Session I.5 - Geometric Integration and Computational Mechanics
Talks
Monday, June 12, 14:00 ~ 14:30
r-adaptivity, deep learning and the deep Ritz method
Chris Budd
University of Bath, UK - This email address is being protected from spambots. You need JavaScript enabled to view it.
Deep learning based methods such as PINNS are becoming increasingly popular as a means of solving differential equations. They are advertised as easy to use, mesh free methods, which work in high dimensions. However significant questions remain as to how reliable they are and what convergence rates they have, and how they compare with more established numerical analysis methods such as finite elements. In this talk I will explore some of these questions in the context of the solution of elliptic PDES with singularities. I will demonstrate that there is much to be gained from combining ideas from structure preserving numerical analysis (including geometric integration) and approximation theory, with the use of deep learning methods.
Joint work with Simone Appela, Tristan Pryer, Lisa Kreusser and Teo Deveney and (University of Bath).
Monday, June 12, 14:30 ~ 15:00
Structure-preserving model order reduction of parametric Hamiltonian systems
Cecilia Pagliantini
University of Pisa, Italy - This email address is being protected from spambots. You need JavaScript enabled to view it.
Model order reduction of parametric differential equations aims at constructing low-complexity high-fidelity surrogate models that allow rapid and accurate solutions under parameter variation. The development of reduced order models for Hamiltonian systems is challenged by several factors: (i) failing to preserve the geometric structure encoding the physical properties of the dynamics might lead to instabilities and unphysical behaviors of the resulting approximate solutions; (ii) the slowly decaying Kolmogorov n-width of transport-dominated and non-dissipative phenomena demands large reduced spaces to achieve sufficiently accurate approximations; and (iii) nonlinear operators require hyper-reduction techniques that preserve the gradient structure of the flow velocity. We will discuss how to address these aspects via a structure-preserving nonlinear reduced basis approach based on dynamical low-rank approximation. The gist of the proposed method is to adapt in time an approximate low-dimensional phase space endowed with the geometric structure of the full model and to ensure that the reduced flow is still Hamiltonian.
Monday, June 12, 15:00 ~ 15:30
Learning of symmetric models for variational dynamical systems from data
Christian Offen
Paderborn University, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
Equations of motions of variational dynamical systems can be derived from an action functional defined by a Lagrangian. When the Lagrangian is not known, it can be identified from dynamical data using machine learning techniques. However, Lagrangians are not uniquely determined by the dynamics. In this talk, I will show a framework to learn symmetric models of Lagrangians. The system’s symmetries and conservation laws do not need to be known a priori but are identified automatically based on a Lie group framework. Learning symmetric over non-symmetric Lagrangians improves qualitative aspects of the model, helps the numerical integration of the data-driven model, and informs the user about important geometric properties of the system.
Joint work with Eva Dierkes (University of Bremen, Germany), Kathrin Flaßkamp (Saarland University, Germany), Yana Lishkova (University of Oxford, UK), Steffen Ridderbusch (University of Oxford, UK), Sina Ober-Blöbaum (Paderborn University, Germany) and Paul Scherer (Cambridge, UK).
Monday, June 12, 16:30 ~ 17:00
Transverse Symplectic Foliation Structure of Souriau Dissipative Statistical Mechanics with Entropy as Casimir Function
Frédéric Barbaresco
THALES, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
Foliation theory is a natural generalized qualitative theory of differential equations, initiated by H. Poincaré, and developed by C.Ehresmann and G. Reeb, with contribution by A. Haefliger, P. Molino, B.L. Reinhart. Riemannian foliations generated by metric functions were developed by Ph. Tondeur. Notion of foliation in thermodynamics appears as soon as 1900 C. Carathéodory paper where horizontal curves roughly correspond to adiabatic processes, performed in the language of Carnot cycles. The properties of the couple of Poisson manifolds was also previously explored by C. Carathéodory in 1935, under the name of “function groups, polar to each other”, where he observed that the two families of differentiable functions formed by the first integrals of F (a completely integrable vector subbundle of TM) and its orthogonal orthF, respectively, called "function groups", are "polar" of the other. This seminal work of C. Caratheodory leads to the concept of a Poisson structure which was first defined and treated