Well-posedness of the EPDiff equation with a pseudo-differential inertia operator

In this article we study the class of right-invariant, fractional order Sobolev-type metrics on groups of diffeomorphisms of a compact manifold M. Our main result concerns well-posedness properties for the corresponding Euler-Arnold equations, also called the EPDiff equations, which are of importance in mathematical physics and in the field of shape analysis and template registration. Depending on the order of the metric, we will prove both local and global well-posedness results for these equations. As a result of our analysis we will also obtain new commutator estimates for elliptic pseudo-differential operators

Optimized broad-histogram simulations for strong first-order phase transitions: Droplet transitions in the large-Q Potts model

The numerical simulation of strongly first-order phase transitions has remained a notoriously difficult problem even for classical systems due to the exponentially suppressed (thermal) equilibration in the vicinity of such a transition. In the absence of efficient update techniques, a common approach to improve equilibration in Monte Carlo simulations is to broaden the sampled statistical ensemble beyond the bimodal distribution of the canonical ensemble. Here we show how a recently developed feedback algorithm can systematically optimize such broad-histogram ensembles and significantly speed up equilibration in comparison with other extended ensemble techniques such as flat-histogram, multicanonical or Wang-Landau sampling. As a prototypical example of a strong first-order transition we simulate the two-dimensional Potts model with up to Q=250 different states on large systems. The optimized histogram develops a distinct multipeak structure, thereby resolving entropic barriers and their associated phase transitions in the phase coexistence region such as droplet nucleation and annihilation or droplet-strip transitions for systems with periodic boundary conditions. We characterize the efficiency of the optimized histogram sampling by measuring round-trip times tau(N,Q) across the phase transition for samples of size N spins. While we find power-law scaling of tau vs. N for small Q \lesssim 50 and N \lesssim 40^2, we observe a crossover to exponential scaling for larger Q. These results demonstrate that despite the ensemble optimization broad-histogram simulations cannot fully eliminate the supercritical slowing down at strongly first-order transitions.Comment: 11 pages, 12 figure

Dynamical mean-field theory for bosons

We discuss the recently developed bosonic dynamical mean-field (B-DMFT) framework, which maps a bosonic lattice model onto the selfconsistent solution of a bosonic impurity model with coupling to a reservoir of normal and condensed bosons. The effective impurity action is derived in several ways: (i) as an approximation to the kinetic energy functional of the lattice problem, (ii) using a cavity approach, and (iii) by using an effective medium approach based on adding a one-loop correction to the selfconsistently defined condensate. To solve the impurity problem, we use a continuous-time Monte Carlo algorithm based on a sampling of a perturbation expansion in the hybridization functions and the condensate wave function. As applications of the formalism we present finite temperature B-DMFT phase diagrams for the bosonic Hubbard model on a 3d cubic and 2d square lattice, the condensate order parameter as a function of chemical potential, critical exponents for the condensate, the approach to the weakly interacting Bose gas regime for weak repulsions, and the kinetic energy as a function of temperature.Comment: 26 pages, 19 figure

Digital phenotyping: Towards replicable findings with comprehensive assessments and integrative models in bipolar disorders

Background: Digital phenotyping promises to unobtrusively obtaining a continuous and objective input of symptomatology from patients\u27 daily lives. The prime example are bipolar disorders, as smartphone parameters directly reflect bipolar symptomatology. Empirical studies, however, have yielded inconsistent findings. We believe that three main shortcomings have to be addressed to fully leverage the potential of digital phenotyping: short assessment periods, rare outcome assessments, and an extreme fragmentation of parameters without an integrative analytical strategy. Methods: To demonstrate how to overcome these shortcomings, we conducted frequent (biweekly) dimensional and categorical expert ratings and daily self-ratings over an extensive assessment period (12 months) in 29 patients with bipolar disorder. Digital phenotypes were monitored continuously. As an integrative analytical strategy, we used structural equation modelling to build latent psychopathological outcomes (mania, depression) and latent digital phenotype predictors (sleep, activity, communicativeness). Outcomes: Combining gold-standard categorical expert ratings with dimensional self and expert ratings resulted in two latent outcomes (mania and depression) with statistically meaningful factor loadings that dynamically varied over 299 days. Latent digital phenotypes of sleep and activity were associated with same-day latent manic psychopathology, suggesting that psychopathological alterations in bipolar disorders relate to domains (latent variables of sleep and activity) and not only to specific behaviors (such as the number of declined incoming calls). The identification of latent psychopathological outcomes that dimensionally vary on a daily basis will enable to empirically determine which combination of digital phenotypes at which days prior to an upcoming episode are viable as digital prodromal predictors. (DIPF/Orig.

Digital phenotyping: towards replicable findings with comprehensive assessments and integrative models in bipolar disorders

Background: Digital phenotyping promises to unobtrusively obtaining a continuous and objective input of symptomatology from patients’ daily lives. The prime example are bipolar disorders, as smartphone parameters directly reflect bipolar symptomatology. Empirical studies, however, have yielded inconsistent findings. We believe that three main shortcomings have to be addressed to fully leverage the potential of digital phenotyping: short assessment periods, rare outcome assessments, and an extreme fragmentation of parameters without an integrative analytical strategy. Methods: To demonstrate how to overcome these shortcomings, we conducted frequent (biweekly) dimensional and categorical expert ratings and daily self-ratings over an extensive assessment period (12 months) in 29 patients with bipolar disorder. Digital phenotypes were monitored continuously. As an integrative analytical strategy, we used structural equation modelling to build latent psychopathological outcomes (mania, depression) and latent digital phenotype predictors (sleep, activity, communicativeness). Outcomes: Combining gold-standard categorical expert ratings with dimensional self and expert ratings resulted in two latent outcomes (mania and depression) with statistically meaningful factor loadings that dynamically varied over 299 days. Latent digital phenotypes of sleep and activity were associated with same-day latent manic psychopathology, suggesting that psychopathological alterations in bipolar disorders relate to domains (latent variables of sleep and activity) and not only to specific behaviors (such as the number of declined incoming calls). The identification of latent psychopathological outcomes that dimensionally vary on a daily basis will enable to empirically determine which combination of digital phenotypes at which days prior to an upcoming episode are viable as digital prodromal predictors

Aims and structure of the German Research Consortium BipoLife for the study of bipolar disorder

Background: Bipolar disorder is a severe and heterogeneous mental disorder. Despite great advances in neuroscience over the past decades, the precise causative mechanisms at the transmitter, cellular or network level have so far not been unraveled. As a result, individual treatment decisions cannot be tailor-made and the uncertain prognosis is based on clinical characteristics alone. Although a subpopulation of patients have an excellent response to pharmacological monotherapy, other subpopulations have been less well served by the medical system and therefore require more focused attention. In particular individuals at high risk of bipolar disorder, young patients in the early stages of bipolar disorder, patients with an unstable highly relapsing course and patients with acute suicidal ideation have been identified as those in need. Structure: A research consortium of ten universities across Germany has therefore implemented a 4 year research agenda including three randomized controlled trials, one epidemiological trial and one cross-sectional trial to address these areas of unmet needs. The topics under investigation will be the improvement of early recognition, specific psychotherapy, and smartphones as an aid for early episode detection and biomarkers of lithium response. A subset of patients will be investigated utilizing neuroimaging (fMRI), neurophysiology (EEG), and biomaterials (genomics, transcriptomics). Conclusions: This article aims to outline the rationale, design, and methods of these individual studies