approach. Moreover, we shall discuss a number of
perspectives on the possible applications of the proposed framework in
parabolic problems on moving domains as well as on multiscale problems. The
above is an overview of results from joint works with A. Cangiani (Nottingham,
UK), Z. Dong (FORTH, Greece / Cardiff UK) and T. Kappas (Leicester, UK). 

GCS 
22nd October 2019 13:05 to 13:50 
Paul Houston 
An AgglomerationBased, Massively Parallel NonOverlapping Additive Schwarz Preconditioner for HighOrder Discontinuous Galerkin Methods on Polytopic Grids
In
this talk we design and analyze a class of twolevel nonoverlapping additive Schwarz preconditioners for the solution of the linear
system of equations stemming from highorder/hp version discontinuous Galerkin
discretizations of secondorder elliptic partial differential equations on
polytopic meshes. The preconditioner is based on a coarse space and a
nonoverlapping partition of the computational domain where local solvers are
applied in parallel. In particular, the coarse space can potentially be chosen
to be nonembedded with respect to the finer space; indeed it can be obtained
from the fine grid by employing agglomeration and edge coarsening techniques.
We investigate the dependence of the condition number of the preconditioned
system with respect to the diffusion coefficient and the discretization parameters, i.e., the mesh size and the polynomial degree of the fine and coarse spaces. Numerical examples are presented which confirm the theoretical bounds. 

GCS 
22nd October 2019 13:55 to 14:40 
Jinchao Xu 
UPWIND FINITE ELEMENT METHODS FOR H(grad), H(curl) AND H(div) CONVECTIONDIFFUSION PROBLEMS
This talk is devoted to the construction and analysis of the finite element approximations for the H(grad), H(curl) and H(div) convectiondiffusion problems. An essential feature of these constructions is to properly average the PDE coefficients on subsimplexes from the underlying simplicial finite element meshes. The schemes are of the class of exponential fitting methods that result in special upwind schemes when the diffusion coefficient approaches to zero. Their wellposedness are established for sufficiently small mesh size assuming that the convectiondiffusion problems are uniquely solvable. Convergence of first order is derived under minimal smoothness of the solution. Some numerical examples are given to demonstrate the robustness and effectiveness for general convectiondiffusion problems. This is a joint work with Shounan Wu.


GCS 
28th October 2019 10:00 to 10:45 
Erwan Faou 
Highorder splitting for the VlasovPoisson equation
We consider the Vlasov{Poisson equation in a Hamiltonian framework and derive time splitting methods based on the decomposition of the Hamiltonian functional between the kinetic and electric energy. We also apply a similar strategy to the Vlasov{ Maxwell system. These are joint works with N. Crouseilles, F. Casas, M. Mehrenberger and L. Einkemmer. 

GCS 
28th October 2019 10:45 to 11:30 
Katharina Kormann 
On structurepreserving particleincell methods for the VlasovMaxwell equations
Numerical schemes that preserve the structure of the kinetic equations can provide new insight into the long time behavior of fusion plasmas. An electromagnetic particleincell solver for the Vlasov{Maxwell equations that preserves at the discrete level the noncanonical Hamiltonian structure of the Vlasov{Maxwell equations has been presented in [1]. In this talk, the framework of this geometric particleincell method will be presented and extension to curvilinear coordinates will be discussed. Moreover, various options for the temporal discretizations will be proposed and compared. [1] M. Kraus, K. Kormann, P. J. Morrison, and E. Sonnendrucker. GEMPIC: geometric electromag netic particleincell methods. Journal of Plasma Physics, 83(4), 2017. 

GCS 
28th October 2019 12:00 to 12:45 
Ernst Hairer 
Numerical treatment of charged particle dynamics in a magnetic field
Combining the Lorentz force equations with Newton's law gives a secondorder dierential equation in space for the motion of a charged particle in a magneticeld. The most natural and widely used numerical discretization is the Boris algorithm,which is explicit, symmetric, volumepreserving, and of order 2.In a rst part we discuss geometric properties (longtime behaviour, and in particularnear energy conservation) of the Boris algorithm. This is achieved by applying standardbackward error analysis. Near energy conservation can be obtained also in situations,where the method is not symplectic.In a second part we consider the motion of a charged particle in a strong magnetic eld.Backward error analysis can no longer be applied, and the accuracy (order 2) breaksdown. To improve accuracy we modify the Boris algorithm in the spirit of exponentialintegrators. Theoretical estimates are obtained with the help of modulated Fourierexpansions of the exact and numerical solutions.This talk is based on joint work with Christian Lubich, and Bin Wang.Related publications (2017{2019) can be downloaded from