in depth by A. Lichnerowicz and independently by A. Kirillov. We introduce a symplectic bifoliation model of Statistical Mechanics, Information Geometry and Heat Theory based on Jean-Marie Souriau's Lie Groups Thermodynamics to describe transverse Poisson structure of metriplectic flow for dissipative phenomena. This model gives a cohomological characterization of Entropy, as an invariant Casimir function in coadjoint representation. The dual space of the Lie algebra foliates into coadjoint orbits identified with the Entropy level sets. In the framework of Thermodynamics, we associate a symplectic bifoliation structure to describe non-dissipative dynamics on symplectic leaves (on level sets of Entropy as constant Casimir function on each leaf), and transversal dissipative dynamics, given by Poisson transverse structure (Entropy production from leaf to leaf). The symplectic foliation orthogonal to the level sets of moment map is the foliation determined by hamiltonian vector fields generated by functions on dual Lie algebra. The orbits of a Hamiltonian action and the level sets of its moment map are polar to each other. The space of Casimir functions on a neighborhood of a point is isomorphic to the space of Casimirs for the transverse Poisson structure. Souriau’s model could be then interpreted by Mademoiselle Paulette Libermann's foliations, clarified as dual to Poisson Gamma-structure of Haefliger, which is the maximum extension of the notion of moment in the sense of J.M. Souriau, as introduced by P. Molino, M. Condevaux and P. Dazord in papers of “Séminaire Sud-Rhodanien de Geometrie ». The symplectic duality to a symplectically complete foliation, in the sense of Libermann, associates an orthogonal foliation. Paulette Libermann proved that a Legendre foliation on a contact manifold is complete if and only if the pseudo-orthogonal distribution is completely integrable, and that the contact form is locally equivalent to the Poincaré-Cartan integral invariant. Paulette Libermann proved a classical theorem relating to co-isotropic foliations, which notably gives a proof of Darboux's theorem. We conclude with link to Cartan foliation and Edmond Fedida works on Cartan's mobile frame-based foliation. As observed by Georges Reeb "Thermodynamics has long accustomed mathematical physics [see DUHEM P.] to the consideration of completely integrable Pfaff forms: the elementary heat dQ [notation of thermodynamicists] representing the elementary heat yielded in an infinitesimal reversible modification is such a completely integrable form. This point does not seem to have been explored since then."
References:
[1] Souriau, J.M.,, Structure des systèmes dynamiques. Dunod (1969)
[2] Souriau, J.M., Mécanique statistique, groupes de Lie et cosmologie. In: Colloque Inter-national du CNRS "Géométrie symplectique et physique Mathématique", 1974. Aix-en-Provence (1976)
[3] Haefliger, A.: Naissance des feuilletages d’Ehresmann-Reeb à Novikov. Journal 2(5), 99–110 (2016)
[4] Libermann, P: Problèmes d’équivalence et géométrie symplectique, Astérisque, tome 107-108, p. 43-68 (1983)
[5] Dazord, P., Molino, P. :Gamma-Structures poissonniennes et feuilletages de Libermann, Pu-blications du Département de Mathématiques de Lyon, fascicule 1B, « Séminaire Sud-Rhodanien 1ère partie », chapitre II , p. 69-89 (1988)
[6] Condevaux, M., Dazord, P., Molino, P. : Géométrie du moment, Publications du Dépar-tement de Mathématiques de Lyon, fascicule 1B, « Séminaire Sud-Rhodanien 1ère par-tie », chapitre V , p. 131-160 (1988)
[7] Molino, P. : Dualité symplectique, feuilletage et géométrie du moment, Publicacions Matemátiques, Vol 33, 533-541 (1989)
[8] Fedida, E. ,Sur la theorie des feuilletages associee au repere mobile : cas des feuilletages de lie. In: Schweitzer, P.A. (eds) Differential Topology, Foliations and Gelfand-Fuks Cohomology. Lecture Notes in Mathematics, vol 652. Springer (1978)
[9] Reeb, G., Structures feuilletées, Differential Topology, Foliations and Gelfand-Fuks cohomology, Rio de Janeiro, 1976, Springer Lecture Notes in Math. 652, 104-113, (1978)
[10] Barbaresco, F.,Symplectic theory of heat and information geometry, chapter 4, Handbook of Statistics, Volume 46, Pages 107-143, Elsevier (2022)
[11] Barbaresco, F. , Jean-Marie Souriau’s Symplectic Model of Statistical Physics: Seminal Papers on Lie Groups Thermodynamics - Quod Erat Demonstrandum. Geometric Struc-tures of Statistical Physics, Information Geometry, and Learning. SPIGL 2020. Spring-er Proceedings in Mathematics & Statistics, vol361. Springer (2021)
[12] Barbaresco, F. Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Nonlinear Lindblad Quantum Master Equation. Entropy, 24, 1626 (2022)
Monday, June 12, 17:00 ~ 17:30
Structured neural networks and some applications to dynamical systems
Davide Murari
Norwegian University of Science and Technology, Norway - This email address is being protected from spambots. You need JavaScript enabled to view it.
