Geometric Integration

Relations to computer science and physics

Olivier Verdier

Who am I?

  • Olivier (Olle) Verdier
    • French, Swedish
    • PhD in Lund (2009)
    • Postdocs in Cologne, Trondheim, Bergen, Umeå
  • Future associate professor at HiB

Book

This talk

Collaborators

R. McLachlan H. Munthe-Kaas K. Modin

McLachlan        Munthe-Kaas        Modin     

Talk available at olivierverdier.com/s/hib-2014-10

Part 1: Polymorphism

Polymorphism

Length of a list (of String)

length :: [String] -> Int

What about length of list of Double?: 

length :: [Double] -> Int

Polymorphism: 

length :: forall a. [a] -> Int

Important: No type introspection

Type specification

Find a function with the type specification 

f :: forall a. a -> a

One possibility

f x = x

It is the only possibility

...but how to prove this?

“No type introspection”?

[Reynolds '83]

JC Reynolds

Choose arbitrary relation \(\simeq \subset A_1\times A_2\)

\(x_1 \colon A_1\), \(x_2\colon A_2\), \(x_1 \simeq x_2\implies f(x_1) \simeq f(x_2)\)

So \(f \colon \forall A. A \to A\) preserves all relations!

Differential Equation

[Newton 1671]

One of the oldest differential equation in history!

Backward error analysis

An integrator maps a vector field to a vector field

Integrators

Types: \(X_n\) vector fields on \(\mathbb{R}^n\)

Integrator: map vf to vf \[\phi \colon X \to X \] “Polymorphic in the dimension”

Specifications?

\(R\colon \mathbb{R^n}\to\mathbb{R^m}\)

Induces a relation \(\simeq_R\) by

\(f \simeq_R g \iff R'.f(x) = g(R(x))\)

Specification: \(f \simeq_R g\implies φ(f) \simeq_R φ(g)\)

Identity is the only possibility

Relaxing specifications

[McLachlan, Modin, Munthe-Kaas, V. '14]

Only allow affine relations \(R\).

The method is immune to:

  • Change of origin, units, rotations, shearing
  • Respects decoupled systems

Only possibility: Runge–Kutta methods

Take home message?

New discipline: specification based numerical analysis

Part 2: Spin systems

Equation of motion

Newton:\[\frac{d^2}{dt^2}q = -\nabla U(q)\]

Differential equation: \[ \frac{d}{dt}q = p/m \quad \frac{d}{dt}p = -\nabla U(x)\]

General form: \[\frac{d}{dt}x = f(x)\]

Symplecticity

Variables are paired in \((q,p)\): sum of 2D planes

Symplecticity: the sum of projected areas is constant

Symplectic integrator

An integrator which solves exactly another mechanical system

Kepler problem

Leap-frog

aka Verlet

Verlet

\[ q_1 = q_{-1} -∆t^2\,\nabla U(q_0)\]

Symplectic, uses the \((q,p)\) decomposition

Midpoint

\[x_1 = x_0 + ∆t\,f((x_0+x_1)/2)\]

Symplectic, but does not use the \((q,p)\) decomposition. Magic!

Spin systems

2D planes replaced by 2D spheres

Hurricanes on gas planets

Hurricane interaction

Point vortices

\[ \frac{d}{dt} r^i = \sum_{j\neq i} Γ^{j} \frac{r^i \times r^j}{\|r^j - r^i\|} \]

\(r^i\) position of i-th hurricane

\(Γ^i\): “force” of the i-th hurricane

Symplectic system

“Midpoint” on spheres

[McLachlan, Modin, V. '14]

\[r_1 = r_0 + Δt \, f\Big(\frac{r_0+r_1}{\|r_0+r_1\|}\Big)\]

Properties:

  • Stays on the spheres
  • Symplectic!

Note: proof of symplecticity has nothing to do with usual midpoint

Simulation

[McLachlan, Modin, V. '14]

Conclusion

  1. Runge–Kutta methods are the only affine compatible integrator
  2. Symplectic “midpoint” method on sphere

Thank You

Bibliography

  • “B-series methods are exactly the local, affine equivariant methods” with McLachlan, Modin, Munthe-Kaas
    arXiv:1409.1019
  • “Aromatic Butcher Series” with Munthe-Kaas
    FoCM, arXiv:1308.5824 (2015)
  • “Symplectic integrators for spin systems” with McLachlan, Modin
    Phys. Rev. E., arXiv:1402.4114 (2014)
  • “Discrete time Hamiltonian spin systems” with McLachlan, Modin
    arXiv:1402.3334 (2014)