GCS 
28th October 2019 14:00 to 14:45 
Wayne Arter 
Challenges for modelling fusion plasmas
Modelling fusion plasmas presents many challenges, so that it is reasonablethat many modelling codes still use simple nite dierence representations that makeit relatively easy to explore new physical processes and preserve numerical stability [1].However, October 2019 announcements by UK government have given UKAEA thechallenge of designing a nuclear fusion reactor in the next 5 years, plus a funding elementfor upgrading existing software both for Exascale and to meet the design challenge. Oneoption under examination is the use of high order 'spectrally accurate' elements.The biggest modelling problem is still that of turbulence mostly at relatively low plasmacollisionality. Specically nondissipative issues are the tracking of plasma particle orbitsbetween collisions, sometimes reducing to tracing lines of divergencefree magneticeld. These particles then build into a Maxwell{Vlasov solver, for which many dierentnumerical representations, exploiting low collisonality and the presence of a strong,directed magnetic eld have been explored [2]. One such is ideal MHD, where I haveexplored the use of the Lie derivative [3, 4]. Some further speculations as to the likelyrole of Lie (and spectral accuracy) in solving Vlasov{Maxwell, its approximations andtheir ensembles, and interactions between the dierent approximations, in the Exascaleera will be presented.This work was funded by the RCUK Energy Programme and the European Communities under thecontract of Association between EURATOM and CCFE.[1] B.D. Dudson, A. Allen, G. Breyiannis, E. Brugger, J. Buchanan, L. Easy, S. Farley, I. Joseph,M. Kim, A.D. McGann, et al. BOUT++: Recent and current developments. Journal of PlasmaPhysics, 81(01):365810104, 2015.[2] W. Arter. Numerical simulation of magnetic fusion plasmas. Reports on Progress in Physics,58:1{59, 1995.[3] W. Arter. Potential Vorticity Formulation of Compressible Magwnetohydrodynamics. PhysicalReview Letters, 110(1):015004, 2013.[4] W. Arter. Beyond Linear Fields: the Lie{Taylor Expansion. Proc Roy Soc A, 473:20160525, 2017.


GCS 
28th October 2019 15:15 to 16:00 
Jitse Niesen 
Spectral deferred correction in particleincell methods
Particleincell methods solve the Maxwell equations for the electromagneticeld in combination with the equation of motion for the charged particles in aplasma. The motion of charegd particles is usually computed using the Boris algorithm,a variant of Stormer{Verlet for Lorentz force omputations, which has impressiveperformance and order two (like Stormer{Verlet). Spectral deferred correction is aniterative time stepping method based on collocation, which in each time step performsmultiple sweeps of a loworder method (here, the Boris method) in order to obtain ahighorder approximation. This talk describes the ongoing eorts of Kristoer Smedt,Daniel Ruprecht, Steve Tobias and the speaker to embed a spectral deferred correctiontime stepper based on the Boris method in a particleincell method.


GCS 
30th October 2019 14:05 to 15:05 
Jinchao Xu 
Deep Neural Networks and Multigrid Methods
In this talk, I will first give an introduction to
several models and algorithms from two different fields: (1) machine learning,
including logistic regression, support vector machine and deep neural networks,
and (2) numerical PDEs, including finite element and multigrid methods. I will then explore mathematical
relationships between these models and algorithms and demonstrate how such
relationships can be used to understand, study and improve the model
structures, mathematical properties and relevant training algorithms for deep
neural networks. In particular, I will demonstrate how a new convolutional
neural network known as MgNet, can be derived by making very minor
modifications of a classic geometric multigrid method for the Poisson equation
and then explore the theoretical and practical potentials of MgNet.


GCS 
31st October 2019 16:00 to 17:00 
Patrick Farrell 
A Reynoldsrobust preconditioner for the 3D stationary NavierStokes equations
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 NavierStokes has
proven challenging: LU factorisation is Reynoldsrobust 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 ideas of Schöberl, Xu, Zikatanov, 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 Reynoldsrobustness. The scheme
combines augmented Lagrangian stabilisation to control the Schur complement,
the convection stabilisation proposed by Douglas & Dupont, a divergencecapturing
additive Schwarz relaxation method on each level, and a specialised
prolongation operator involving nonoverlapping local Stokes solves. The
properties of the preconditioner are tailored to the divergencefree
CG(k)DG(k1) discretisation and the appropriate relaxation is derived from
considerations of finite element exterior calculus.
We present 3D simulations with over one billion degrees
of freedom with robust performance from Reynolds numbers 10 to 5000.