Neural networks have gained much attention for their effectiveness in various applications. However, they typically lack a predetermined structure, and their properties are often not well understood. To address this issue, it may be desirable to incorporate properties of the target function or data being processed into the design of the neural network. Having a systematic approach to designing structured networks can be highly beneficial. In this talk, we present a framework combining ODEs, and suitable numerical methods can be employed to model neural networks with specific properties. Additionally, we offer some particular applications of this approach to data-driven modelling and the problem of approximating solutions of ODEs.
Monday, June 12, 17:30 ~ 18:30
Dissipative and stochastic systems, Lyapunov functions and adaptive algorithms
Benedict Leimkuhler
University of Edinburgh, UK - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will discuss the convergence of a wide collection of algorithms for both optimization and sampling based on discretization of extended systems of ordinary and stochastic differential equations. These methods introduce control laws to guide convergence and build in a form of adaptivity through dynamics. They are particularly well adapted to ``stiff systems'' in which steep gradients in some components can dramatically slow or destabilize standard algorithms.
Joint work with Katerina Karoni (University of Edinburgh) and Gabriel Stoltz (Ecole des Ponts).
Tuesday, June 13, 14:00 ~ 14:30
On the contractivity of ODEs and numerical integrators on Riemannian manifolds
Brynjulf Owren
NTNU, Norway - This email address is being protected from spambots. You need JavaScript enabled to view it.
On the contractivity of ODEs and numerical integrators on Riemannian manifolds
In the theory of differential equations concepts related to stability are very well established and understood. Parallel theories for numerical schemes in Euclidean spaces are also well developed. From the beginning, the notion of absolute stability, using a linear test equation was the prevalent tool. Later, Dahlquist, Burrage and Butcher were leading the development of a numerical stability theory that also makes sense for nonlinear problems, that of B-stable or contractive methods, set in Hilbert spaces. In more recent times, it has become popular to construct and analyse numerical schemes for differentiable manifolds, methods that are entirely intrinsic, examples are the Lie group integrators. For the study of nonlinear stability, Riemannian manifolds, or even Finsler manifolds, seem to be a useful framework for studying contractive methods. Building on work by Kunzinger et al. [1] and Simpson-Porco & Bullo [2] we shall suggest a way to generalise the notion of B-stability to Riemannian manifolds. We study the geodesic implicit Lie-Euler method as a model method and show some first theoretical and numerical results.
[1] Michael Kunzinger, Hermann Schichl, Roland Steinbauer, and James A. Vickers. Global Gronwall estimates for integral curves on Riemannian man- ifolds. Rev. Mat. Complut., 19(1):133–137, 2006.
[2] John W. Simpson-Porco and Francesco Bullo. Contraction theory on Riemannian manifolds, Systems & Control Letters 65 (2014), 74-80.
Joint work with Martin Arnold (Martin Luther University, Halle-Wittenberg, Germany), Elena Celledoni (NTNU, Norway), Ergys Cokaj (NTNU, Norway) and Denise Tumiotto (Martin Luther University, Halle-Wittenberg, Germany).
Tuesday, June 13, 14:30 ~ 15:00
A framework for stable geometric spectral methods
Arieh Iserles
University of Cambridge, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.
Spectral methods are extraordinarily powerful for boundary-value problems with regular boundary conditions and `nice' geometries, more problematic for initial-value problems because of stability issues. In this talk we briefly review a recent theory of T-functions and W-functions, whereby evolutionary PDEs can be discretised stably in different geometries and while conserving, as necessary, the $L_2$ norm. We also discuss the conservation of the Hamiltonian in this setting.
Joint work with Marcus Webb (University of Manchester, UK) and Jing Gao (Jia-Tong University, X'ian, China).
