Geometry, compatibility and structure preservation in computational differential equations

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Programme theme

Computations of differential equations are a fundamental activity in applied mathematics. While historically the main quest was to derive all-purpose algorithms such as finite difference, finite volume and finite element methods for space discretization, Runge–Kutta and linear multistep methods for time integration, in the last 25 years the focus has shifted to special classes of differential equations and purpose-built algorithms that are tailored to preserve special features of each class. This has given rise to the new fields of geometric numerical integration and of structure preserving discretization. In addition to being quantitatively accurate, these novel methods have the advantage of also being qualitatively accurate as they inherit the key structural properties of their continuum counterparts. This has meant a large-scale introduction of geometric and topological thinking into modern numerical mathematics.

During this scientific programme at the Isaac Newton Institute for Mathematical Sciences, we will address fundamental questions in the field of structure preserving discretizations of differential equations on manifolds in space and time. We will bring together two communities that have been pursuing their science along parallel tracks to endeavour breakthroughs in some major scientific applications, which call for advanced numerical simulation techniques. This will lead to the development of a new generation of space-time discretizations for evolutionary equations.

During the programme we intend to organise three workshops and two focused study periods lasting two weeks on selected application areas.

The core themes of the programme are:

Compatible discretizations.
Geometric numerical integration.
Structure preservation and numerical relativity.
Applications to computations in quantum mechanics.

Click here to download the programme's final scientific report

GCS programme: interview with organisers Arieh Iserles, Elena Celledoni and Elizabeth Mansfield

Preprints

Title	Year	Programme
Conservative Galerkin methods for dispersive Hamiltonian problems Authors: James Jackaman, Tristan Pryer Download PDF	2019	GCS
Discrete conservation laws for finite element discretisations of multisymplectic PDEs Authors: James Jackaman, Elena Celledoni Download PDF	2019	GCS
A surface nite element method for computational modelling of cell blebbing Authors: Andreas Dedner, Bjorn Stinner, Adam Nixon Download PDF	2019	GCS
DISCRETE VARIATIONAL METHODS AND SYMPLECTIC GENERALIZED ADDITIVE RUNGE-KUTTA METHODS Authors: Antonella Zanna Download PDF	2019	GCS
SYMPLECTIC P-STABLE ADDITIVE RUNGE-KUTTA METHODS Authors: Antonella Zanna Download PDF	2019	GCS
Symmetry and invariant bases in finite element exterior calculus Authors: Martin Licht Download PDF	2019	GCS
ON BASIS CONSTRUCTIONS IN FINITE ELEMENT EXTERIOR CALCULUS Authors: Martin Licht Download PDF	2019	GCS
A converging nite element scheme for motion by curvature of a network with a triple junction Authors: Paola Pozzi, Bjorn Stinner Download PDF	2019	GCS
NEWEST VERTEX BISECTION OVER GENERAL TRIANGULATIONS Authors: Michael Holst, Zhao Lyu, Martin Licht Download PDF	2019	GCS
Numerical preservation of multiple local conservation laws Authors: Peter E. Hydon, Gianluca Frasca-Caccia Download PDF	2019	GCS
Locally conservative finite difference schemes for the modified KdV equation Authors: Gianluca Frasca-Caccia, Peter E. Hydon Download PDF	2019	GCS

Workshops

GCSW01

Seminars

Wednesday 17th July 2019
11:00 to 12:00	Christian Offen Massey University Detection of high codimensional bifurcations in variational PDEs	Room 1

Monday 22nd July 2019
14:00 to 15:00	Ralf Hiptmair ETH Zürich Exterior Shape Calculus	Room 2

Tuesday 23rd July 2019
14:00 to 15:00	Annalisa Buffa EPFL - Ecole Polytechnique Fédérale de Lausanne; Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI) Dual complexes and mortaring for regular approximations of electromagnetic fields	Room 1

Wednesday 24th July 2019
15:00 to 16:00	Onno Bokhove University of Leeds A new wave-to-wire wave-energy model: from variational principle to compatible space-time discretisation	Room 1

Tuesday 30th July 2019
14:00 to 15:00	Melvin Leok University of California, San Diego The Connections Between Discrete Geometric Mechanics, Information Geometry and Machine Learning	Room 2
15:00 to 16:00	Benjamin Tapley Norwegian University of Science and Technology Computational methods for simulating inertial particles in discrete incompressible flows.	Room 2