GCS 
11th November 2019 14:00 to 15:00 
Shi Jin 
Random Batch Methods for Interacting Particle Systems and Consensusbased Global Nonconvex Optimization in Highdimensional Machine Learning (copy)
We develop random batch methods for 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. Then, we apply these methods to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian motion from random matrix theory, Thomson's problem, distribution of wealth, opinion dynamics and clustering. Numerical results show that the methods can capture both the transient solutions and the global equilibrium in these problems. We also apply this method and improve the consensusbased global optimization algorithm for high dimensional machine learning problems. This method does not require taking gradient in finding global minima for nonconvex functions in high dimensions. 

GCS 
11th November 2019 16:00 to 17:00 
Sergio Blanes 
Magnus, splitting and composition techniques for solving nonlinear Schrödinger equations
In this talk I will consider several nonautonomous nonlinear Schrödinger equations (the GrossPitaevskii equation, the KohnSham equation and an Quantum Optimal Control equation) and some of the numerical methods that have been used to solve them.With a proper linearization of these equations we end up with nonautonomous linear systems where many of the algebraic techniques from Magnus, splitting and composition algorithms can be used. This will be an introductory talk to stimulate some collaboration between participants of the program at the INI.


GCS 
12th November 2019 16:00 to 17:00 
Erwan Faou 
Some results in the long time analysis of Hamiltonian PDEs and their numerical approximations
I will review some results concerning the long time behavior of Hamiltonian PDEs, and address similar questions for their numerical approximation. I will show numerical resonances can appear both in space and time. I will also discuss the long time stability of solitary waves evolving on a discret set of lattice points. 

GCS 
13th November 2019 14:00 to 15:00 
Caroline Lasser 
What it takes to catch a wave packet
Wave packets describe the quantum vibrations of a molecule. They are highly oscillatory, highly localized and move in high dimensional configuration spaces. The talk addresses three meshless numerical methods for catching them: single Gaussian beams, superpositions of them, and the socalled linearized initial value representation. 

GCS 
13th November 2019 16:00 to 17:00 
Alexander Ostermann 
Lowregularity time integrators
Nonlinear Schrödinger equations are usually solved by pseudospectral methods, where the time integration is performed by splitting schemes or exponential integrators. Notwithstanding the benefits of this approach, its successful application requires additional regularity of the solution. For instance, secondorder Strang splitting requires four additional derivatives for the solution of the cubic nonlinear Schrödinger equation. Similar statements can be made about other dispersive equations like the Kortewegde Vries or the Boussinesq equation. In this talk, we introduce lowregularity Fourier integrators as an alternative. They are obtained from Duhamel's formula in the following way: first, a Lawsontype transformation eliminates the leading linear term and second, the dominant nonlinear terms are integrated exactly in Fourier space. For cubic nonlinear Schrödinger equations, firstorder convergence of such methods only requires the boundedness of one additional derivative of the solution, and secondorder convergence the boundedness of two derivatives. Similar improvements can also be obtained for other dispersive problems. This is joint work with Frédéric Rousset (Université ParisSud), Katharina Schratz (HariotWatt, UK), and Chunmei Su (Technical University of Munich). 

GCS 
20th November 2019 13:05 to 13:45 
CANCELLED  
GCS 
20th November 2019 13:50 to 14:30 
Tony Lelievre 
title tba
Various applications require the sampling of probability measures
restricted to submanifolds defined as the level set of some functions, in
particular in computational statistical physics. We will present recent results
on socalled Hybrid Monte Carlo methods, which consists in adding an extra
momentum variable to the state of the system, and discretizing the associated
Hamiltonian dynamics with some stochastic perturbation in the extra variable.
In order to avoid biases in the invariant probability measures sampled by
discretizations of these stochastically perturbed Hamiltonian dynamics, a
Metropolis rejection procedure can be considered. The soobtained scheme
belongs to the class of generalized Hybrid Monte Carlo (GHMC) algorithms, and
we will discuss how to ensure that the sampling method is unbiased in practice. References:  T. Lelièvre, M. Rousset and G. Stoltz, Langevin dynamics with constraints and computation of free energy differences, Mathematics of Computation, 81(280), 2012.  T. Lelièvre, M. Rousset and G. Stoltz, Hybrid Monte Carlo methods for sampling probability measures on submanifolds, to appear in Numerische Mathematik, 2019.  E. Zappa, M. HolmesCerfon, and J. Goodman. Monte Carlo on manifolds: sampling densities and integrating functions. Communications in Pure and Applied Mathematics, 71(12), 2018. 

