We develop Riemannian Stein Variational Gradient Descent (RSVGD), a Bayesian inference method that generalizes Stein Variational Gradient Descent (SVGD) to Riemann manifold. The benefits are two-folds: (i) for inference tasks in Euclidean spaces, RSVGD has the advantage over SVGD of utilizing information geometry, and (ii) for inference tasks on Riemann manifolds, RSVGD brings the unique advantages of SVGD to the Riemannian world. To appropriately transfer to Riemann manifolds, we conceive novel and non-trivial techniques for RSVGD, which are required by the intrinsically different characteristics of general Riemann manifolds from Euclidean spaces. We also discover Riemannian Stein's Identity and Riemannian Kernelized Stein Discrepancy. Experimental results show the advantages over SVGD of exploring distribution geometry and the advantages of particle-efficiency, iteration-effectiveness and approximation flexibility over other inference methods on Riemann manifolds.

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We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity,is the delta function supported at the origin. In the L 2 supercritical setting p > 3, we construct self-similar blow-up solutions belonging to the energy space L ∞x ∩Ḣ 1 x . This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic Weber functions and solving the jump condition at x = 0 imposed by the δ term in (0.1). This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case 0 < p − 3 1 using the log Binet formula for the gamma function, and contour deformation and stationary phase/Laplace method in the integral formulae for the parabolic Weber functions.

Input–output analysis of transitional channel flows has proven to be a valuable analytical tool for identifying important flow structures and energetic motions. The traditional approach abstracts the nonlinear terms as forcing that is unstructured, in the sense that this forcing is not directly tied to the underlying nonlinearity in the dynamics. This paper instead employs a structured-singular-value-based approach that preserves certain input–output properties of the nonlinear forcing function in an effort to recover the larger range of key flow features identified through nonlinear analysis, experiments and direct numerical simulation (DNS) of transitional channel flows. Application of this method to transitional plane Couette and plane Poiseuille flows leads to not only the identification of the streamwise coherent structures predicted through traditional input–output approaches, but also the characterization of the oblique flow structures as those requiring the least energy to induce transition, in agreement with DNS studies, and nonlinear optimal perturbation analysis. The proposed approach also captures the recently observed oblique turbulent bands that have been linked to transition in experiments and DNS with very large channel size. The ability to identify the larger amplification of the streamwise varying structures predicted from DNS and nonlinear analysis in both flow regimes suggests that the structured approach allows one to maintain the nonlinear effects associated with weakening of the lift-up mechanism, which is known to dominate the linear operator. Capturing this key nonlinear effect enables the prediction of a wider range of known transitional flow structures within the analytical input–output modelling paradigm.

Sixteen American-English vowels including 12 monothongs and 4 diphthongs were recorded in a phonetic context of /hVd/ from young English (E), Chinese (C), and Korean (K) talkers. The Chinese and Korean talkers were bilingual and had stayed in United States up to 6 years. Two sets of experiments will be discussed: acoustic analysis and intelligibility of English vowels produced by the three groups of talkers. Results of acoustic analysis showed that there was no significant difference in F1×F2 vowel space among the three groups of talkers. In addition, the three groups of talkers showed great similarity in F2/F1 ratio across the 12 monothongs. Vowel durations had significantly greater variability across vowel categories for the Chinese and Korean talkers than for the English talkers, indicating that, besides producing spectral differences among vowels, Chinese and Korean talkers also attempted to generate durational difference among vowels to make each vowel distinguishable from others. More acoustic features such as spectral tilt and formant transition in the diphthongs and the effects of acoustic features on vowel perception by native English listeners will be discussed. Furthermore, the relationship between the vowel intelligibility and the second language experience of non-native talkers will be examined.