Wednesday 31st July 2019
15:00 to 16:00	Anthony Bloch University of Michigan Optimal control and the geometry of integrable systems	Room 2

Wednesday 7th August 2019
14:00 to 15:00	Arieh Iserles University of Cambridge Fast approximation on the real line	Room 2
15:00 to 16:00	Robert McLachlan Massey University The Lie algebra of classical mechanics	Room 2

Wednesday 14th August 2019
14:00 to 15:00	David Garfinkle Oakland University Numerical General Relativity	Room 2
15:00 to 16:00	Ari Stern Washington University in St. Louis Finite element methods for Hamiltonian PDEs	Room 2

Wednesday 21st August 2019
14:00 to 15:00	Fernando Casas Universitat Jaume I More on composition methods: error estimation and pseudo-symmetry	Room 2
15:00 to 16:00	Yajuan Sun Chinese Academy of Sciences Contact Hamiltonian system and its application in solving Vlasov-Poisson Fokker-Planck system	Room 2

Wednesday 28th August 2019
14:00 to 15:00	Gerard Awanou University of Illinois at Chicago Computational geometric optics: Monge-Ampere	Room 2
15:00 to 16:00	Milo Viviani Chalmers University of Technology Lie-Poisson methods for isospectral flows and their application to long-time simulation of spherical ideal hydrodynamics	Room 2

Wednesday 4th September 2019
14:00 to 15:00	Sina Ober-Blöbaum University of Oxford Variational formulations for dissipative systems	Room 2
15:00 to 16:00	Sigrid Leyendecker Friedrich-Alexander-Universität Erlangen-Nürnberg Mixed order and multirate variational integrators for the simulation of dynamics on different time scales	Room 2

Thursday 5th September 2019
14:15 to 15:15	Chus Sanz-Serna Universidad Carlos III de Madrid Numerical Integrators for the Hamiltonian Monte Carlo Method	Room 2

Wednesday 11th September 2019
14:00 to 15:00	Peter Hydon University of Kent Conservation laws and Euler operators	Room 2
15:00 to 16:00	Gianluca Frasca-Caccia University of Kent Numerical preservation of local conservation laws	Room 2

Wednesday 18th September 2019
14:05 to 14:50	Blanca Ayuso De Dios Università degli Studi di Milano - Bicocca; Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI) Constructing Discontinuous Galerkin methods for Vlasov-type systems	Room 2
15:05 to 15:50	Linyu Peng Waseda University Variational systems on the variational bicomplex	Room 2

Thursday 19th September 2019
16:00 to 17:00	Marcus Webb University of Manchester Energy preserving spectral methods on the real line whose analysis strays into the complex plane (copy)	Room 2

Wednesday 25th September 2019
14:05 to 14:50	Martin Licht UC San Diego Newest Results in Newest Vertex Bisection	Room 2

Wednesday 9th October 2019
14:05 to 14:50	Richard Falk Rutgers, The State University of New Jersey Numerical Computation of Hausdorff Dimension	Room 2
15:05 to 15:50	Daniele Boffi Università degli Studi di Pavia Approximation of eigenvalue problems arising from partial differential equations: examples and counterexamples	Room 2

Monday 21st October 2019
16:00 to 17:00	Donatella Marini Università degli Studi di Pavia; Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI) Kirk Lecture: A recent technology for Scientific Computing: the Virtual Element Method	Room 1

Tuesday 22nd October 2019
09:05 to 09:50	Franco Brezzi Institute for applied mathematics and information technologies (IMATI) Serendipity Virtual Elements	Room 1
09:55 to 10:40	Andrea Cangiani A posteriori error estimation for discontinuous Galerkin methods on general meshes and adaptivity	Room 1
11:10 to 11:55	Emmanuil Georgoulis University of Leicester; National Technical University of Athens Discontinuous Galerkin methods on arbitrarily shaped elements.	Room 1
13:05 to 13:50	Paul Houston University of Nottingham An Agglomeration-Based, Massively Parallel Non-Overlapping Additive Schwarz Preconditioner for High-Order Discontinuous Galerkin Methods on Polytopic Grids	Room 1
13:55 to 14:40	Jinchao Xu Pennsylvania State University UPWIND FINITE ELEMENT METHODS FOR H(grad), H(curl) AND H(div) CONVECTION-DIFFUSION PROBLEMS	Room 1

