Efficient spatial modelling using the SPDE approach with bivariate splines

Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associated computational cost can be a bottleneck, especially in realistic applications. It has been shown that computational efficiency can be gained by doing the computations using Gaussian Markov random fields (GMRFs) as the GFs can be seen as weak solutions to corresponding stochastic partial differential equations (SPDEs) using piecewise linear finite elements. We introduce a new class of representations of GFs with bivariate splines instead of finite elements. This allows an easier implementation of piecewise polynomial representations of various degrees. It leads to GMRFs that can be inferred efficiently and can be easily extended to non-stationary fields. The solutions approximated with higher order bivariate splines converge faster, hence the computational cost can be alleviated. Numerical simulations using both real and simulated data also demonstrate that our framework increases the flexibility and efficiency.Comment: 26 pages, 7 figures and 3 table

Lookahead Strategies for Sequential Monte Carlo

Based on the principles of importance sampling and resampling, sequential Monte Carlo (SMC) encompasses a large set of powerful techniques dealing with complex stochastic dynamic systems. Many of these systems possess strong memory, with which future information can help sharpen the inference about the current state. By providing theoretical justification of several existing algorithms and introducing several new ones, we study systematically how to construct efficient SMC algorithms to take advantage of the "future" information without creating a substantially high computational burden. The main idea is to allow for lookahead in the Monte Carlo process so that future information can be utilized in weighting and generating Monte Carlo samples, or resampling from samples of the current state.Comment: Published in at http://dx.doi.org/10.1214/12-STS401 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Symmetry Reduction and Boundary Modes for Fe-Chains on an s-wave Superconductor

We investigate the superconducting phase diagram and boundary modes for a quasi-1D system formed by three Fe-Chains on an s-wave superconductor, motivated by the recent Princeton experiment. The $\vec l\cdot\vec s$ onsite spin-orbit term, inter-chain diagonal hopping couplings, and magnetic disorders in the Fe-chains are shown to be crucial for the superconducting phases, which can be topologically trivial or nontrivial in different parameter regimes. For the topological regime a single Majorana and multiple Andreew bound modes are obtained in the ends of the chain, while for the trivial phase only low-energy Andreev bound states survive. Nontrivial symmetry reduction mechanism induced by the $\vec l\cdot\vec s$ term, diagonal hopping couplings, and magnetic disorder is uncovered to interpret the present results. Our study also implies that the zero-bias peak observed in the recent experiment may or may not reflect the Majorana zero modes in the end of the Fe-chains.Comment: 5 pages, 4 figures; some minor errors are correcte