The success of symplectic integrators for Hamiltonian ODEs has led to a decades-long program of research seeking analogously structure-preserving numerical methods for Hamiltonian PDEs. In this paper, we construct a large class of such methods by combining finite element exterior calculus (FEEC) for spatial semidiscretization with symplectic integrators for time discretization. The resulting methods satisfy a local multisymplectic conservation law in space and time, which generalizes the symplectic conservation law of Hamiltonian ODEs, and which carries finer information about Hamiltonian structure than other approaches based on global function spaces. We give particular attention to conforming FEEC methods and hybridizable discontinuous Galerkin (HDG) methods. The theory and methods are illustrated by application to the semilinear Hodge wave equation.

We consider the application of finite element exterior calculus (FEEC) methods to a class of canonical Hamiltonian PDE systems involving differential forms. Solutions to these systems satisfy a local multisymplectic conservation law, which generalizes the more familiar symplectic conservation law for Hamiltonian systems of ODEs, and which is connected with physically-important reciprocity phenomena, such as Lorentz reciprocity in electromagnetics. We characterize hybrid FEEC methods whose numerical traces satisfy a version of the multisymplectic conservation law, and we apply this characterization to several specific classes of FEEC methods, including conforming Arnold-Falk-Winther-type methods and various hybridizable discontinuous Galerkin (HDG) methods. Interestingly, the HDG-type and other nonconforming methods are shown, in general, to be multisymplectic in a stronger sense than the conforming FEEC methods. This substantially generalizes previous work of McLachlan and Stern [Found. Comput. Math., 20 (2020), pp. 35-69] on the more restricted class of canonical Hamiltonian PDEs in the de Donder-Weyl "grad-div" form.

This paper examines functional equivariance, recently introduced by McLachlan and Stern [Found. Comput. Math. (2022)], from the perspective of backward error analysis. We characterize the evolution of certain classes of observables (especially affine and quadratic) by structure-preserving numerical integrators in terms of their modified vector fields. Several results on invariant preservation and symplecticity of modified vector fields are thereby generalized to describe the numerical evolution of non-invariant observables.

We develop a new, coordinate-free formulation of Hamiltonian mechanics on the dual of a Lie algebroid. Our approach uses a connection, rather than coordinates in a local trivialization, to obtain global expressions for the horizontal and vertical dynamics. We show that these dynamics can be obtained in two equivalent ways: (1) using the canonical Lie–Poisson structure, expressed in terms of the connection; or (2) using a novel variational principle that generalizes Hamilton’s phase space principle.

We hybridize the methods of finite element exterior calculus for the Hodge-Laplace problem on differential $k$-forms in $\mathbb{R}^n$. In the cases $k = 0$ and $k = n$, we recover well-known primal and mixed hybrid methods for the scalar Poisson equation, while for $0 < k < n$, we obtain new hybrid finite element methods, including methods for the vector Poisson equation in $n = 2$ and $n = 3$ dimensions. We also generalize Stenberg postprocessing from $k = n$ to arbitrary $k$, proving new superconvergence estimates. Finally, we discuss how this hybridization framework may be extended to include nonconforming and hybridizable discontinuous Galerkin methods.

We introduce a nonconforming hybrid finite element method for the two-dimensional vector Laplacian, based on a primal variational principle for which conforming methods are known to be inconsistent. Consistency is ensured using penalty terms similar to those used to stabilize hybridizable discontinuous Galerkin (HDG) methods, with a carefully chosen penalty parameter due to Brenner, Li, and Sung [Math. Comp., 76 (2007), pp. 573-595]. Our method accommodates elements of arbitrarily high order and, like HDG methods, it may be implemented efficiently using static condensation. The lowest-order case recovers the $P_1$-nonconforming method of Brenner, Cui, Li, and Sung [Numer. Math., 109 (2008), pp. 509-533], and we show that higher-order convergence is achieved under appropriate regularity assumptions. The analysis makes novel use of a family of weighted Sobolev spaces, due to Kondrat'ev, for domains admitting corner singularities.

This paper is motivated by a practical problem: many U.S. states have public hearings on "communities of interest" as part of their redistricting process, but no state has as yet adopted a concrete method of spatializing and aggregating community maps in order to take them into account in the drawing of new boundaries for electoral districts. Below, we describe a year-long project that collected and synthesized thousands of community maps through partnerships with grassroots organizations and/or government offices. The submissions were then aggregated by geographical clustering with a modified Hausdorff distance; then, the text from the narrative submissions was classified with semantic labels so that short runs of a Markov chain could be used to form semantic sub-clusters. The resulting dataset is publicly available, including the raw data of submitted community maps as well as post-processed community clusters and a scoring system for measuring how well districting plans respect the clusters. We provide a discussion of the strengths and weaknesses of this methodology and conclude with proposed directions for future work.

Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy evolution equations expressing important properties of the system. For example, a time-evolution PDE may have an observable that satisfies a local conservation law, such as the multisymplectic conservation law for Hamiltonian PDEs. We introduce the concept of functional equivariance , a natural sense in which a numerical integrator may preserve the dynamics satisfied by certain classes of observables, whether or not they are invariant. After developing the general framework, we use it to obtain results on methods preserving local conservation laws in PDEs. In particular, integrators preserving quadratic invariants also preserve local conservation laws for quadratic observables, and symplectic integrators are multisymplectic.

We investigate, both analytically and numerically, dispersive fractalisation and quantisation of solutions to periodic linear and nonlinear Fermi–Pasta–Ulam–Tsingou systems. When subject to periodic boundary conditions and discontinuous initial conditions, e.g., a step function, both the linearised and nonlinear continuum models for FPUT exhibit fractal solution profiles at irrational times (as determined by the coefficients and the length of the interval) and quantised profiles (piecewise constant or perturbations thereof) at rational times. We observe a similar effect in the linearised FPUT chain at timestwhere these models have validity, namelyt= O(h−2), wherehis proportional to the intermass spacing or, equivalently, the reciprocal of the number of masses. For nonlinear periodic FPUT systems, our numerical results suggest a somewhat similar behaviour in the presence of small nonlinearities, which disappears as the nonlinear force increases in magnitude. However, these phenomena are manifested on very long time intervals, posing a severe challenge for numerical integration as the number of masses increases. Even with the high-order splitting methods used here, our numerical investigations are limited to nonlinear FPUT chains with a smaller number of masses than would be needed to resolve this question unambiguously.

INTRODUCTION:Machine learning models were used to discover novel disease trajectories for autosomal dominant Alzheimer's disease. METHODS:Longitudinal structural magnetic resonance imaging, amyloid positron emission tomography (PET), and fluorodeoxyglucose PET were acquired in 131 mutation carriers and 74 non-carriers from the Dominantly Inherited Alzheimer Network; the groups were matched for age, education, sex, and apolipoprotein ε4 (APOE ε4). A deep neural network was trained to predict disease progression for each modality. Relief algorithms identified the strongest predictors of mutation status. RESULTS:The Relief algorithm identified the caudate, cingulate, and precuneus as the strongest predictors among all modalities. The model yielded accurate results for predicting future Pittsburgh compound B (R2  = 0.95), fluorodeoxyglucose (R2  = 0.93), and atrophy (R2  = 0.95) in mutation carriers compared to non-carriers. DISCUSSION:Results suggest a sigmoidal trajectory for amyloid, a biphasic response for metabolism, and a gradual decrease in volume, with disease progression primarily in subcortical, middle frontal, and posterior parietal regions.

For all n, we define the n-dimensional critical catenoid $$M_n$$ to be the unique rotationally symmetric, free boundary minimal hypersurface of non-trivial topology embedded in the closed unit ball in $$\mathbb {R}^{n+1}$$ . We show that the Morse index $$\mathrm{MI}(n)$$ of $$M_n$$ satisfies the following asymptotic estimate as n tends to infinity. $$\begin{aligned} \mathop {{\text {Lim}}}_{n\rightarrow +\infty }\frac{\text {Log}(\mathrm{MI}(n))}{\sqrt{n}\text {Log}(\sqrt{n})} = 1. \end{aligned}$$ We illustrate our results with an in-depth study of the numerical problem, providing exact values for the Morse index for $$n=2,\ldots ,100$$ , together with qualitative studies of $$\mathrm{MI}(n)$$ and related geometric quantities for large values of n.

The Yang-Mills equations generalize Maxwell's equations to nonabelian gauge groups, and a quantity analogous to charge is locally conserved by the nonlinear time evolution. Christiansen and Winther observed that, in the nonabelian case, the Galerkin method with Lie algebra-valued finite element differential forms appears to conserve charge globally but not locally, not even in a weak sense. We introduce a new hybridization of this method, give an alternative expression for the numerical charge in terms of the hybrid variables, and show that a local, per-element charge conservation law automatically holds.

