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Brad Bell

Senior Principal Mathematician



Research Interests

Optimization, Statistics, Numerical Methods, Software Engineering


Dr. Bell's graduate research was in optimization. Since then much of his research has concerned applications of optimization and numerical methods to data analysis. This includes procedures for properly weighting data from various sources, Kalman smoothing algorithms for the nonlinear case, and procedures for estimating signal arrival times.
He designed a compartmental modeling program that is used by researchers around the world to analyze the kinetics of drugs and naturally occurring compounds in humans and other animals. He is currently working on algorithms that estimate the variation between realizations of scientific models, with special emphasis on modeling pharmacokinetics. He joined APL-UW in 1978.


B.A. Mathematics and Physics, Saint Lawrence University, 1973

M.A. Mathematics, University of Washington, 1976

Ph.D. Mathematics, University of Washington, 1984


2000-present and while at APL-UW

The connection between Bayesian estimation of a Gaussian random field and RKHS

Aravkin, A.Y., B.M. Bell, J.V. Burke, and G. Pillonetto, "The connection between Bayesian estimation of a Gaussian random field and RKHS," IEEE Trans. Neural Networks Learn. Syst., 26, 15-18, doi:10.1109/TNNLS.2014.2337939, 2015.

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1 Jul 2015

Reconstruction of a function from noisy data is key in machine learning and is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution suitably balances adherence to the observed data and the corresponding RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian random field with covariance proportional to the kernel associated with the RKHS. In this brief, we provide a statistical interpretation when more general losses are used, such as absolute value, Vapnik or Huber. Specifically, for any finite set of sampling locations (that includes where the data were collected), the maximum a posteriori estimate for the signal samples is given by the RKHS estimate evaluated at the sampling locations. This connection establishes a firm statistical foundation for several stochastic approaches used to estimate unknown regularization parameters. To illustrate this, we develop a numerical scheme that implements a Bayesian estimator with an absolute value loss. This estimator is used to learn a function from measurements contaminated by outliers.

Distributed Kalman smoothing in static Bayesian networks

Pillonetto, G., B.M. Bell, and S. Del Favero, "Distributed Kalman smoothing in static Bayesian networks," Automatica, 49, 1001-1011, doi:10.1016/j.automatica.2013.01.016, 2013.

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1 Apr 2013

This paper considers the state-space smoothing problem in a distributed fashion. In the scenario of sensor networks, we assume that the nodes can be ordered in space and have access to noisy measurements relative to different but correlated states. The goal of each node is to obtain the minimum variance estimate of its own state conditional on all the data collected by the network using only local exchanges of information. We present a cooperative smoothing algorithm for Gauss–Markov linear models and provide an exact convergence analysis for the algorithm, also clarifying its advantages over the Jacobi algorithm. Extensions of the numerical scheme able to perform field estimation using a grid of sensors are also derived.

An l1-Laplace robust Kalman smoother

Aravkin, A., B. Bell, J. Burke, and G. Pillonetto, "An l1-Laplace robust Kalman smoother," IEEE Trans. Autom. Control, 56, 2898-2911, doi: 10.1109/TAC.2011.2141430, 2011.

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1 Dec 2011

Robustness is a major problem in Kalman filtering and smoothing that can be solved using heavy tailed distributions; e.g., l1-Laplace. This paper describes an algorithm for finding the maximum a posteriori (MAP) estimate of the Kalman smoother for a nonlinear model with Gaussian process noise and l1-Laplace observation noise. The algorithm uses the convex composite extension of the Gauss-Newton method. This yields convex programming subproblems to which an interior point path-following method is applied. The number of arithmetic operations required by the algorithm grows linearly with the number of time points because the algorithm preserves the underlying block tridiagonal structure of the Kalman smoother problem. Excellent fits are obtained with and without outliers, even though the outliers are simulated from distributions that are not l1-Laplace. It is also tested on actual data with a nonlinear measurement model for an underwater tracking experiment. The l1-Laplace smoother is able to construct a smoothed fit, without data removal, from data with very large outliers.

More Publications

Acoustics Air-Sea Interaction & Remote Sensing Center for Environmental & Information Systems Center for Industrial & Medical Ultrasound Electronic & Photonic Systems Ocean Engineering Ocean Physics Polar Science Center