Optimal transport in competition with reaction: the Hellinger-Kantorovich distance and geodesic curves

We discuss a new notion of distance on the space of finite and nonnegative measures which can be seen as a generalization of the well-known Kantorovich-Wasserstein distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and its properties. In fact the distance can be equivalently described by an optimal transport problem on the cone space over the underlying metric space. We give a construction of geodesic curves and discuss their properties

Gradient structures and geodesic convexity for reaction-diffusion systems

We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic \lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory. We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic \lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory

An evolutionary elastoplastic plate model derived via Gamma convergence

This paper is devoted to dimension reduction for linearized elastoplasticity in the rate-independent case. The reference configuration of the three-dimensional elastoplastic body has a two-dimensional middle surface and a positive but small thickness. Under suitable scalings we derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations (linear Kirchhoff--Love plate), which are coupled via plastic strains. We establish strong convergence of the solutions in the natural energy space. The analysis uses an abstract Gamma-convergence theory for rate-independent evolutionary systems that is based on the notion of energetic solutions. This concept is formulated via an energy-storage functional and a dissipation functional, such that energetic solutions are defined in terms of a stability condition and an energy balance. The Mosco convergence of the quadratic energy-storage functional follows the arguments of the elastic case. To handle the evolutionary situation the interplay with the dissipation functional is controlled by cancellation properties for Mosco-convergent quadratic energies

On microscopic origins of generalized gradient structures

Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary $\Gamma$-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.Comment: Keywords: Generalized gradient structure, gradient system, evolutionary \Gamma-convergence, energy-dissipation principle, variational evolution, relative entropy, large-deviation principl

Fine properties of geodesics and geodesic λ\lambda-convexity for the Hellinger-Kantorovich distance

We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger--Kantorovich problem (HK), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton--Jacobi equation arising in the dual dynamic formulation of HK, which are sufficiently strong to construct a characteristic transport-dilation flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of HK geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic $\lambda$-convexity with respect to the Hellinger--Kantorovich distance. Examples of geodesically convex functionals are provided.Comment: Hellinger-Kantorovich distance, regularity geodesic curves, optimality conditions for dual potentials, geodesic semiconvexit

Coexistence of Hamiltonian-like and dissipative dynamics in chains of coupled phase oscillators with skew-symmetric coupling

We consider rings of coupled phase oscillators with anisotropic coupling. When the coupling is skew-symmetric, i. e. when the anisotropy is balanced in a specific way, the system shows robustly a coexistence of Hamiltonian-like and dissipative regions in the phase space. We relate this phenomenon to the time-reversibility property of the system. The geometry of low-dimensional systems up to five oscillators is described in detail. In particular, we show that the boundary between the dissipative and Hamiltonian-like regions consists of families of heteroclinic connections. For larger chains with skew-symmetric coupling, some sufficient conditions for the coexistence are provided, and in the limit of N → ∞ oscillators, we formally derive an amplitude equation for solutions in the neighborhood of the synchronous solution. It has the form of a nonlinear Schrödinger equation and describes the Hamiltonian-like region existing around the synchronous state similarly to the case of finite rings

Spectrum and amplitude equations for scalar delay-differential equations with large delay

The subject of the paper are scalar delay-differential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delay-differential equations close to the destabilization threshold

Fine properties of geodesics and geodesic lambda-convexity for the Hellinger--Kantorovich distance

We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger--Kantorovich problem (HK), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton--Jacobi equation arising in the dual dynamic formulation of HK, which are sufficiently strong to construct a characteristic transport-dilation flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of HK geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic lambda-convexity with respect to the Hellinger--Kantorovich distance

Effects of an exercise programme for chronically ill and mobility-restricted elderly with structured support by the general practitioner's practice (HOMEfit) - study protocol of a randomised controlled trial

<p>Abstract</p> <p>Background</p> <p>Exercise programmes can be administered successfully as therapeutic agents to patients with a number of chronic diseases and help to improve physical functioning in older adults. Usually, such programmes target either healthy and mobile community-dwelling seniors or elderly individuals living in nursing institutions or special residences. Chronically ill or mobility-restricted individuals, however, are difficult to reach when they live in their own homes.</p> <p>A pilot study has shown good feasibility of a home-based exercise programme that is delivered to this target group through cooperation between general practitioners and exercise therapists. A logical next step involves evaluation of the effects of the programme.</p> <p>Methods/design</p> <p>The study is designed as a randomised controlled trial. We plan to recruit 210 patients (≥ 70 years) in about 15 general practices.</p> <p>The experimental intervention (duration 12 weeks)-a multidimensional home-based exercise programme-is delivered to the participant by an exercise therapist in counselling sessions at the general practitioner's practice and on the telephone. It is based on methods and strategies for facilitating behaviour change according to the Health Action Process Approach (HAPA). The control intervention-baseline physical activities-differs from the experimental intervention with regard to content of the counselling sessions as well as to content and frequency of the promoted activities.</p> <p>Primary outcome is functional lower body strength measured by the "chair-rise" test. Secondary outcomes are: physical function (battery of motor tests), physical activity (step count), health-related quality of life (SF-8), fall-related self-efficacy (FES-I), and exercise self-efficacy (SSA-Scale).</p> <p>The hypothesis that there will be differences between the two groups (experimental/control) with respect to post-interventional chair-rise time will be tested using an ANCOVA with chair-rise time at baseline, treatment group, and study centre effects as explanatory variables. Analysis of the data will be undertaken using the principle of intention-to-treat.</p> <p>Trial registration</p> <p>Current Controlled Trials <a href="http://www.controlled-trials.com/ISRCTN17727272">ISRCTN17727272</a>.</p

Recruiting Hard-to-Reach Subjects for Exercise Interventions: A Multi-Centre and Multi-Stage Approach Targeting General Practitioners and Their Community-Dwelling and Mobility-Limited Patients

The general practitioner (GP)’s practice appears to be an ideal venue for recruiting community-dwelling older adults with limited mobility. This study (Current Controlled Trials ISRCTN17727272) aimed at evaluating the recruiting process used for a multi-centre exercise intervention (HOMEfit). Each of six steps resulted in an absolute number of patients (N1–N6). Sex and age (for N4–N6) and reasons for dropping out were assessed. Patient database screening (N1–N3) at 15 GP practices yielded N1 = 5,990 patients aged 70 and above who had visited their GP within the past 6 months, N2 = 5,467 after exclusion of institutionalised patients, N3 = 1,545 patients eligible. Using a pre-defined limitation algorithm in order to conserve the practices’ resources resulted in N4 = 1,214 patients (80.3 ± 5.6 years, 68% female), who were then officially invited to the final assessment of eligibility at the GP’s practice. N5 = 434 patients (79.5 ± 5.4 years, 69% female) attended the practice screening (n = 13 of whom had not received an official invitation). Finally, N6 = 209 (79.8 ± 5.2 years, 74% female) were randomised after they were judged eligible and had given their written informed consent to participate in the randomised controlled trial (overall recruitment rate: 4.4%). The general strategy of utilising a GP’s practice to recruit the target group proved beneficial. The data and experiences presented here can help planners of future exercise-intervention studies