Klas Modin

Erik Jansson, Jonathan Krook, Klas Modin et al.

We address recovery of the three-dimensional backbone structure of single polypeptide proteins from single-particle cryo-electron microscopy (Cryo-SPA) data. Cryo-SPA produces noisy tomographic projections of electrostatic potentials of macromolecules. From these projections, we use methods from shape analysis to recover the three-dimensional backbone structure. Thus, we view the reconstruction problem as an indirect matching problem, where a point cloud representation of the protein backbone is deformed to match 2D tomography data. The deformations are obtained via the action of a matrix Lie group. By selecting a deformation energy, the optimality conditions are obtained, which lead to computational algorithms for optimal deformations. We showcase our approach on synthetic data, for which we recover the three-dimensional structure of the backbone.

Robert I McLachlan, Klas Modin, Hans Munthe-Kaas et al.

Butcher series appear when Runge-Kutta methods for ordinary differential equations are expanded in power series of the step size parameter. Each term in a Butcher series consists of a weighted elementary differential, and the set of all such differentials is isomorphic to the set of rooted trees, as noted by Cayley in the mid 19th century. A century later Butcher discovered that rooted trees can also be used to obtain the order conditions of Runge-Kutta methods, and he found a natural group structure, today known as the Butcher group. It is now known that many numerical methods also can be expanded in Butcher series; these are called B-series methods. A long-standing problem has been to characterize, in terms of qualitative features, all B-series methods. Here we tell the story of Butcher series, stretching from the early work of Cayley, to modern developments and connections to abstract algebra, and finally to the resolution of the characterization problem. This resolution introduces geometric tools and perspectives to an area traditionally explored using analysis and combinatorics.

Martin Bauer, Sarang Joshi, Klas Modin

In this article we explore an algorithm for diffeomorphic random sampling of nonuniform probability distributions on Riemannian manifolds. The algorithm is based on optimal information transport (OIT)---an analogue of optimal mass transport (OMT). Our framework uses the deep geometric connections between the Fisher-Rao metric on the space of probability densities and the right-invariant information metric on the group of diffeomorphisms. The resulting sampling algorithm is a promising alternative to OMT, in particular as our formulation is semi-explicit, free of the nonlinear Monge--Ampere equation. Compared to Markov Chain Monte Carlo methods, we expect our algorithm to stand up well when a large number of samples from a low dimensional nonuniform distribution is needed.

Klas Modin

The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher--Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry---the differential geometric approach to statistics. The Riemannian structures restrict to the submanifold of multivariate Gaussian distributions, where they induce Riemannian metrics on the space of covariance matrices. Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher--Rao geometries are discussed. The link to matrices is obtained by considering OMT and information geometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the $QR$, Cholesky, spectral, and singular value decompositions. We also give a coherent description of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples. The paper is a combination of previously known and original results. As a survey it covers the Riemannian geometry of OMT and polar decompositions (smooth and linear category), entropy gradient flows, and the Fisher--Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions. The original contributions include new gradient flows associated with various matrix decompositions, new geometric interpretations of previously studied isospectral flows, and a new proof of the polar decomposition of matrices based an entropy gradient flow.

Klas Modin, Gustaf Söderlind

Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter $ε$, and the schemes under study preserve the symplectic structure in the case $ε=0$. In the case $0<ε\ll 1$ the energy dissipation rate is shown to be asymptotically correct by backward error analysis. Theoretical results on monotone decrease of the modified Hamiltonian function for small enough step sizes are given. Further, an analysis proving near conservation of relative equilibria for small enough step sizes is conducted. Numerical examples, verifying the analyses, are given for a planar pendulum and an elastic 3--D pendulum. The results are superior in comparison with a conventional explicit Runge-Kutta method of the same order.

Stig Larsson, Takayasu Matsuo, Klas Modin et al.

The aim of this paper is the derivation of structure preserving schemes for the solution of the EPDiff equation, with particular emphasis on the two dimensional case. We develop three different schemes based on the Discrete Variational Derivative Method (DVDM) on a rectangular domain discretized with a regular, structured, orthogonal grid. We present numerical experiments to support our claims: we investigate the preservation of energy and linear momenta, the reversibility, and the empirical convergence of the schemes. The quality of our schemes is finally tested by simulating the interaction of singular wave fronts.

James Benn, Stephen Marsland, Robert I McLachlan et al.