Tuesday, June 13, 15:00 ~ 15:30
A new Lagrangian approach to control affine systems with a quadratic Lagrange term
Sigrid Leyendecker
Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work, we consider optimal control problems for mechanical systems with fixed initial and free final state and a quadratic Lagrange term. Specifically, the dynamics is described by a second order ODE containing an affine control term. Classically, Pontryagin’s maximum principle gives necessary optimality conditions for the optimal control problem. For smooth problems, alternatively, a variational approach based on an augmented objective can be followed. Here, we propose a new Lagrangian approach leading to equivalent necessary optimality conditions in the form of Euler-Lagrange equations. Thus, the differential geometric structure (simular to classical Lagrangian dynamics) can be exploited in the framework of optimal control problems. In particular, the formulation enables the symplectic discretisation of the optimal control problem via variational integrators in a straighforward way.
Joint work with Sina Ober-Blöbaum (Universität Paderborn, Germany), Sofya Maslovskaya (Universität Paderborn, Germany) and Flora Szemenyei (Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany).
Tuesday, June 13, 15:30 ~ 16:00
Half-explicit integrators for constrained mechanical systems on Lie groups
Martin Arnold
Martin Luther University Halle-Wittenberg, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
Configuration space with Lie group structure have become a quasi-standard in the dynamical simulation of flexible multibody systems with large rotations (O. Brüls, A. Cardona, M. Arnold: Lie group generalized-$\alpha$ time integration of constrained flexible multibody systems. Mechanism and Machine Theory 48:121-137, 2012). The equations of motion form constrained systems on Lie groups that are isomorphic to Cartesian products of (direct or semi-direct) products of $\mathbb{R}^3$ and SO(3). For these Lie groups, the exponential map $\exp$, its right trivialized tangent $d\exp$ and the inverse $d\exp^{-1}$ may be evaluated in closed form with a computational complexity that compares to the classical Rodrigues formula for exaluating $\exp$ on SO(3). Therefore, local parametrizations in terms of elements of the corresponding Lie algebra may be used efficiently.
Combining the classical Munthe-Kaas approach for Runge-Kutta Lie group time integration and a half-explicit strategy for solving constrained systems (M. Arnold: Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2 . BIT Numerical Mathematics 38:415-438, 1998), we end up with a half-explicit Lie group integrator for constrained mechanical systems. Up to order $p=5$, methods with a reasonable number of half-explicit Runge-Kutta stages have been constructed including the HELieDOP5 integrator that generalizes the well known 5th order Dormand and Prince method to constrained mechanical systems on Lie groups. The results of the theoretical convergence analysis are verified by numerical tests for some classical benchmark problems.
Joint work with Denise Tumiotto (Martin Luther University Halle-Wittenberg, Germany).
Tuesday, June 13, 17:00 ~ 17:30
Geometric Integration and Numerical Analysis on Symmetric Spaces
Hans Munthe-Kaas
Lie-Størmer Center at The Arctic University of Norway, Norway - This email address is being protected from spambots. You need JavaScript enabled to view it.
Symmetric spaces are spaces equipped with a symmetric product. In differential geometry, prime examples are spheres, hyperbolic spaces and Grassman manifolds. Together with Reinout Quispel and Antonella Zanna, we have explored many properties of spaces with a symmetric product, in works going back two decades. After a brief survey of this body of work, I will discuss a new canonical numerical time integration algorithm on symmetric spaces and discuss recent work on understanding the algebras of canonical connections on symmetric spaces, torsion free, with constant curvature, where the goal is to develop a theory of B-series on symmetric spaces.
Joint work with Jonatan Stava, University of Bergen.
Tuesday, June 13, 17:30 ~ 18:00
Functional equivariance and modified vector fields
Ari Stern
Washington University in St. Louis, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
In a recent paper with Robert McLachlan (Found. Comput. Math., 2022), we introduced the notion of functional equivariance for numerical integrators. This generalized previous work on numerical preservation of linear and quadratic invariants by characterizing methods that preserve the evolution of not-necessarily-invariant linear and quadratic observables, with applications to local conservation laws in numerical PDEs. This talk discusses some new work relating functional equivariance to properties of modified vector fields, generalizing previous results for invariant-preserving methods.
Joint work with Sanah Suri (Washington University in St. Louis).
Tuesday, June 13, 18:00 ~ 18:30
Structure-preserving Reduced Complexity Modelling
Michael Kraus
Max Planck Institute for Plasma Physics, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
In many applications, such as optimisation, uncertainty quantification and inverse problems, it is required to perform repeated simulations of high-dimensional physical systems for different choices of parameters. In order to save computational cost, surrogate models can be constructed by expressing the solution in a low-dimensional basis, obtained from training data. This is referred to as model reduction.
