The recursive Hessian sketch for adaptive filtering

We introduce in this paper the recursive Hessian sketch, a new adaptive filtering algorithm based on sketching the same exponentially weighted least squares problem solved by the recursive least squares algorithm. The algorithm maintains a number of sketches of the inverse autocorrelation matrix and recursively updates them at random intervals. These are in turn used to update the unknown filter estimate. The complexity of the proposed algorithm compares favorably to that of recursive least squares. The convergence properties of this algorithm are studied through extensive numerical experiments. With an appropriate choice or parameters, its convergence speed falls between that of least mean squares and recursive least squares adaptive filters, with less computations than the latter

Approximate Methods for State-Space Models

State-space models provide an important body of techniques for analyzing time-series, but their use requires estimating unobserved states. The optimal estimate of the state is its conditional expectation given the observation histories, and computing this expectation is hard when there are nonlinearities. Existing filtering methods, including sequential Monte Carlo, tend to be either inaccurate or slow. In this paper, we study a nonlinear filter for nonlinear/non-Gaussian state-space models, which uses Laplace's method, an asymptotic series expansion, to approximate the state's conditional mean and variance, together with a Gaussian conditional distribution. This {\em Laplace-Gaussian filter} (LGF) gives fast, recursive, deterministic state estimates, with an error which is set by the stochastic characteristics of the model and is, we show, stable over time. We illustrate the estimation ability of the LGF by applying it to the problem of neural decoding and compare it to sequential Monte Carlo both in simulations and with real data. We find that the LGF can deliver superior results in a small fraction of the computing time.Comment: 31 pages, 4 figures. Different pagination from journal version due to incompatible style files but same content; the supplemental file for the journal appears here as appendices B--E

Discriminative Bayesian filtering lends momentum to the stochastic Newton method for minimizing log-convex functions

To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective's gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. We establish matrix-based conditions under which the effect of older observations diminishes over time, in a manner analogous to Polyak's heavy ball momentum. We illustrate various aspects of our approach with an example and review other relevant innovations for the stochastic Newton method

Geometry Modeling for Unstructured Mesh Adaptation

The quantification and control of discretization error is critical to obtaining reliable simulation results. Adaptive mesh techniques have the potential to automate discretization error control, but have made limited impact on production analysis workflow. Recent progress has matured a number of independent implementations of flow solvers, error estimation methods, and anisotropic mesh adaptation mechanics. However, the poor integration of initial mesh generation and adaptive mesh mechanics to typical sources of geometry has hindered adoption of adaptive mesh techniques, where these geometries are often created in Mechanical Computer- Aided Design (MCAD) systems. The difficulty of this coupling is compounded by two factors: the inherent complexity of the model (e.g., large range of scales, bodies in proximity, details not required for analysis) and unintended geometry construction artifacts (e.g., translation, uneven parameterization, degeneracy, self-intersection, sliver faces, gaps, large tolerances be- tween topological elements, local high curvature to enforce continuity). Manual preparation of geometry is commonly employed to enable fixed-grid and adaptive-grid workflows by reducing the severity and negative impacts of these construction artifacts, but manual process interaction inhibits workflow automation. Techniques to permit the use of complex geometry models and reduce the impact of geometry construction artifacts on unstructured grid workflows are models from the AIAA Sonic Boom and High Lift Prediction are shown to demonstrate the utility of the current approach

CMOS-3D smart imager architectures for feature detection

This paper reports a multi-layered smart image sensor architecture for feature extraction based on detection of interest points. The architecture is conceived for 3-D integrated circuit technologies consisting of two layers (tiers) plus memory. The top tier includes sensing and processing circuitry aimed to perform Gaussian filtering and generate Gaussian pyramids in fully concurrent way. The circuitry in this tier operates in mixed-signal domain. It embeds in-pixel correlated double sampling, a switched-capacitor network for Gaussian pyramid generation, analog memories and a comparator for in-pixel analog-to-digital conversion. This tier can be further split into two for improved resolution; one containing the sensors and another containing a capacitor per sensor plus the mixed-signal processing circuitry. Regarding the bottom tier, it embeds digital circuitry entitled for the calculation of Harris, Hessian, and difference-of-Gaussian detectors. The overall system can hence be configured by the user to detect interest points by using the algorithm out of these three better suited to practical applications. The paper describes the different kind of algorithms featured and the circuitry employed at top and bottom tiers. The Gaussian pyramid is implemented with a switched-capacitor network in less than 50 μs, outperforming more conventional solutions.Xunta de Galicia 10PXIB206037PRMinisterio de Ciencia e Innovación TEC2009-12686, IPT-2011-1625-430000Office of Naval Research N00014111031

