- Olivier (Olle) Verdier
- French, Swedish
- PhD in Lund (2009)
- Postdocs in Cologne, Trondheim, Bergen, Umeå

- Future associate professor at HiB

- “Polymorphism”: arXiv:1409.1019
- Spin systems: arXiv:1402.4114

Collaborators

McLachlan Munthe-Kaas Modin

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

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**

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?

[Reynolds '83]

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!

[Newton 1671]

One of the oldest differential equation in history!

An integrator maps a vector field to a vector field

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

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

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

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

[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**

New discipline: **specification based numerical analysis**

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

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

Symplecticity: the sum of projected areas is constant

An integrator which solves exactly **another mechanical system**

aka Verlet

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

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

\[x_1 = x_0 + ∆t\,f((x_0+x_1)/2)\]

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

2D planes replaced by **2D spheres**

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

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

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

**Symplectic system**

[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

[McLachlan, Modin, V. '14]

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

- “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)