Regularized Principal Component Analysis for Spatial Data

In many atmospheric and earth sciences, it is of interest to identify dominant spatial patterns of variation based on data observed at $p$ locations and $n$ time points with the possibility that $p>n$. While principal component analysis (PCA) is commonly applied to find the dominant patterns, the eigenimages produced from PCA may exhibit patterns that are too noisy to be physically meaningful when $p$ is large relative to $n$. To obtain more precise estimates of eigenimages, we propose a regularization approach incorporating smoothness and sparseness of eigenimages, while accounting for their orthogonality. Our method allows data taken at irregularly spaced or sparse locations. In addition, the resulting optimization problem can be solved using the alternating direction method of multipliers, which is easy to implement, and applicable to a large spatial dataset. Furthermore, the estimated eigenfunctions provide a natural basis for representing the underlying spatial process in a spatial random-effects model, from which spatial covariance function estimation and spatial prediction can be efficiently performed using a regularized fixed-rank kriging method. Finally, the effectiveness of the proposed method is demonstrated by several numerical example

Dynamics of drop impact on solid surfaces: evolution of impact force and self-similar spreading

We investigate the dynamics of drop impacts on dry solid surfaces. By synchronising high-speed photography with fast force sensing, we simultaneously measure the temporal evolution of the shape and impact force of impacting drops over a wide range of Reynolds numbers (Re). At high Re, when inertia dominates the impact processes, we show that the early-time evolution of impact force follows a square-root scaling, quantitatively agreeing with a recent self-similar theory. This observation provides direct experimental evidence on the existence of upward propagating self-similar pressure fields during the initial impact of liquid drops at high Re. When viscous forces gradually set in with decreasing Re, we analyse the early-time scaling of the impact force of viscous drops using a perturbation method. The analysis quantitatively matches our experiments and successfully predicts the trends of the maximum impact force and the associated peak time with decreasing Re. Furthermore, we discuss the influence of viscoelasticity on the temporal signature of impact forces. Last but not least, we also investigate the spreading of liquid drops at high Re following the initial impact. Particularly, we find an exact parameter-free self-similar solution for the inertia-driven drop spreading, which quantitatively predicts the height of spreading drops at high Re. The limit of the self-similar approach for drop spreading is also discussed. As such, our study provides a quantitative understanding of the temporal evolution of impact forces across the inertial, viscous and viscoelastic regimes and sheds new light on the self-similar dynamics of drop impact processes.Comment: 24 pages, 9 figures, accepted by Journal of Fluid Mechanic

Analyses of celestial pole offsets with VLBI, LLR, and optical observations

This work aims to explore the possibilities of determining the long-period part of the precession-nutation of the Earth with techniques other than very long baseline interferometry (VLBI). Lunar laser ranging (LLR) is chosen for its relatively high accuracy and long period. Results of previous studies could be updated using the latest data with generally higher quality, which would also add ten years to the total time span. Historical optical data are also analyzed for their rather long time-coverage to determine whether it is possible to improve the current Earth precession-nutation model

Engineered spin phase diagram of two interacting electrons in semiconductor nanowire quantum dots

Spin properties of two interacting electrons in a quantum dot (QD) embedded in a nanowire with controlled aspect ratio and longitudinal magnetic fields are investigated by using a configuration interaction (CI) method and exact diagonalization (ED) techniques. The developed CI theory based on a three-dimensional (3D) parabolic model provides explicit formulations of the Coulomb matrix elements and allows for straightforward and efficient numerical implementation. Our studies reveal fruitful features of spin singlet-triplet transitions of two electrons confined in a nanowire quantum dot (NWQD), as a consequence of the competing effects of geometry-controlled kinetic energy quantization, the various Coulomb interactions, and spin Zeeman energies. The developed theory is further employed to study the spin phase diagram of two quantum-confined electrons in the regime of "cross over" dimensionality, from quasi-two-dimensional (disk-like) QDs to finite one-dimensional (rod-like) QDs.Comment: 9 pages, 6 figure