GCS 
20th November 2019 15:05 to 15:35 
Jonathan Goodman 
Step size control for Newton type MCMC samplers Jonathan Goodman
ABSTRACT: MCMC sampling
can use ideas from the optimization community. Optimization via Newton’s
method can fail without line search, even for smooth strictly convex problems.
Affine invariant Newton based MCMC sampling uses a Gaussian proposal
based on a quadratic model of the potential using the local gradient and
Hessian. This can fail (conjecture: give a transient Markov chain) even
for smooth strictly convex potentials. We describe a criterion that
allows a sequence of proposal distributions from X_n with decreasing “step
sizes” until (with probability 1) a proposal is accepted. “Very detailed
balance” allows the whole process to preserve the target distribution.
The method works in experiments but the theory is missing.


GCS 
20th November 2019 15:40 to 16:10 
Miranda HolmesCefron 
A Monte Carlo method to sample a Stratification
Many problems in materials science and biology involve particles
interacting with strong, shortranged bonds, that can break and form on
experimental timescales. Treating such bonds as constraints can significantly
speed up sampling their equilibrium distribution, and there are several methods
to sample subject to fixed constraints. We introduce a Monte Carlo method to handle
the case when constraints can break and form. Abstractly, the method samples a
probability distribution on a stratification: a collection of
manifolds of different dimensions, where the lowerdimensional manifolds lie on
the boundaries of the higherdimensional manifolds. We show several
applications in polymer physics, selfassembly of colloids, and volume
calculation. 

GCS 
21st November 2019 13:05 to 13:45 
Alessandro Barp  Hamiltonian Monte Carlo on Homogeneous Manifolds for QCD and Statistics.  
GCS 
21st November 2019 13:50 to 14:30 
Benedict Leimkuhler 
Some thoughts about constrained sampling algorithms
I will survey our work on algorithms for
sampling diffusions on manifolds, including isokinetic methods and constrained Langevin
dynamics methods. These have mostly been introduced and tested in the
setting of molecular dynamics. It is interesting to consider possible
uses of these ideas in other types of sampling computations, like neural
network parameterization and training of generative models. 

GCS 
21st November 2019 15:05 to 15:45 
Elena Celledoni 
Deep learning as optimal control problems and Riemannian discrete gradient descent.
We consider
recent work where deep learning neural networks have been interpreted as
discretisations of an optimal control problem subject to an ordinary
differential equation constraint. We review the first order conditions for
optimality, and the conditions ensuring optimality after discretisation. This
leads to a class of algorithms for solving the discrete optimal control problem
which guarantee that the corresponding discrete necessary conditions for
optimality are fulfilled. The differential equation setting lends itself to
learning additional parameters such as the time discretisation. We explore this
extension alongside natural constraints (e.g. time steps lie in a simplex). We
compare these deep learning algorithms numerically in terms of induced flow and
generalisation ability.
References
 M Benning, E Celledoni, MJ Ehrhardt, B Owren, CB Schönlieb, Deep learning as optimal control problems: models and numerical methods, JCD.


GCS 
21st November 2019 16:00 to 17:00 
Peter Clarkson 
Symmetric Orthogonal Polynomials
In this talk I will discuss symmetric orthogonal polynomials on the real line. Such polynomials give rise to orthogonal systems which have important applications in spectral methods, with several important advantages if their differentiation matrix is skewsymmetric and highly structured. Such orthogonal systems, where the differentiation matrix is skewsymmetric, tridiagonal and irreducible, have recently been studied by Iserles and Webb. The symmetric orthogonal polynomials studied will include generalisations of the classical Hermite weight and generalisations of the Freud weight.