Past investigations have shown that, when performing model reduction of Lagrangian or Hamiltonian systems, it is crucial to preserve the symplectic structure associated with the system in order to ensure long-term numerical stability and restrict error growth.
In this talk, we will review structure-preserving model reduction and machine learning techniques for the construction of reduced bases, hyper-reduction and flow approximation.
Wednesday, June 14, 14:00 ~ 14:30
Explicit Energy-Preserving Momentum-Scaling Schemes for Hamiltonian Systems
Andy Wan
University of Northern British Columbia, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
We introduce a novel class of explicit energy-preserving momentum-scaling (EPMS) schemes for Hamiltonian systems of the form $H(\boldsymbol q,\boldsymbol p)=\frac{1}{2}{\boldsymbol p}^T M^{-1}(\boldsymbol q)\boldsymbol p+U(\boldsymbol q)$. EPMS schemes consist of two main steps: first, utilize an explicit scheme satisfying a non-degenerate condition; second, follow by scaling of momentum variables to achieve exact energy preservation. We show that EPMS schemes are consistent. Moreover, we give a sufficient condition for explicit Runge-Kutta methods to satisfy the non-degenerate condition, showing that a wide class of explicit Runge-Kutta methods can be turned into EPMS schemes. Numerical experiments showcasing computational efficiency of EPMS schemes versus implicit energy-preserving schemes are presented.
Joint work with Molei Tao (Georgia Institute of Technology).
Wednesday, June 14, 14:30 ~ 15:00
Curvature approximation in arbitrary dimension with Regge finite elements
Evan Gawlik
University of Hawaii, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We study the curvature of Regge metrics on simplicial triangulations of dimension $N$. Here, a Riemannian metric is called a Regge metric if it is piecewise smooth and its tangential-tangential components are single-valued on every codimension-1 simplex in the triangulation. When such a metric is piecewise polynomial, it belongs to a finite element space called the Regge finite element space. Regge metrics are not classically differentiable, but it turns out that one can still make sense of their curvature in a distributional sense. In the lowest-order setting, the distributional curvature of a Regge metric is a linear combination of Dirac delta distributions supported on codimension-2 simplices $S$, weighted by the angle at $S$: $2\pi$ minus the sum of the dihedral angles incident at $S$. For piecewise polynomial Regge metrics of higher degree, the distributional curvature includes additional contributions involving the curvature in the interior of each $N$-simplex and the jump in the mean curvature across each codimension-1 simplex.
We study the convergence of the distributional curvature under refinement of the triangulation. We show that in the $H^{-2}$-norm, this convergence takes place at a rate of $O(h^{r+1})$ when a smooth Riemannian metric is interpolated by a piecewise polynomial Regge metric of degree $r \ge 0$ on a triangulation whose maximum simplex diameter is $h$, provided that either $N=2$ or $r \ge 1$.
Joint work with Yakov Berchenko-Kogan (Florida Institute of Technology, USA) and Michael Neunteufel (TU Wien, Austria).
Wednesday, June 14, 15:00 ~ 16:00
Variational and Thermodynamically Consistent Discretization for Heat Conducting Fluids
François J. M. Gay-Balmaz
CNRS Ecole Normale Supérieure, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
We construct a structure-preserving and thermodynamically consistent finite element method and time-stepping scheme for heat conducting viscous fluids. The method is deduced by discretizing a variational formulation for nonequilibrium thermodynamics that extends Hamilton's principle for fluids to systems with irreversible processes. The resulting scheme preserves the balance of energy and mass to machine precision, as well as the second law of thermodynamics, both at the spatially and temporally discrete levels. The method is shown to apply both with insulated and prescribed heat flux boundary conditions, as well as with prescribed temperature boundary conditions.
Joint work with Evan Gawlik (University of Hawaii).
Wednesday, June 14, 16:30 ~ 17:00
Eulerian and Lagrangian stability in Zeitlin's model of hydrodynamics
Klas Modin
Chalmers and University of Gothenburg, Sweden - This email address is being protected from spambots. You need JavaScript enabled to view it.
The 2-D Euler equations of a perfect fluid possess a beautiful geometric description: they are reduced geodesic equations on the infinite-dimensional Lie group of symplectomorphims with respect to a right-invariant Riemannian metric. This structure enables insights to Eulerian and Lagrangian stability via sectional curvature and Jacobi equations.