Monday 28th October 2019
10:00 to 10:45	Erwan Faou INRIA Rennes - Bretagne Atlantique High-order splitting for the Vlasov-Poisson equation	Room 2
10:45 to 11:30	Katharina Kormann Technische Universität München On structure-preserving particle-in-cell methods for the Vlasov-Maxwell equations	Room 2
12:00 to 12:45	Ernst Hairer Université de Genève Numerical treatment of charged particle dynamics in a magnetic field	Room 2
14:00 to 14:45	Wayne Arter Culham Centre for Fusion Energy Challenges for modelling fusion plasmas	Room 2
15:15 to 16:00	Jitse Niesen University of Leeds Spectral deferred correction in particle-in-cell methods	Room 2

Wednesday 30th October 2019
14:05 to 15:05	Jinchao Xu Pennsylvania State University Deep Neural Networks and Multigrid Methods	Room 2

Thursday 31st October 2019
16:00 to 17:00	Patrick Farrell University of Oxford A Reynolds-robust preconditioner for the 3D stationary Navier-Stokes equations	Room 2

Monday 11th November 2019
14:00 to 15:00	Shi Jin Shanghai Jiao Tong University Random Batch Methods for Interacting Particle Systems and Consensus-based Global Non-convex Optimization in High-dimensional Machine Learning (copy)	Room 2
16:00 to 17:00	Sergio Blanes Universidad Politécnica de Valencia Magnus, splitting and composition techniques for solving non-linear Schrödinger equations	Room 2

Tuesday 12th November 2019
16:00 to 17:00	Erwan Faou INRIA Rennes - Bretagne Atlantique Some results in the long time analysis of Hamiltonian PDEs and their numerical approximations	Room 2

Wednesday 13th November 2019
14:00 to 15:00	Caroline Lasser Technische Universität München What it takes to catch a wave packet	Room 2
16:00 to 17:00	Alexander Ostermann Leopold-Franzens Universtät Innsbruck Low-regularity time integrators	Room 2

Wednesday 20th November 2019
13:05 to 13:45	CANCELLED	Room 2
13:50 to 14:30	Tony Lelievre ENPC - École des Ponts ParisTech title tba	Room 2
15:05 to 15:35	Jonathan Goodman Step size control for Newton type MCMC samplers Jonathan Goodman	Room 2
15:40 to 16:10	Miranda Holmes-Cefron A Monte Carlo method to sample a Stratification	Room 2

Thursday 21st November 2019
13:05 to 13:45	Alessandro Barp Imperial College London Hamiltonian Monte Carlo on Homogeneous Manifolds for QCD and Statistics.	Room 2
13:50 to 14:30	Benedict Leimkuhler University of Edinburgh Some thoughts about constrained sampling algorithms	Room 2
15:05 to 15:45	Elena Celledoni Norwegian University of Science and Technology Deep learning as optimal control problems and Riemannian discrete gradient descent.	Room 2
16:00 to 17:00	Peter Clarkson University of Kent Symmetric Orthogonal Polynomials	Room 2

Tuesday 26th November 2019
14:05 to 14:50	Evelyne Hubert INRIA Sophia Antipolis Symmetry Preserving Interpolation	Room 1
15:05 to 16:05	Anders Hansen University of Cambridge On the Solvability Complexity Index (SCI) hierarchy - Establishing the foundations of computational mathematics	Room 1

Thursday 28th November 2019
16:00 to 17:00	Elizabeth Mansfield University of Kent On the nature of mathematical joy	Room 1

Wednesday 4th December 2019
14:05 to 14:50	Balázs Kovács Universität Tübingen Energy estimates: proving stability for evolving surface PDEs and geometric flows	Room 2

Thursday 5th December 2019
13:30 to 14:15	Vanessa Styles University of Sussex Numerical approximations of a tractable mathematical model for tissue growth	Room 2
14:15 to 15:00	Bjorn Stinner University of Warwick Phase field modelling of free boundary problems	Room 2
15:15 to 16:00	Bertram Düring University of Sussex Structure-preserving variational schemes for nonlinear partial differential equations with a Wasserstein gradient flow structure	Room 2

Friday 6th December 2019
16:00 to 17:00	Chus Sanz-Serna Universidad Carlos III de Madrid Rothschild Lecture: Hamiltonian Monte Carlo and geometric integration	Room 1

