Abstract. This paper presents regularity results and associated high-order numerical methods for one-dimensional Fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight ω times a "regular" unknown, a characterization of the regularity of solutions is obtained in terms of the smoothness of the corresponding right-hand sides. In particular, for right-hand sides which are analytic in a Bernstein Ellipse, analyticity in the same Bernstein Ellipse is obtained for the "regular" unknown. Moreover, a sharp Sobolev regularity result is presented which completely characterizes the co-domain of the Fractional-Laplacian operator in terms of certain weighted Sobolev spaces introduced in (Babuška and Guo, SIAM J. Numer. Anal. 2002). The present theoretical treatment relies on a full eigendecomposition for a certain weighted integral operator in terms of the Gegenbauer polynomial basis. The proposed Gegenbauer-based Nyström numerical method for the Fractional-Laplacian Dirichlet problem, further, is significantly more accurate and efficient than other algorithms considered previously. The sharp error estimates presented in this paper indicate that the proposed algorithm is spectrally accurate, with convergence rates that only depend on the smoothness of the right-hand side. In particular, convergence is exponentially fast (resp. faster than any power of the mesh-size) for analytic (resp. infinitely smooth) right-hand sides. The properties of the algorithm are illustrated with a variety of numerical results.

Soil moisture retrieval from SAR images is always affected by speckle noise and uncertainities associated to soil parameters, which impact negatively on the accuracy of soil moisture estimates. In this paper a Bayesian model is proposed to address these issues. A soil moisture Bayesian estimator from polarimetric SAR images is presented. This estimator is based on a set of stochastic equations for the polarimetric soil backscattering coefficients, which naturally includes models for the soil scattering, the speckle and the soil spatial heterogeneity. Since it is a Bayesian estimator, it may extensively use a priori information about soil condition, enhancing the performance of the retrieval. The Oh model is used as scattering model, although it presents a limiting range of validity for retrieving. After fully stating the mathematical modeling, numerical simulations are presented. First, traditional minimization-based retrieval using Oh model is investigated. The Bayesian retrieval scheme is then compared with Oh's retrieval. The results indicate that Bayesian model enlarge the validity region of Oh's retrieval. Moreover, as speckle effects are reduced by multilooking, Bayesian retrieval approachs to Oh's retrieval. On the other hand, an improvement in the accuracy of the retrieval is achieved by using a precise prior when speckle effects are large. The proposed algorithm can be applied to investigate which are the optimum parameters regarding multi-loking process and prior information required to perform a precise retrieval in a given soil type/condition.

Shifted equivalent sources and FFT acceleration for periodic scattering problems, including Wood anomalies,