Adaptive ensemble PTV

Ensemble particle tracking velocimetry (EPTV) is a method to extract high-resolution statistical information on flow fields from particle image velocimetry (PIV) images. The process is based on tracking particles and extracting the velocity probability distribution functions of the image ensemble in averaging-regions deemed to contain a sufficient number of particle pairs/tracks. The size of the averaging regions depends on the particle density and the number of snapshots. An automatic adaptive variation of the ensemble PTV is presented to further push the spatial resolution of the method. The proposed adaptive-EPTV is based on stretching and orienting the averaging regions along the direction of maximum curvature of the velocity fields. The process requires a predictor calculation with isotropic-window EPTV to compute the second derivatives of the mean velocity components. In a second step, the principal directions of the Hessian tensor are calculated to tune the optimal orientation and stretch of the averaging regions. The stretching and orientation are achieved using a Gaussian windowing with different standard deviation along the local principal direction of the Hessian tensor. The algorithm is first validated using three different synthetic datasets: a sinusoidal displacement field, a channel flow and the flow around a NACA 0012 airfoil. An experimental test case of an impinging jet equipped with a fractal grid at the nozzle outlet is also carried out

Sequential Monte Carlo Methods for System Identification

One of the key challenges in identifying nonlinear and possibly non-Gaussian state space models (SSMs) is the intractability of estimating the system state. Sequential Monte Carlo (SMC) methods, such as the particle filter (introduced more than two decades ago), provide numerical solutions to the nonlinear state estimation problems arising in SSMs. When combined with additional identification techniques, these algorithms provide solid solutions to the nonlinear system identification problem. We describe two general strategies for creating such combinations and discuss why SMC is a natural tool for implementing these strategies.Comment: In proceedings of the 17th IFAC Symposium on System Identification (SYSID). Added cover pag

Dual-State Kalman Filter Forecasting and Control Theory Applications for Proactive Ramp Metering

Deterioration of freeway traffic flow condition due to bottlenecks can be ameliorated with ramp metering. A challenge in ramp metering is that it is not possible to process data in real-time and use the output in a control algorithm. This is due to the fact that by the time processing is completed and a control measure applied, the traffic state will have changed. A solution to this problem is to forecast the traffic state and implement a control measure based on the forecast. A dual-state Kalman filter was used to forecast traffic data at two locations on a freeway (I-84). A Kalman filter is an optimal recursive data processing algorithm; predictions are based on only the previous time-step’s prediction and all previous data do not need to be stored and reprocessed with new measurements. A coordinated feedback ramp metering control logic was implemented. The closed-loop system seeks to control the traffic density on the mainline while minimizing on-ramp queues through weighting functions. The integration of the Kalman filter with the ramp meter control logic accomplishes the ramp meter algorithmic scheme in which is proactive to changes in freeway conditions by controlling a forecasted state. In this closed-loop framework, real-time forecasts are produced with a continuously updated prediction that minimizes errors and recursively improves with each successive measurement. MATLAB was used to model the closed-loop control system as well as modify the input output constraints to evaluate and tune controller performance

A SLAM Algorithm Based on Adaptive Cubature Kalman Filter

We need to predict mathematical model of the system and a priori knowledge of the noise statistics when traditional simultaneous localization and mapping (SLAM) solutions are used. However, in many practical applications, prior statistics of the noise are unknown or time-varying, which will lead to large estimation errors or even cause divergence. In order to solve the above problem, an innovative cubature Kalman filter-based SLAM (CKF-SLAM) algorithm based on an adaptive cubature Kalman filter (ACKF) was established in this paper. The novel algorithm estimates the statistical parameters of the unknown system noise by introducing the Sage-Husa noise statistic estimator. Combining the advantages of the CKF-SLAM and the adaptive estimator, the new ACKF-SLAM algorithm can reduce the state estimated error significantly and improve the navigation accuracy of the SLAM system effectively. The performance of this new algorithm has been examined through numerical simulations in different scenarios. The results have shown that the position error can be effectively reduced with the new adaptive CKF-SLAM algorithm. Compared with other traditional SLAM methods, the accuracy of the nonlinear SLAM system is significantly improved. It verifies that the proposed ACKF-SLAM algorithm is valid and feasible