Wednesday 11th December 2019
14:05 to 14:50	Antonella Zanna Universitetet i Bergen On the construction of some symplectic P-stable additive Runge—Kutta methods	Room 2
15:05 to 15:50	Brynjulf Owren Norwegian University of Science and Technology Equivariance and structure preservation in numerical methods; some cases and viewpoints	Room 2

Geometry, compatibility and structure preservation in computational differential equations

Organisers

Douglas Arnold University of Minnesota
Franco Brezzi Institute for applied mathematics and information technologies (IMATI)
Elena Celledoni Norwegian University of Science and Technology
Arieh Iserles University of Cambridge
Elizabeth Mansfield University of Kent
Reinout Quispel La Trobe University

Participants

Douglas Arnold University of Minnesota
Wayne Arter Culham Centre for Fusion Energy
Gerard Awanou University of Illinois at Chicago
Blanca Ayuso De Dios Università degli Studi di Milano - Bicocca; Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI)
Mary Barker Washington University in St. Louis
Alessandro Barp Imperial College London
Soeren Bartels Albert-Ludwigs-Universität Freiburg
Lourenco Beirao da Veiga Università degli Studi di Milano - Bicocca
Paul Bergold Technische Universität München
Sergio Blanes Universidad Politécnica de Valencia
Anthony Bloch University of Michigan
Daniele Boffi Università degli Studi di Pavia
Geir Bogfjellmo Norwegian University of Science and Technology
Onno Bokhove University of Leeds
Franco Brezzi Institute for applied mathematics and information technologies (IMATI)
Maria Bruna University of Oxford
Olivier Brüls Université de Liège
Christopher Budd University of Bath
Annalisa Buffa EPFL - Ecole Polytechnique Fédérale de Lausanne; Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI)
Erik Burman University College London
Andrea Cangiani
Fernando Casas Universitat Jaume I
Elena Celledoni Norwegian University of Science and Technology
Quentin Chauleur Université de Rennes 1
Snorre Christiansen University of Oslo
Jan L. Cieśliński Uniwersytet w Bialymstoku
Colin Cotter Imperial College London
Charles Curry Norwegian University of Science and Technology
Andreas Dedner University of Warwick
Alan Demlow Texas A&M University
Bertram Düring University of Sussex
Kurusch Ebrahimi-Fard Norwegian University of Science and Technology
Solve Eidnes Norwegian University of Science and Technology
Charlie Elliott University of Warwick
Richard Falk Rutgers, The State University of New Jersey
Erwan Faou INRIA Rennes - Bretagne Atlantique
Patrick Farrell University of Oxford
Jason Frank Universiteit Utrecht
Gianluca Frasca-Caccia University of Kent
Daisuke Furihata Osaka University
David Garfinkle Oakland University
Evan Gawlik University of Hawaii
Emmanuil Georgoulis University of Leicester; National Technical University of Athens
Håkon Gimse Universitetet i Bergen
Volker Grimm Karlsruhe Institute of Technology (KIT)
Johnny Guzma´n Brown University
Ernst Hairer Université de Genève
Ralf Hiptmair ETH Zürich
Anil Hirani University of Illinois at Urbana-Champaign
Michael Holst University of California, San Diego
Jialin Hong Chinese Academy of Sciences
Paul Houston University of Nottingham
Kaibo Hu University of Minnesota
Evelyne Hubert INRIA Sophia Antipolis
Peter Hydon University of Kent
Arieh Iserles University of Cambridge
James Jackaman Memorial University of Newfoundland
Shi Jin Shanghai Jiao Tong University
Katharina Kormann Technische Universität München
Balázs Kovács Universität Tübingen
Lisa Maria Kreusser University of Cambridge
Karolina Kropielnicka University of Gdansk
Caroline Lasser Technische Universität München
Luis Lehner Perimeter Institute for Theoretical Physics
Benedict Leimkuhler University of Edinburgh
Tony Lelievre ENPC - École des Ponts ParisTech
Melvin Leok University of California, San Diego
Sigrid Leyendecker Friedrich-Alexander-Universität Erlangen-Nürnberg
Lu Li NTNU
Martin Licht UC San Diego; University of California, San Diego
Christian Lubich Eberhard Karls Universität Tübingen
Pål Erik Lystad NTNU
Elizabeth Mansfield University of Kent
Gloria Mari-Beffa University of Wisconsin-Madison
Donatella Marini Università degli Studi di Pavia; Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI)
Robert McLachlan Massey University
Hans Munthe-Kaas Universitetet i Bergen
Jitse Niesen University of Leeds
Nilima Nigam Simon Fraser University
Sina Ober-Blöbaum University of Oxford
Christian Offen Massey University
Alexander Ostermann Leopold-Franzens Universtät Innsbruck
Brynjulf Owren Norwegian University of Science and Technology
Linyu Peng Waseda University
Ilaria Perugia Universität Wien
Timothee Pouchon University of Edinburgh
Reinout Quispel La Trobe University
Francesca Rapetti Université de Nice Sophia Antipolis
Sebastian Reich Universität Potsdam; University of Reading
Marcel Rød NTNU
Giancarlo Sangalli Università degli Studi di Pavia
Chus Sanz-Serna Universidad Carlos III de Madrid
Alexander Schmeding Universitetet i Bergen
Katharina Schratz Karlsruhe Institute of Technology (KIT)
Carola-Bibiane Schönlieb University of Cambridge
Pranav Singh University of Bath
Ari Stern Washington University in St. Louis
Bjorn Stinner University of Warwick
Vanessa Styles University of Sussex
Chunmei Su Technischen Universität München
Yajuan Sun Chinese Academy of Sciences
Endre Süli University of Oxford
Nikolas Tapia Weierstraß-Institut für Angewandte Analysis und Stochastik; Technische Universität Berlin
Benjamin Tapley Norwegian University of Science and Technology
Milo Viviani Chalmers University of Technology
Tiffany Vlaar University of Edinburgh
Andy Wan University of Northern British Columbia
Beth Wingate University of Exeter
Ragnar Winther University of Oslo
Jinchao Xu Pennsylvania State University
Antonella Zanna Universitetet i Bergen