The Zeitlin model is a finite-dimensional analog of the 2-D Euler equations; the only known discretization that preserves the rich geometric structure. Theoretical and numerical studies indicate that Zeitlin's model provides consistent long-time behaviour on large scales, but to which extent it truly reflects the Euler equations is mainly open. Towards progress, I present here two new results. First, convergence of the sectional curvature in the Euler--Zeitlin equations on the Lie algebra $\mathfrak{su}(N)$ to that of the Euler equations on the sphere. Second, $L^2$-convergence of the corresponding Jacobi equations for Lagrangian and Eulerian stability. The new results allow geometric conclusions about Zeitlin's model to be transferred to Euler's equations and vice versa.
Joint work with Manolis Perrot (Univ. Grenoble Alpes, France).
Wednesday, June 14, 17:00 ~ 17:30
Parallel iterative methods for variational integration and related problems
Rodrigo Takuro Sato Martín de Almagro
Friedrich-Alexander-Universität Erlangen-Nürnberg - Institute of Applied Dynamics, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
Discrete variational methods show excellent performance in numerical simulations of different mechanical systems. In this talk, we present an iterative procedure for the solution of discrete variational equations for boundary value problems. More concretely, we explore a parallelization strategy that leverages the capabilities of multicore CPUs and GPUs (graphics cards).
We study this parallel method for higher-order Lagrangian systems, which appear in fully-actuated problems and beyond. The study of the convergence conditions of these methods poses interesting challenges that have led us to the study of, among other things, the discrete Jacobi equation.
Joint work with Sebastián J. Ferraro (Universidad Nacional del Sur & CONICET) and David Martín de Diego (Instituto de Ciencias Matemáticas - CSIC-UAM-UC3M-UCM).
Wednesday, June 14, 17:30 ~ 18:00
Conservative iterative methods for structure-preserving discretisations
James Jackaman
NTNU, Norway - This email address is being protected from spambots. You need JavaScript enabled to view it.
There has been substantial work on the development of structure-preserving discretisations of differential equations, where the numerical approximation reflects key conservation laws observed of the continuum solution. While such discretisations are unquestionable valuable, their practical utility can be limited by the fact that standard iterative methods for solution of the resulting linear and nonlinear systems only resolve the underlying conserved quantities when solved to near-machine precision. In this talk, we present a generalisation of the (preconditioned) flexible GMRES algorithm that can preserve arbitrarily many such conserved quantities exactly at (nearly) any stopping tolerance, with a small additional cost. Numerical results are presented for several structure-preserving finite-element discretisations of linear parabolic and hyperbolic model problems.
Joint work with Scott MacLachlan (Memorial University of Newfoundland, Canada).
Wednesday, June 14, 18:00 ~ 18:30
Type II Hamiltonian Lie Group Variational Integrators for Geometric Adjoint Sensitivity Analysis
Brian Tran
UC San Diego, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
We present our construction of continuous and discrete Type II variational principles for Hamiltonian systems on cotangent bundles of Lie groups, which allows for Type II boundary conditions, i.e., fixed initial position and terminal momenta boundary conditions. The motivation for these boundary conditions arises from the adjoint sensitivity method, which is ubiquitous in dynamically-constrained optimization and optimal control problems. Traditionally, such Type II variational principles are only defined locally. However, for dynamics on the cotangent bundle of a Lie group, left-trivialization allows us to define this variational principle globally. Our discrete variational principle leads to an intrinsic, symplectic, and momentum-preserving integrator for Lie group Hamiltonian systems that allows for Type II boundary conditions and maximally degenerate Hamiltonians. We show how this method can be used to exactly compute sensitivities for optimization problems subject to dynamics on a Lie group. We conclude with numerical examples of optimal control problems on SO(3).
Joint work with Melvin Leok (UC San Diego).
Posters
Weak approximation of stochastic differential equations
Leroy Alix
University of Edinburgh, Uk - This email address is being protected from spambots. You need JavaScript enabled to view it.
The work focussed on weak approximation of stochastic differential equation and develop a method of computing solutions of Langevin dynamics using variable stepsize. The method assume a knowledge of the problem allowing to establish a good `monitor function' which locates points of rapid change in solutions of stochastic differential equations. Using time-transformation we show that it is possible to integrate a rescaled system using fixed-stepsize numerical discretization effectively placing more timesteps where needed.
Simulating diffusions by Lie group integration
Curry Charles HA
Norwegian University of Science and Technology, Norway - This email address is being protected from spambots. You need JavaScript enabled to view it.
We explain how Lie group and other geometric integration techniques can be applied to efficient simulation of diffusions, both on manifolds and Euclidean space.
