Klas Modin

B-series methods are exactly the affine equivariant methods

Butcher series, also called B-series, are a type of expansion, fundamental in the analysis of numerical integration. Numerical methods that can be expanded in B-series are defined in all dimensions, so they correspond to sequences of maps—one map for each dimension. A long-standing problem has been to characterise those sequences of maps that arise from B-series. This problem is solved here: we prove that a sequence of smooth maps between vector fields on affine spaces has a B-series expansion if and only if it is affine equivariant, meaning it respects all affine maps between affine spaces.

Diffeomorphic Density Matching by Optimal Information Transport

We address the following problem: given two smooth densities on a manifold, find an optimal diffeomorphism that transforms one density into the other. Our framework builds on connections between the Fisher–Rao information metric on the space of probability densities and right-invariant metrics on the infinite-dimensional manifold of diffeomorphisms. This optimal information transport, and modifications thereof, allow us to construct numerical algorithms for density matching. The algorithms are inherently more efficient than those based on optimal mass transport or diffeomorphic registration. Our methods have applications in medical image registration, texture mapping, image morphing, nonuniform random sampling, and mesh adaptivity. Some of these applications are illustrated in examples.

Geometric Generalisations of SHAKE and RATTLE

A geometric analysis of the shake and rattle methods for constrained Hamiltonian problems is carried out. The study reveals the underlying differential geometric foundation of the two methods, and the exact relation between them. In addition, the geometric insight naturally generalises shake and rattle to allow for a strictly larger class of constrained Hamiltonian systems than in the classical setting.In order for shake and rattle to be well defined, two basic assumptions are needed. First, a nondegeneracy assumption, which is a condition on the Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy assumption, which is a condition on the geometry of the constrained phase space. Non-trivial examples of systems fulfilling, and failing to fulfill, these assumptions are given.

Symplectic integrators for spin systems

We present a symplectic integrator, based on the implicit midpoint method, for classical spin systems where each spin is a unit vector in R{3}. Unlike splitting methods, it is defined for all Hamiltonians and is O(3)-equivariant, i.e., coordinate-independent. It is a rare example of a generating function for symplectic maps of a noncanonical phase space. It yields a new integrable discretization of the spinning top.

Collective Lie-Poisson integrators on R3

We develop Lie-Poisson integrators for general Hamiltonian systems on

Integrability of Nonholonomically Coupled Oscillators

\mathbf{R}^{3}

Collective symplectic integrators

equipped with the rigid body bracket. The method uses symplectic realisation of

Geodesic Warps by Conformal Mappings

\mathbf{R}^{3}

Geometry of matrix decompositions seen through optimal transport and information geometry

on

Collective Lie-Poisson integrators on

T^{*}\mathbf{R}^{2}