Show All

Geometry, compatibility and structure preservation in computational differential equations

Programme theme

GCS programme: interview with organisers Arieh Iserles, Elena Celledoni and Elizabeth Mansfield

Preprints

Conservative Galerkin methods for dispersive Hamiltonian problems

Discrete conservation laws for finite element discretisations of multisymplectic PDEs

A surface nite element method for computational modelling of cell blebbing

DISCRETE VARIATIONAL METHODS AND SYMPLECTIC GENERALIZED ADDITIVE RUNGE-KUTTA METHODS

SYMPLECTIC P-STABLE ADDITIVE RUNGE-KUTTA METHODS

Symmetry and invariant bases in finite element exterior calculus

ON BASIS CONSTRUCTIONS IN FINITE ELEMENT EXTERIOR CALCULUS

A converging nite element scheme for motion by curvature of a network with a triple junction

NEWEST VERTEX BISECTION OVER GENERAL TRIANGULATIONS

Numerical preservation of multiple local conservation laws

Locally conservative finite difference schemes for the modified KdV equation

Workshops

Tutorial workshop

Structure preservation and general relativity

The future of structure-preserving algorithms

Seminars

Organisers

Participants

Supported By

Geometry, compatibility and structure preservation in computational differential equations

Programme theme

GCS programme: interview with organisers Arieh Iserles, Elena Celledoni and Elizabeth Mansfield

Preprints

Conservative Galerkin methods for dispersive Hamiltonian problems

Discrete conservation laws for finite element discretisations of multisymplectic PDEs

A surface nite element method for computational modelling of cell blebbing

DISCRETE VARIATIONAL METHODS AND SYMPLECTIC GENERALIZED ADDITIVE RUNGE-KUTTA METHODS

SYMPLECTIC P-STABLE ADDITIVE RUNGE-KUTTA METHODS

Symmetry and invariant bases in finite element exterior calculus

ON BASIS CONSTRUCTIONS IN FINITE ELEMENT EXTERIOR CALCULUS

A converging nite element scheme for motion by curvature of a network with a triple junction

NEWEST VERTEX BISECTION OVER GENERAL TRIANGULATIONS

Numerical preservation of multiple local conservation laws

Locally conservative finite difference schemes for the modified KdV equation

Workshops

Tutorial workshop

Structure preservation and general relativity

The future of structure-preserving algorithms

Seminars

Organisers

Participants

Supported By

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