Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems

This work continues a line of work on developing partially explicit methods for multiscale problems. In our previous works, we considered linear multiscale problems where the spatial heterogeneities are at the subgrid level and are not resolved. In these works, we have introduced contrast-independent, partially explicit time discretizations for linear equations. The contrast-independent, partially explicit time discretization divides the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Following this decomposition, temporal splitting was proposed, which treats fast components implicitly and slow components explicitly. The space decomposition and temporal splitting are chosen such that they guarantees stability, and we formulated a condition for the time stepping. This condition was formulated as a condition on slow spaces. In this paper, we extend this approach to nonlinear problems. We propose a splitting approach and derive a condition that guarantees stability. This condition requires some type of contrast-independent spaces for slow components of the solution. We present numerical results and show that the proposed methods provide results similar to implicit methods with a time step that is independent of the contrast.

Soil moisture retrieval under wheat crop using RISAT-1 hybrid polarimetric SAR data

The study aims to retrieve soil moisture from RISAT-1 hybrid polarimetric SAR data. Although the use of linear polarimetric SAR data has been well understood and documented, but hybrid polarimetric SAR data is grossly under explored and under reported for this purpose. Regression analysis has been carried to develop soil moisture retrieval models and validated the same. The retrieval models have been developed from back scattering coefficients (RH & RV) and m- space decomposition parameters (even bounce, odd bounce, and volume component) generated from RISAT-1 hybrid polarimetric SAR data. A total of three models are analyzed in this work, (i) using both RH &RV, (ii) volume component, and (iii) using even bounce, odd bounce and volume component. The study results showed that the model using m- decomposition derived parameters can provide better accuracy with R2 and RMSE of 0.92 and 2.45 per cent respectively in comparison to other two models.

Space decomposition based on visible objects in an indoor environment

When a person moves, the set of objects in their visual range changes. Hence, the set of objects perceived from a specific range of locations may be considered as a signature (possibly non-unique) of this region and be used for the localization of this person. In the case of fixed objects, the number of regions with a set of specific visible objects is limited. A verbal description containing references to elements of this set of visible objects could then be used to localize a person in the space. This paper proposes an approach for decomposing a space into regions that are characterized by such sets of visible objects. In our approach, at least a portion of partial surfaces of an object must be visible (beyond single points) to make part of the signature. Our method calculates two-dimensional visibility polygons for a portion of an object’s surface. Overlaying these polygons, we partition the space in regions of visibility signatures. The approach has been implemented, and we demonstrate how to represent space by qualitative locations using these visibility signatures. We further show how this representation can be used to locate a person within a space by a set of visible objects.

Decomposition Space Theory and the Bing Shrinking Criterion

‘Decomposition Space Theory and the Bing Shrinking Criterion’ gives a proof of the central Bing shrinking criterion and then provides an introduction to the key notions of the field of decomposition space theory. The chapter begins by proving the Bing shrinking criterion, which characterizes when a given map between compact metric spaces is approximable by homeomorphisms. Next, it develops the elements of the theory of decomposition spaces. A key fact is that a decomposition space associated with an upper semi-continuous decomposition of a compact metric space is again a compact metric space. Decomposition spaces are key in the proof of the disc embedding theorem.