Symplectic groupoids for Poisson integrators
Oscar Cosserat
CNRS/La Rochelle Université, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
Geometric technics are developed to build Hamiltonian Poisson integrators for generic Hamiltonian and Poisson structure. Such technics allow to compute precise estimates on the error and are theoretical tools to understand better the stability of such integrators on long run simulations. The main geometric object is the local symplectic groupoid of the considered Poisson structure. Those technics are illustrated on rigid body dynamics and Lotka-Volterra equations. The poster is based on the following preprints : Symplectic Groupoids for Poisson Integrators, O. C., arXiv:2303.15883 Numerical Methods in Poisson Geometry and their Application to Mechanics , O. C., C. Laurent-Gengoux, V. Salnikov, arXiv:2205.04838
Joint work with Camille Laurent-Gengoux (Institut Élie Cartan de Lorraine, France) and Vladimir Salnikov (CNRS/La Rochelle Unviersité, France).
Error Estimates of Numerical Methods for the Long-time Dynamics of the Nonlinear Klein-Gordon Equation
Yue Feng
Sorbonne University, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will present the error estimates of numerical methods for the long-time dynamics of the nonlinear Klein-Gordon equation (NKGE) with weak nonlinearity, which is characterized by $\varepsilon^2$ with $\varepsilon \in (0, 1]$ a dimensionless parameter. Different numerical methods are adopted to discretize the NKGE and rigorous error bounds are established for the long-time dynamics. Numerical methods include finite difference methods, exponential wave integrators and time-splitting methods with particular attentions paid on error bounds of different numerical methods explicitly depending on the mesh size $h$, time step $\tau$ as well as the parameter $\varepsilon$ up to the time $t= T/\varepsilon^2$ with $T \gt 0$ fixed. As a by-product, our results are extended to an oscillatory NKGE whose solution propagates waves with wavelength at $O(1)$ in space and $O(\varepsilon^2)$ in time. Extensive numerical examples are provided to confirm our error bounds and demonstrate that they are sharp.
Joint work with Weizhu Bao (National University of Singapore), Yongyong Cai (Beijing Normal University), Chunmei Su (Tsinghua University) and Wenfan Yi (Hunan University).
Optimisation schemes on manifolds obtained via discretisations of conformal Hamiltonian systems
Marta Ghirardelli
NTNU, Norway - This email address is being protected from spambots. You need JavaScript enabled to view it.
Over the last few decades, optimisation methods for problems on manifolds have been gaining more and more popularity [1], [2]. In many cases, they can be seen as an alternative to solving constrained optimisation problems in Euclidean spaces. More precisely, the manifold setting allows to include the constraints in a more natural geometric way. When considering applications to deep learning, constraints on the parameter space may sometimes be useful to prevent the problem of exploding or vanishing gradients, which often occurs e.g. during the training of recurrent neural networks. It would then be desirable to choose compact manifolds, such as the Stiefel manifold or the orthogonal group.
Still in the fields of machine learning and deep learning, Gradient Descent (GD) algorithms are widely used as they require only first order information of the function $f$ to be minimised, and they are easy to implement. They are hence suitable for minimising e.g. cost or loss function in such contexts, where one has to deal with large amounts of data. Unfortunately, these methods are proved to achieve linear convergence only, as long as $f$ is smooth and strongly convex, and can consequently be rather slow. Accelerated versions have been studied, especially in the Euclidean setting where we find Nesterov's accelerated gradient, Polyaks's heavy ball and a relativistic gradient descent method. The last two of these, in particular, can be seen as the discretisation of some conformal Hamiltonian systems [3] where $f$ is seen as the potential energy. We want to consider the corresponding systems when $f$ is defined on a manifold $M$. To do so we first formulate the Hamiltonian function as the sum of the potential energy $f$ and a momentum dependent kinetic energy. The related Hamiltonian vector field is defined on the cotangent bundle of $M$, namely $T^*M$, where an exact natural symplectic form $\omega$ can be defined. The manifold $(T^*M, \omega)$ is then called exact, and the conformal Hamiltonian system is recovered by adding to the conservative vector field, a dissipative one, the Liouville vector field [4]. The flow of this vector field can then be approximated via splitting methods, composing the conservative flow with the dissipative one.
[1] P-A Absil, Robert Mahony, and Rodolphe Sepulchre. Optimisation algorithms on matrix manifolds, 2009.
[2] Nicolas Boumal. An introduction to optimisation on smooth manifolds, 2023.