Motivated by the Hodgkin-Huxley model of neuronal dynamics, we study explicit numerical integrators for "conditionally linear" systems of ordinary differential equations. We show that splitting and composition methods, when applied to the Van der Pol oscillator and to the Hodgkin-Huxley model, do a better job of preserving limit cycles of these systems for large time steps, compared with the "Euler-type" methods (including Euler's method, exponential Euler, and semi-implicit Euler) commonly used in computational neuroscience, with no increase in computational cost. These limit cycles are important to preserve, due to their role in neuronal spiking. Splitting methods even compare favorably to the explicit exponential midpoint method, which is twice as expensive per step. The second-order Strang splitting method is seen to perform especially well across a range of non-stiff and stiff dynamics.

Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant connections on algebroids. This has fundamental consequences for the theory of numerical integrators, giving a characterization of the spaces on which Butcher and Lie--Butcher series methods, which generalize Runge--Kutta methods, may be applied.

In this paper, we prove necessary and sufficient conditions for a hybridizable discontinuous Galerkin method to satisfy a multisymplectic conservation law, when applied to a canonical Hamiltonian system of partial differential equations. We show that these conditions are satisfied by the “hybridized” versions of several of the most commonly used finite element methods, including mixed, nonconforming, and discontinuous Galerkin methods. (Interestingly, for the continuous Galerkin method in dimension greater than one, we show that multisymplecticity only holds in a weaker sense.) Consequently, these general-purpose finite element methods may be used for structure-preserving discretization (or semidiscretization) of canonical Hamiltonian systems of ODEs or PDEs. This establishes multisymplecticity for a large class of arbitrarily high-order methods on unstructured meshes.

Maxwell’s equations describe the evolution of electromagnetic fields, together with constraints on the divergence of the magnetic and electric flux densities. These constraints correspond to fundamental physical laws: the nonexistence of magnetic monopoles and the conservation of charge, respectively. However, one or both of these constraints may be violated when one applies a finite element method to discretize in space. This is a well-known and long-standing problem in computational electromagnetics. We use domain decomposition to construct a family of primal hybrid finite element methods for Maxwell’s equations, where the Lagrange multipliers are shown to correspond to a numerical trace of the magnetic field and a numerical flux of the electric flux density. Expressing the charge conservation constraint in terms of this numerical flux, we show that both constraints are strongly preserved. As a special case, these methods include a hybridized version of Nédélec’s method, implying that it preserves the constraints more strongly than previously recognized. These constraint-preserving properties are illustrated using numerical experiments in both the time domain and frequency domain. In 2-D, we also observe a superconvergence phenomenon, where hybrid post-processing yields an improved estimate of the magnetic field.

We show that a cooperative game may be decomposed into a sum of component games, one for each player, using the combinatorial Hodge decomposition on a graph. This decomposition is shown to satisfy certain efficiency, null-player, symmetry, and linearity properties. Consequently, we obtain a new characterization of the classical Shapley value as the value of the grand coalition in each player's component game. We also relate this decomposition to a least-squares problem involving inessential games (in a similar spirit to previous work on least-squares and minimum-norm solution concepts) and to the graph Laplacian. Finally, we generalize this approach to games with weights and/or constraints on coalition formation.

A computational procedure is developed for determining the conversion probability for reaction-diffusion systems in which a first-order catalytic reaction is performed over active particles. We apply this general method to systems on metric graphs, which may be viewed as 1-dimensional approximations of 3-dimensional systems, and obtain explicit formulas for conversion. We then study numerically a class of 3-dimensional systems and test how accurately they are described by model formulas obtained for metric graphs. The optimal arrangement of active particles in a 1-dimensional multiparticle system is found, which is shown to depend on the level of catalytic activity: conversion is maximized for low catalytic activity when all particles are bunched together close to the point of gas injection, and for high catalytic activity when the particles are evenly spaced.

This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases, this theory includes not only Lagrangian reduction (including reduction by stages) for Lie group actions, but also classical Routh reduction, which we show is naturally posed in this fibered setting. Along the way, we also develop some new results for Lagrangian mechanics on Lie algebroids, most notably a new, coordinate-free formulation of the equations of motion. Finally, we extend the foregoing to include fibered and Lie algebroid generalizations of the Hamilton-Pontryagin principle of Yoshimura and Marsden, along with the associated reduction theory.

This paper adapts the techniques of finite element exterior calculus to study and discretize the abstract Hodge--Dirac operator, which is a square root of the abstract Hodge--Laplace operator considered by Arnold, Falk, and Winther [Bull. Amer. Math. Soc., 47 (2010), pp. 281--354]. Dirac-type operators are central to the field of Clifford analysis, where recently there has been considerable interest in their discretization. We prove a priori stability and convergence estimates, and show that several of the results in finite element exterior calculus can be recovered as corollaries of these new estimates. (A corrected version is attached.)