Wind field reconstruction with adaptive random Fourier features

We investigate the use of spatial interpolation methods for reconstructing the horizontal near-surface wind field given a sparse set of measurements. In particular, random Fourier features is compared with a set of benchmark methods including kriging and inverse distance weighting. Random Fourier features is a linear model β ( x ) = ∑ k = 1 K β k e i ω k x approximating the velocity field, with randomly sampled frequencies ω k and amplitudes β k trained to minimize a loss function. We include a physically motivated divergence penalty | ∇ ⋅ β ( x ) | 2 , as well as a penalty on the Sobolev norm of β . We derive a bound on the generalization error and a sampling density that minimizes the bound. We then devise an adaptive Metropolis–Hastings algorithm for sampling the frequencies of the optimal distribution. In our experiments, our random Fourier features model outperforms the benchmark models.

Walking with robot-generated haptic forces in a virtual environment: a new approach to analyze lower limb coordination

Background Walking with a haptic tensile force applied to the hand in a virtual environment (VE) can induce adaptation effects in both chronic stroke and non-stroke individuals. These effects are reflected in spatiotemporal outcomes such as gait speed. However, the concurrent kinematic changes occurring in bilateral lower limb coordination have yet to be explored. Methods Chronic stroke participants were stratified based on overground gait speed into lower functioning (LF < 0.8 m/s, N = 7) and higher functioning (HF ≥ 0.8 m/s, N = 7) subgroups. These subgroups and an age-matched control group (N = 14, CG) walked on a self-paced treadmill in a VE with either robot-generated haptic leash forces delivered to the hand and then released or with an instrumented cane. Walking in both leash (10 and 15 N) and cane conditions were compared to pre-force baseline values to evaluate changes in lower limb coordination outcomes. Results All groups showed some kinematic changes in thigh, leg and foot segments when gait speed increased during force and post-force leash as well as cane walking. These changes were also reflected in intersegmental coordination and 3D phase diagrams, which illustrated increased intersegmental trajectory areas (p < 0.05) and angular velocity. These increases could also be observed when the paretic leg transitions from stance to swing phases while walking with the haptic leash. The Sobolev norm values accounted for both angular position and angular velocity, providing a single value for potentially quantifying bilateral (i.e. non-paretic vs paretic) coordination during walking. These values tended to increase (p < 0.05) proportionally for both limbs during force and post-force epochs as gait speed tended to increase. Conclusions Individuals with chronic stroke who increased their gait speed when walking with tensile haptic forces and immediately after force removal, also displayed moderate concurrent changes in lower limb intersegmental coordination patterns in terms of angular displacement and velocity. Similar results were also seen with cane walking. Although symmetry was less affected, these findings appear favourable to the functional recovery of gait. Both the use of 3D phase diagrams and assigning Sobolev norm values are potentially effective for detecting and quantifying these coordination changes.

A look on some results about Camassa–Holm type equations

We present an overview of some contributions of the author regarding Camassa–Holm type equations. We show that an equation unifying both Camassa–Holm and Novikov equations can be derived using the invariance under certain suitable scaling, conservation of the Sobolev norm and existence of peakon solutions. Qualitative analysis of the two-peakon dynamics is given.

Prescribed mean curvature equation on torus

Prescribed mean curvature problems on the torus have been considered in one dimension. In this paper, we prove the existence of a graph on the n-dimensional torus 𝕋 n {\mathbb{T}^{n}} , the mean curvature vector of which equals the normal component of a given vector field satisfying suitable conditions for a Sobolev norm, the integrated value, and monotonicity.

Sobolev–Kantorovich inequalities under CD(0,∞) condition

We refine and generalize several interpolation inequalities bounding the [Formula: see text] norm of a probability density with respect to the reference measure [Formula: see text] by its Sobolev norm and the Kantorovich distance to [Formula: see text] on a smooth weighted Riemannian manifold satisfying [Formula: see text] condition.

The Convergence of the Legendre–Galerkin Spectral Method for Constructing Atmospheric Acoustic Normal Modes

The asymptotic rate of convergence of the Legendre–Galerkin spectral approximation to an atmospheric acoustic eigenvalue problem is established, as the dimension of the approximating subspace approaches infinity. Convergence is in the [Formula: see text] Sobolev norm and is based on the existing theory [F. Chatelin, Spectral Approximations of Linear Operators (SIAM, 2011)]. The assumption is made that the eigenvalues are simple. Numerical results that help interpret the theory are presented. Eigenvalues corresponding to acoustic modes with smaller [Formula: see text] norms are especially accurately approximated, even with lower dimensioned basis sets of Legendre polynomials. The deficiencies in the potential applications of the theoretical results are noted in connection with the numerical examples.

Sharp estimates of the geometric rigidity on the first Heisenberg group

We prove quantitative stability of isometries on the first Heisenberg group with sub-Riemannian geometry: every (1 + )-quasi-isometry of the John domain of the Heisenberg group is close to some isometry with order of closeness in the uniform norm and with the order of closeness+ in the Sobolev norm. An example demonstrating the asymptotic sharpness of the results is given.