[3] Guilherme França, Jeremias Sulam, Daniel Robinson, and René Vidal. Conformal symplectic and relativistic optimisation, 2020.
[4] Robert McLachlan and Matthew Perlmutter. Conformal hamiltonian systems, 2001.
Symmetry Preservation in Hamiltonian Systems: Simulation and Learning
David Martin de Diego
ICMAT, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
Symmetry has always played a pivotal role in understanding Hamiltonian systems. Whilst the key concept in general Hamiltonian systems is Lagrangian submanifolds, when symmetry is present symmetry-invariant Lagrangian submanifolds play the main role. In this poster we present a novel technique to generate all invariant Lagrangian submanifolds that can be used both to simulate or to learn systems with symmetry. Our constructions result in integrators and learning mechanisms that conserve the corresponding momentum mapping. Extensions to the Poisson setting are also introduced and, finally, a general process to "geometrize" and endow with symmetry non-geometric integrators is presented.
Joint work with Miguel Vaquero (School of Science and Technology, IE University, Spain) and Jorge Cortés (Department of Mechanical and Aerospace Engineering, University of California San Diego).
Variational integrators and frequency-dependent damping
Rodrigo Takuro Sato Martin de Almagro
Friedrich-Alexander-Universität Erlangen-Nürnberg , Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
The generalized-$\alpha$ method is numerical algorithm for the integration of mechanical systems that can be interpreted as a generalization of other popular algorithms such as the Newmark-$\beta$. It is quite popular among those working on flexible multi body dynamics due to its unconditional stability and frequency-dependent dissipation properties, which allows it to eliminate undesirable high-frequency oscillations that may otherwise compromise the accuracy or the convergence speed of a simulation.
We wondered how could variational methods offer similar advantages by including simple additional forcing terms. This poster is an exploration of this, where we use the wave equation as a model problem. By discretising it using variational methods and inserting dissipative terms, we study their behavior and compare them to analogous results from the generalized-$\alpha$.
Joint work with Sigrid Leyendecker.
Structure-preserving MGARK methods for Hamiltonian systems
Kevin Schäfers
University of Wuppertal, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
Generalized additive Runge-Kutta (GARK) schemes have shown to be a suitable tool for solving ordinary differential equations with additively partitioned right-hand sides. This work combines the ideas of symplectic GARK schemes and multirate GARK (MGARK) schemes to solve additively partitioned Hamiltonian systems with multirate behavior more efficiently. In a general setting of non-separable Hamiltonian systems, order conditions, as well as conditions for symplecticity, symmetry and time-reversibility are derived. Moreover, investigations for the special case of separable Hamiltonian systems are carried out. Numerical results underline the performance of the derived schemes.
Joint work with Michael Günther (University of Wuppertal, Germany) and Adrian Sandu (Virginia Polytechnic Institute and State University, USA).
Noncommutative differential geometry on discrete spaces.
Damien Tageddine
McGill University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this poster, we will present how the general framework of noncommutative geometry can be used for the discretization of differential operators. We present the exterior derivative as a commutator with a hermitian operator; the so-called Dirac operator. We show that finite difference expressions can be recovered as convex combinations of eigenvalues of this commutator. In addition, we show that under suitable conditions i.e. when the coefficients of the Dirac operator are determined by a suitable distribution, the Laplace operator on a smooth manifold is recovered at the limit.
Joint work with Jean-Christophe Nave (McGill University. Canada).
A factorization approach to the extrinsic approximation of volume-preserving diffeomorphisms
Seth Taylor
McGill University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
We present a semi-Lagrangian numerical method for diffeomorphism approximation and its application to incompressible hydrodynamics on the sphere. The method approximates the flow of a velocity field using a spatio-temporal discretization formed by a composition of submaps. This technique substitutes the effects of spatial refinement with the operation of composition by adaptively growing the temporal discretization. In turn, the method has the capacity of accurately and sparsely representing the generation of fine scales globally using only a linear increase in the degrees of freedom. Based on a factorization of diffeomorphisms developed by Modin, we design a geometric correction technique to constrain the evolution to a tubular neighbourhood of the volume-preserving diffeomorphism group. An analysis on the adaptive use of this correction technique in conjunction with the submap decomposition is given and supported through numerical experimentation on some canonical geophysical flows. We demonstrate the ability to improve the conservation properties of the method and capture turbulent energy cascades at subgrid scales.
Joint work with Jean-Christophe Nave (McGill University, Canada).