TY - GEN
AB - Many problems in science and engineering require the efficient numerical
approximation of integrals, a particularly important application being the
numerical solution of initial value problems for differential equations. For
complex systems, an equidistant discretization is often inadvisable, as it
either results in prohibitively large errors or computational effort. To this
end, adaptive schemes have been developed that rely on error estimators based
on Taylor series expansions. While these estimators a) rely on strong
smoothness assumptions and b) may still result in erroneous steps for complex
systems (and thus require step rejection mechanisms), we here propose a
data-driven time stepping scheme based on machine learning, and more
specifically on reinforcement learning (RL) and meta-learning. First, one or
several (in the case of non-smooth or hybrid systems) base learners are trained
using RL. Then, a meta-learner is trained which (depending on the system state)
selects the base learner that appears to be optimal for the current situation.
Several examples including both smooth and non-smooth problems demonstrate the
superior performance of our approach over state-of-the-art numerical schemes.
The code is available under https://github.com/lueckem/quadrature-ML.
AU - Dellnitz, Michael
AU - Hüllermeier, Eyke
AU - Lücke, Marvin
AU - Ober-Blöbaum, Sina
AU - Offen, Christian
AU - Peitz, Sebastian
AU - Pfannschmidt, Karlson
ID - 21600
T2 - arXiv:2104.03562
TI - Efficient time stepping for numerical integration using reinforcement learning
ER -
TY - GEN
AU - Ridderbusch, Steffen
AU - Offen, Christian
AU - Ober-Blöbaum, Sina
AU - Goulart, Paul
ID - 21572
TI - Learning ODE Models with Qualitative Structure Using Gaussian Processes
ER -
TY - GEN
AB - In backward error analysis, an approximate solution to an equation is
compared to the exact solution to a nearby "modified" equation. In numerical
ordinary differential equations, the two agree up to any power of the step
size. If the differential equation has a geometric property then the modified
equation may share it. In this way, known properties of differential equations
can be applied to the approximation. But for partial differential equations,
the known modified equations are of higher order, limiting applicability of the
theory. Therefore, we study symmetric solutions of discretized partial
differential equations that arise from a discrete variational principle. These
symmetric solutions obey infinite-dimensional functional equations. We show
that these equations admit second-order modified equations which are
Hamiltonian and also possess first-order Lagrangians in modified coordinates.
The modified equation and its associated structures are computed explicitly for
the case of rotating travelling waves in the nonlinear wave equation.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19941
T2 - arXiv:2006.14172
TI - Backward error analysis for variational discretisations of partial differential equations
ER -
TY - JOUR
AU - Kreusser, Lisa Maria
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19939
IS - 5
JF - Nonlinearity
SN - 0951-7715
TI - Detection of high codimensional bifurcations in variational PDEs
VL - 33
ER -
TY - THES
AB - Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise
in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate
solutions. In order to draw valid conclusions from numerical computations, it is crucial
to understand which qualitative aspects numerical solutions have in common with the
exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity
under discretisation on long-term behaviour of motions is classically well known, in this
thesis
(a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian
boundary value problems is explained. In parameter dependent systems at a bifurcation
point the solution set to a boundary value problem changes qualitatively. Bifurcation
problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to
persistent bifurcations of Hamiltonian boundary value problems. Further results for
symmetric settings are derived.
(b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem.
(c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs
with variational structure. Recognition equations for A-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations.
(d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers’ equation, KdV, fluid equations, . . . ) using Clebsch variables is analysed.
It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation.
(e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating)
travelling waves in the nonlinear wave equation is discussed.
AU - Offen, Christian
ID - 19947
TI - Analysis of Hamiltonian boundary value problems and symplectic integration
ER -
TY - JOUR
AB - We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19938
IS - 6
JF - Foundations of Computational Mathematics
TI - Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation
VL - 20
ER -
TY - JOUR
AB - Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations, …) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected to conservation properties and other geometric features of solutions to the PDE and, therefore, of great interest for numerical integration. For the example of Burgers' equations and related PDEs we use Clebsch variables to lift the original system to a collective Hamiltonian system on a symplectic manifold whose structure is related to the original Lie-Poisson structure. On the collective Hamiltonian system a symplectic integrator can be applied. Our numerical examples show excellent conservation properties and indicate that the disadvantage of an increased phase-space dimension can be outweighed by the advantage of symplectic integration.
AU - McLachlan, Robert I
AU - Offen, Christian
AU - Tapley, Benjamin K
ID - 19945
IS - 1
JF - Journal of Computational Dynamics
SN - 2158-2505
TI - Symplectic integration of PDEs using Clebsch variables
VL - 6
ER -
TY - JOUR
AB - A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for second-order systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19935
JF - Nonlinearity
SN - 0951-7715
TI - Bifurcation of solutions to Hamiltonian boundary value problems
ER -
TY - JOUR
AB - In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19943
JF - New Zealand Journal of Mathematics
KW - Hamiltonian boundary value problems
KW - singularities
KW - conformal symplectic geometry
KW - catastrophe theory
KW - conjugate loci
TI - Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci
VL - 48
ER -
TY - JOUR
AB - Symplectic integrators can be excellent for Hamiltonian initial value problems. Reasons for this include their preservation of invariant sets like tori, good energy behaviour, nonexistence of attractors, and good behaviour of statistical properties. These all refer to {\em long-time} behaviour. They are directly connected to the dynamical behaviour of symplectic maps φ:M→M' on the phase space under iteration. Boundary value problems, in contrast, are posed for fixed (and often quite short) times. Symplecticity manifests as a symplectic map φ:M→M' which is not iterated. Is there any point, therefore, for a symplectic integrator to be used on a Hamiltonian boundary value problem? In this paper we announce results that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19937
JF - Numerical Algorithms
SN - 1017-1398
TI - Symplectic integration of boundary value problems
ER -
TY - GEN
AB - Two smooth map germs are right-equivalent if and only if they generate two
Lagrangian submanifolds in a cotangent bundle which have the same contact with
the zero-section. In this paper we provide a reverse direction to this
classical result of Golubitsky and Guillemin. Two Lagrangian submanifolds of a
symplectic manifold have the same contact with a third Lagrangian submanifold
if and only if the intersection problems correspond to stably right equivalent
map germs. We, therefore, obtain a correspondence between local Lagrangian
intersection problems and catastrophe theory while the classical version only
captures tangential intersections. The correspondence is defined independently
of any Lagrangian fibration of the ambient symplectic manifold, in contrast to
other classical results. Moreover, we provide an extension of the
correspondence to families of local Lagrangian intersection problems. This
gives rise to a framework which allows a natural transportation of the notions
of catastrophe theory such as stability, unfolding and (uni-)versality to the
geometric setting such that we obtain a classification of families of local
Lagrangian intersection problems. An application is the classification of
Lagrangian boundary value problems for symplectic maps.
AU - Offen, Christian
ID - 19940
T2 - arXiv:1811.10165
TI - Local intersections of Lagrangian manifolds correspond to catastrophe theory
ER -