Multilevel Estimation of Normalization Constants Using the Ensemble Kalman-Bucy Filter

In this article we consider the application of multilevel Monte Carlo, for the estimation of normalizing constants. In particular we will make use of the filtering algorithm, the ensemble Kalman-Bucy filter (EnKBF), which is an N-particle representation of the Kalma-Bucy filter (KBF). The EnKBF is of interest as it coincides with the optimal filter in the continuous-linear setting, i.e. the KBF. This motivates our particular setup in the linear setting. The resulting methodology we will use is the multilevel ensemble Kalman-Bucy filter (MLEnKBF). We provide an analysis based on deriving Lq-bounds for the normalizing constants using both the single-level, and the multilevel algorithms. Our results will be highlighted through numerical results, where we firstly demonstrate the error-to-cost rates of the MLEnKBF comparing it to the EnKBF on a linear Gaussian model. Our analysis will be specific to one variant of the MLEnKBF, whereas the numerics will be tested on different variants. We also exploit this methodology for parameter estimation, where we test this on the models arising in atmospheric sciences, such as the stochastic Lorenz 63 and 96 model.Comment: 33 pages, 21 figure

On the Incorporation of Box-Constraints for Ensemble Kalman Inversion

The Bayesian approach to inverse problems is widely used in practice to infer unknown parameters from noisy observations. In this framework, the ensemble Kalman inversion has been successfully applied for the quantification of uncertainties in various areas of applications. In recent years, a complete analysis of the method has been developed for linear inverse problems adopting an optimization viewpoint. However, many applications require the incorporation of additional constraints on the parameters, e.g. arising due to physical constraints. We propose a new variant of the ensemble Kalman inversion to include box constraints on the unknown parameters motivated by the theory of projected preconditioned gradient flows. Based on the continuous time limit of the constrained ensemble Kalman inversion, we discuss a complete convergence analysis for linear forward problems. We adopt techniques from filtering which are crucial in order to improve the performance and establish a correct descent, such as variance inflation. These benefits are highlighted through a number of numerical examples on various inverse problems based on partial differential equations

A Statistical Framework and Analysis for Perfect Radar Pulse Compression

Perfect radar pulse compression coding is a potential emerging field which aims at providing rigorous analysis and fundamental limit radar experiments. It is based on finding non-trivial pulse codes, which we can make statistically equivalent, to the radar experiments carried out with elementary pulses of some shape. A common engineering-based radar experiment design, regarding pulse-compression, often omits the rigorous theory and mathematical limitations. In this work our aim is to develop a mathematical theory which coincides with understanding the radar experiment in terms of the theory of comparison of statistical experiments. We review and generalize some properties of the It\^{o} measure. We estimate the unknown i.e. the structure function in the context of Bayesian statistical inverse problems. We study the posterior for generalized $d$-dimensional inverse problems, where we consider both real-valued and complex-valued inputs for posteriori analysis. Finally this is then extended to the infinite dimensional setting, where our analysis suggests the underlying posterior is non-Gaussian

Expression of the helix-loop-helix protein, Id, during branching morphogenesis in the kidney

Expression of the helix-loop-helix protein, Id, during branching morphogenesis in the kidney. Id, a member of the helix-loop-helix protein family, is an inhibitor of transcriptional activation by basic-helix-loop-helix proteins. In the developing mouse kidney, Id mRNA was observed as early as 12.5 days post-coitum (dpc) specifically in the condensed mesenchyme surrounding the ureteric buds by in situ hybridization. At 14.5 dpc, Id mRNA was localized to the collecting tubules and developing glomeruli while the surrounding mesenchyme lacked Id hybridization. From birth to day 10 postnatal, Id mRNA is were localized in to the collecting tubules, immature glomeruli and renal pelvis. In the adult kidney, Id mRNA was detectable by Northern blot analysis but no cell type-specific localization was noted by in situ hybridization. These results indicate a role for HLH-bHLH proteins in the differentiation of the epithelial structures of the kidney

A Review of the EnKF for Parameter Estimation

The ensemble Kalman filter is a well-known and celebrated data assimilation algorithm. It is of particular relevance as it used for high-dimensional problems, by updating an ensemble of particles through a sample mean and covariance matrices. In this chapter we present a relatively recent topic which is the application of the EnKF to inverse problems, known as ensemble Kalman Inversion (EKI). EKI is used for parameter estimation, which can be viewed as a black-box optimizer for PDE-constrained inverse problems. We present in this chapter a review of the discussed methodology, while presenting emerging and new areas of research, where numerical experiments are provided on numerous interesting models arising in geosciences and numerical weather prediction