GCS 
26th November 2019 14:05 to 14:50 
Evelyne Hubert 
Symmetry Preserving Interpolation
In
this talk I choose to present the PhD work of Erick Rodriguez Bazan. We address
multivariate interpolation in the presence of symmetry as given by a finite
group. Interpolation is a prime tool in algebraic computation while symmetry is
a qualitative feature that can be more relevant to a mathematical model than the numerical accuracy of the parameters. Beside its preservation, symmetry shall also be exploited to alleviate the computational cost. We revisit minimal degree and least interpolation spaces [de Boor & Ron 1990] with symmetry adapted bases (rather than the usual monomial bases). In these bases, the multivariate Vandermonde matrix (a.k.a colocation matrix) is block diagonal as soon as the set of nodes is invariant. These blocks capture the inherent redundancy in the computations. Furthermore any equivariance an interpolation problem might have will be automatically preserved : the output interpolant will have the same equivariance property. The special case of multivariate Hermite interpolation leads us to question the representation of polynomial ideals. Gröbner bases, the preferred tool for algebraic computations, breaks any kind of symmetry. The prior notion of HBases, introduced by Macaulay, appears as more suitable. Reference: https://dl.acm.org/citation.cfm?doid=3326229.3326247 https://hal.inria.fr/hal01994016 Joint work with Erick Rodriguez Bazan 

GCS 
26th November 2019 15:05 to 16:05 
Anders Hansen 
On the Solvability Complexity Index (SCI) hierarchy  Establishing the foundations of computational mathematics
There
are four areas in computational mathematics that have been intensely
investigated over more than half a century: Spectral problems, PDEs,
optimisation and inverse problems. However, despite the matureness of these
fields, the foundations are far from known. Indeed, despite almost 90 years of
quantum mechanics, it is still unknown whether it is possible to compute the
spectrum of a selfadjoint Schrodinger operator with a bounded smooth
potential. Similarly, it is not known which time dependent Schrodinger
equations can be computed (despite well posedness of the equation). Linear
programs (LP) can be solved with rational inputs in polynomial time, but can
LPs be solved with irrational inputs? Problems in signal and image processing
tend to use irrational numbers, so what happens if one plugs in the discrete
cosine transform in one's favourite LP solver? Moreover, can one always compute
the solutions to wellconditioned infinitedimensional inverse problems, and if
not, which inverse problems can then be solved?
In
this talk we will discuss solutions to many of the questions above, and some of
the results may seem paradoxical. Indeed, despite being an open problem for
more than half a century, computing spectra of Schrodinger operators with
a bounded potential is not harder than computing spectra of infinite diagonal
matrices, the simplest of all infinitedimensional spectral problems. Moreover,
computing spectra of compact operators, for which the method has been known for
decades, is strictly harder than computing spectra of such Schrodinger
operators. Regarding linear programs (and basis pursuit, semidefinite programs
and LASSO) we have the following. For any integer K > 2 and any norm, there
exists a family of well conditioned inputs containing irrational numbers so
that no algorithm can compute K correct digits of a minimiser, however, there
exists an algorithm that can compute K1 correct digits. But any algorithm
producing K1 correct digits will need arbitrarily long time. Finally,
computing K2 correct digits can be done in polynomial time in the number of
variables. As we will see, all of these problems can be solved via the the
Solvability Complexity Index (SCI) hierarchy, which is a theoretical program
for establishing the boundaries of what computers can achieve in the
sciences.


GCS 
28th November 2019 16:00 to 17:00 
Elizabeth Mansfield 
On the nature of mathematical joy
Elizabeth Mansfield will discuss seven levels of mathematical joy based on her mathematical travels. This is a talk for a general audience.


GCS 
4th December 2019 14:05 to 14:50 
Balázs Kovács 
Energy estimates: proving stability for evolving surface PDEs and geometric flows
In this talk we will give some details on the main steps
and ideas behind energy estimates used to prove stability of backward
difference
semi and full discretisations of parabolic evolving
surface problems, or geometric flows (e.g. mean curvature flow).
We will give details on how the Gstability result of
Dahlquist and the multiplier techniques of Nevanlinna and Odeh will be used.