The nonlinear spaces of shapes (unparameterized immersed curves or submanifolds) are of interest for many applications in image analysis, such as the identification of shapes that are similar modulo the action of some group. In this paper we study a general representation of shapes that is based on linear spaces and is suitable for numerical discretization, being robust to noise. We develop the theory of currents for shape spaces by considering both the analytic and numerical aspects of the problem. In particular, we study the analytical properties of the current map and the $H^{-s}$ norm that it induces on shapes. We determine the conditions under which the current determines the shape. We then provide a finite element discretization of the currents that is a practical computational tool for shapes. Finally, we demonstrate this approach on a variety of examples.

Klas Modin, Olivier Verdier

We study a family of nonholonomic mechanical systems. These systems consist of harmonic oscillators coupled through nonholonomic constraints. In particular, the family includes the so called contact oscillator, which has been used as a test problem for numerical methods for nonholonomic mechanics. Furthermore, the systems under study constitute simple models for continuously variable transmission gearboxes. The main result is that each system in the family is integrable reversible with respect to the canonical reversibility map on the cotangent bundle. This result may explain previous numerical observations, that some discretisations of the contact oscillator have favourable structure preserving properties.

Robert I McLachlan, Klas Modin, Olivier Verdier et al.

We show that symplectic Runge-Kutta methods provide effective symplectic integrators for Hamiltonian systems with index one constraints. These include the Hamiltonian description of variational problems subject to position and velocity constraints nondegenerate in the velocities, such as those arising in sub-Riemannian geometry and control theory.

Geir Bogfjellmo, Klas Modin, Olivier Verdier

Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, require the solution of a coupled set of non-linear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De~Casteljau's algorithm, which leads to generalized Bézier curves. To construct C2-splines from such curves is a complicated non-linear problem, until now lacking numerical methods. Here we provide an iterative algorithm for C2-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel, thus suitable for multi-core implementation. We demonstrate the algorithm for three geometries of interest: the $n$-sphere, complex projective space, and the real Grassmannian.

Paolo Cifani, Klas Modin, Cecilia Pagliantini et al.

We compare three approaches for structure preserving numerical integration of isospectral flows on quadratic Lie algebras. Such flows originate from Hamiltonian dynamics on the cotangent bundle of the Lie group. It is known, via discrete reduction theory, that symplectic Runge--Kutta methods applied to the cotangent bundle formulation induce isospectral symplectic Runge--Kutta (ISOSYRK) schemes on the Lie algebra. Here, we show that the same symplectic Runge--Kutta method, but applied to the transport formulation of the flow on the Lie group, is equivalent to the corresponding ISOSYRK scheme. We also give numerical results suggesting that the formulation on the Lie group is more efficient for schemes with two or more intermediate stages.

Klas Modin

Geometric integration of non-autonomous classical engineering problems, such as rotor dynamics, is investigated. It is shown, both numerically and by backward error analysis, that geometric (structure preserving) integration algorithms are superior to conventional Runge-Kutta methods.

Caleb Rottman, Martin Bauer, Klas Modin et al.

In this article we study the problem of thoracic image registration, in particular the estimation of complex anatomical deformations associated with the breathing cycle. Using the intimate link between the Riemannian geometry of the space of diffeomorphisms and the space of densities, we develop an image registration framework that incorporates both the fundamental law of conservation of mass as well as spatially varying tissue compressibility properties. By exploiting the geometrical structure, the resulting algorithm is computationally efficient, yet widely general.

Robert I. McLachlan, Klas Modin, Olivier Verdier

Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space $(S^2)^n$. In this paper we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features, that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on $T^*\mathbf{R}^{2n}$ for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with Kähler geometry, this provides another geometric proof of symplecticity.

Robert I. McLachlan, Klas Modin, Hans Munthe-Kaas et al.

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 \emph{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 \emph{affine equivariant}, meaning it respects all affine maps between affine spaces.

Robert I. McLachlan, Klas Modin, Olivier Verdier

We present a symplectic integrator, based on the canonical midpoint rule, for classical spin systems in which each spin is a unit vector in $\mathbb{R}^3$. Unlike splitting methods, it is defined for all Hamiltonians, and is $O(3)$-equivariant. It is a rare example of a generating function for symplectic maps of a noncanonical phase space. It yields an integrable discretization of the reduced motion of a free rigid body.

Robert I. McLachlan, Klas Modin, Olivier Verdier et al.

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.