This paper is devoted to the study of metric subregularity and strong subregularity of any positive order q for set-valued mappings in finite and infinite dimensions. While these notions have been studied and applied earlier for q = 1 and-to a much lesser extentfor q ∈ (0, 1), no results are available for the case q > 1. We derive characterizations of these notions for subgradient mappings, develop their sensitivity analysis under small perturbations, and provide applications to the convergence rate of Newton-type methods for solving generalized equations.Keywords Variational analysis · Metric subregularity and strong subregularity of higher order · Newton and quasi-Newton methods · Generalized normals and subdifferentials Mathematics Subject Classification 49J52 · 90C30 · 90C31

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In this paper, we present maximal and area integral characterizations of Bergman spaces in the unit ball of C n . The characterizations are in terms of maximal functions and area integral functions on Bergman balls involving the radial derivative, the complex gradient, and the invariant gradient. As an application, we obtain new maximal and area integral characterizations of Besov spaces. Moreover, we give an atomic decomposition of real-variable type with respect to Carleson tubes for Bergman spaces.

Linearized inversion algorithms are the main techniques in seismic imaging that apply the single-scattering (Born) approximation to the scattered field, and therefore, have difficulty handling the strong perturbation of model parameters and nonlinear multiple-scattering effects. To theoretically overcome these drawbacks in the linearization of the inverse scattering problem, we have developed an approach to deal with nonlinear double-scattering inversion. We first used an integral equation formulation associated with the scattered field consisting of single and double scattering in an acoustic earth model based on the second-order Born approximation, and we found that the approximation of the scattered field can be naturally related to the generalized Radon transform (GRT). We then adopted the inverse GRT to obtain the corresponding quadratic approximate inversion solution. Our inversion scheme can appropriately handle the first-order transmission effects from double scattering in a local area, which gives a significant amplitude correction for the inversion results and ultimately results in a more accurate image with true amplitude. We conducted numerical experiments that showed the conventional single-scattering inversion was good in amplitude only for perturbation up to 10% of the background medium, but our approach can work for up to 40% or more. Test results indicated that our inversion scheme can effectively relax the requirement of the weak perturbation of the model parameter in the Born approximation and can deal with the complex model directly. The computational complexity of our new scheme is almost the same as conventional linear scattering inversion processing. Therefore, the cost of our approach is at a similar level to that of linear scattering inversion.

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