GCS 
5th December 2019 13:30 to 14:15 
Vanessa Styles 
Numerical approximations of a tractable mathematical model for tissue growth
We consider a free boundary problem representing
one of the simplest mathematical descriptions of the growth and death of a
tumour. The mathematical model takes the form of a closed interface evolving
via forced mean curvature flow where the forcing depends on the solution of a
PDE that holds in the domain enclosed by the interface. We derive sharp
interface and diffuse interface finite element approximations of this model and
present some numerical results 

GCS 
5th December 2019 14:15 to 15:00 
Bjorn Stinner 
Phase field modelling of free boundary problems
Diffuse interface models based on the phase
field methodology have been developed and investigated in various applications
such as solidification processes, tumour growth, or multiphase flow. The
interfaces are represented by thin layers, across which quantities rapidly but
smoothly change their values. These interfacial layers are described in terms
of order parameters, the equations for which can be solved using relatively
straightforward methods, such as finite elements with adaptive mesh refinement,
as no tracking of any interface is required. The interface motion is usually
coupled to other fields and equations adjacent or on the interface, for
instance, diffusion equations in alloys or the momentum equation in fluid flow.
We discuss how such systems can be incorporated into phase field models in a
generic way. Furthermore, we present a computational framework where specific
models can be implemented and later on conveniently amended, if desired, in a
highlevel language, and which then bind to efficient software backends. A
couple of code listings and numerical simulations serve to illustrate the
approach 

GCS 
5th December 2019 15:15 to 16:00 
Bertram Düring 
Structurepreserving variational schemes for nonlinear partial differential equations with a Wasserstein gradient flow structure
A wide range of diffusion equations can be interpreted as gradient
flow with respect to Wasserstein distance of an energy functional. Examples
include the heat equation, the porous medium equation, and the fourthorder
DerridaLebowitzSpeerSpohn equation. When it comes to solving equations of
gradient flow type numerically, schemes that respect the equation's special
structure are of particular interest. The gradient flow structure gives rise to
a variational scheme by means of the minimising movement scheme (also called
JKO scheme, after the seminal work of Jordan, Kinderlehrer and Otto) which
constitutes a timediscrete minimization problem for the energy.
While the scheme has been used originally for
analytical aspects, more recently a number of authors have explored the
numerical potential of this scheme. In this talk we present some results on
Lagrangian schemes for Wasserstein gradient flows in one spatial dimension and
then discuss extensions to higher approximation order and to higher spatial
dimensions 

GCS 
6th December 2019 16:00 to 17:00 
Chus SanzSerna 
Rothschild Lecture: Hamiltonian Monte Carlo and geometric integration
Many application fields require samples from an arbitrary probability distribution. Hamiltonian Monte Carlo is a sampling algorithm that originated in the physics literature and has later gained much popularity among statisticians. This is a talk addressed to a general audience, where I will describe the algorithm and some of its applications. The exposition requires basic ideas from different fields, from statistical physics to geometric integration of differential equations and from Bayesian statistics to Hamiltonian dynamics and I will provide the necessary background, albeit superficially. 

GCS 
11th December 2019 14:05 to 14:50 
Antonella Zanna 
On the construction of some symplectic Pstable additive Runge—Kutta methods
Symplectic partitioned Runge–Kutta methods can be
obtained from a variational formulation treating all the terms in the
Lagrangian with the same quadrature formula. We construct a family of
symplectic methods allowing the use of different quadrature formula for
different parts of the Lagrangian. In particular, we study a family of methods
using Lobatto quadrature (with corresponding Lobatto IIIAIIIB symplectic
method) and Gauss–Legendre quadrature combined in an appropriate way. The
resulting methods are similar to additive RungeKutta methods. The IMEX method,
using the Verlet and IMR combination is a particular case of this family.
The methods have the same favourable implicitness as the
underlying Lobatto IIIAIIIB pair. Differently from the Lobatto IIIAIIIB,
which are known not to be Pstable, we show that the new methods satisfy the
requirements for Pstability.


GCS 
11th December 2019 15:05 to 15:50 
Brynjulf Owren 
Equivariance and structure preservation in numerical methods; some cases and viewpoints
Our point of departure is the situation when there is a group of transformations acting both on our problem space and on the space in which our computations are produced. Equivariance happens when the map from the problem space to the computation space, i.e. our numerical method, commutes with the group action. This is a rather general and vague definition, but we shall make it precise and consider a few concrete examples in the talk. In some cases, the equivariance property is natural, in other cases it is something that we want to impose in the numerical method in order to obtain computational schemes with certain desired structure preserving qualities. Many of the examples we present will be related to the numerical solution of differential equations and we may also present some recent examples from artificial neural networks and discrete integrable systems. This is work in progress and it summarises some of the ideas the speaker has been discussing with other participants this autumn. 