Scalable Hierarchical Multilevel Sampling of Lognormal Fields Conditioned on Measured Data

We explore the problem of drawing posterior samples from a lognormal permeability field conditioned by noisy measurements at discrete locations. The underlying unconditioned samples are based on a scalable PDE-sampling technique that shows better scalability for large problems than the traditional Karhunen-Loeve sampling, while still allowing for consistent samples to be drawn on a hierarchy of spatial scales. Lognormal random fields produced in this scalable and hierarchical way are then conditioned to measured data by a randomized maximum likelihood approach to draw from a Bayesian posterior distribution. The algorithm to draw from the posterior distribution can be shown to be equivalent to a PDE-constrained optimization problem, which allows for some efficient computational solution techniques. Numerical results demonstrate the efficiency of the proposed methods. In particular, we are able to match statistics for a simple flow problem on the fine grid with high accuracy and at much lower cost on a scale of coarser grids.

Improvements to the use of the Trajectory-Adaptive Multilevel Sampling algorithm for the study of rare events

The Trajectory-Adaptive Multilevel Sampling (TAMS) is a promising method to determine probabilities of noise-induced transition in multi-stable high-dimensional dynamical systems. In this paper, we focus on two improvements of the current algorithm related to (i) the choice of the target set and (ii) the formulation of the score function. In particular, we use confidence ellipsoids determined from linearised dynamics in the choice of the target set. Furthermore, we define a score function based on empirical transition paths computed at relatively high noise levels. The suggested new TAMS method is applied to two typical problems illustrating the benefits of the modifications.

Improvements to the use of the Trajectory-Adaptive Multilevel Sampling algorithm for the study of rare events

The Trajectory-Adaptive Multilevel Sampling (TAMS) is a promising method to determine probabilities of noise induced transition in multi-stable high-dimensional dynamical systems. In this paper, we focus on two improvements of the current algorithm related to (i) the choice of the target set and (ii) the formulation of the score function. In particular, we use confidence ellipsoids determined from linearized dynamics in the choice of the target set. Furthermore, we define a score function based on empirical transition paths computed at relatively high noise levels. The suggested new TAMS method is applied to two typical problems illustrating the benefits of the modifications.

Random Spatial and Systematic Random Sampling Approach to Development Survey Data: Evidence from Field Application in Malawi

Implementing development surveys in developing countries can be challenging. Limited time, high survey costs, lack of information, and technical difficulties are some of the general constraints that plague development researchers. These constraints can hinder data collection and introduce selection bias into the survey data. We outline a multilevel sampling approach for use in areas where comprehensive information on geographical or household characteristics of local population are not readily available. Our approach includes the use of geographical information systems (GIS) for random spatial sampling and personal digital assistants (PDAs) with a global positioning system (GPS) for household systematic random sampling with random walk. Evidence from our field application in Malawi show that the multilevel sampling approach yields relevant survey data which is comparable to historical and nationally representative values; and supports rapid aggregation of preliminary results after the survey. This multilevel design is cost-effective in implementation and reduces bias avenues in the household selection. Overall, this multilevel sampling approach can be used to generate survey data in developing countries where detailed geographical information and household characteristics data are not readily available. It also presents ways of reducing bias in survey data given budget constraints.