Reduced Rank Square Root Research Articles

Computational reduction for noninvasive transmural electrophysiological imaging

Noninvasive transmural electrophysiological imaging (TEPI) combines body-surface electrocardiograms and image-derived anatomic data to compute subject-specific electrical activity and the relevant diseased substrates deep into the ventricular myocardium. Based on the Bayesian estimation where the priors come from probabilistic simulations of high dimensional EP models, TEPI engages intensive computation that hinders its clinical translation. We present a reduced-rank square-root (RRSR) algorithm for TEPI that reduces computational time by neglecting minor components of estimation uncertainty and improves numerical stability by the square-root structure. Phantom and real-data experiments demonstrate the ability of RRSR-TEPI to bring notable computational reduction without significant sacrifice of diagnostic efficacy, particularly in imaging and quantifying post-infarct substrates.

A Hybrid Kalman Filter Algorithm for Large-Scale Atmospheric Chemistry Data Assimilation

Abstract In the past, a number of algorithms have been introduced to solve data assimilation problems for large-scale applications. Here, several Kalman filters, coupled to the European Operational Smog (EUROS) atmospheric chemistry transport model, are used to estimate the ozone concentrations in the boundary layer above Europe. Two Kalman filter algorithms, the reduced-rank square root (RRSQRT) and the ensemble Kalman filter (EnKF), were implemented in a prior study. To combine the best features of these two filters, a hybrid filter was constructed by making use of the reduced-rank approximation of the covariance matrix as a variance reducer for the EnKF. This hybrid algorithm, complementary orthogonal subspace filter for efficient ensembles (COFFEE), is coupled to the EUROS model. The performance of all algorithms is compared in terms of residual errors and number of EUROS model evaluations. The COFFEE results score somewhere between the EnKF and RRSQRT results for less than approximately 30 model evaluations. For more than approximately 30 model evaluations, the COFFEE results are, in all cases, better than the EnKF and RRSQRT results. The results of the COFFEE simulations with more than about 60 model evaluations proved to be significantly better than all the EnKF and RRSQRT simulations (even better than those with 100 and 200 modes or ensemble members). The performance of both the EnKF- and RRSQRT-type filters is affected by the nonlinear properties of the model and observation operator, because both rely on linearization to some extent. To further study this aspect, several measures of nonlinearity were calculated and linked with the performance of these algorithms.

Data assimilation of ground‐level ozone in Europe with a Kalman filter and chemistry transport model

A Kalman filter coupled to the atmospheric chemistry transport model EUROS has been used to estimate the ozone concentrations in the boundary layer above Europe. Two Kalman filter algorithms, the reduced rank square root (RRSQRT) and the ensemble Kalman filter (ENKF), were implemented in this study. Both required, in general, a large number of EUROS model simulations for an assimilation. The observations consisted of hourly ozone data in a set of 135 ground‐based stations in Europe for the period, June 1996. Half of these stations were used for the assimilation and the other half only for validation of the results. The combination between data assimilation (Kalman filter) and the atmospheric chemistry transport model, EUROS, gave more accurate results for boundary layer ozone than the EUROS model or measurements used separately. The average difference between assimilated and measured ozone concentrations decreased from 27.4 to 20.5 μg m−3 for the average of the stations used for validation in Europe. Both algorithms tend to converge to about the same accuracy, with an increasing number of EUROS model runs. About 10–20 EUROS model calculations were found sufficient for a good assimilation. The results are supported by a number of simulations that also reveal a local character for the assimilation process.

On the construction of a reduced rank square-root Kalman filter for efficient uncertainty propagation

The Kalman filter is a sequential estimation procedure that combines a stochastic dynamical model with observations in order to update the model state and the associated uncertainty. In the situation where no measurements are available the filter works as an uncertainty propagator. The most computationally demanding part of the Kalman filter is to propagate the covariance through the dynamical system, which may be completely infeasible in high-dimensional models. The reduced rank square-root (RRSQRT) filter is a special formulation of the Kalman filter for large-scale applications. In this formulation, the covariance matrix of the model state is expressed in a limited number of modes M. In the classical implementation of the RRSQRT filter the computational costs of the truncation step grow very fast with the number of modes (> M 3). In this work, a new approach based on the Lanzcos algorithm is formulated. It provides a more cost-efficient scheme and includes a precision coefficient that can be tuned for specific applications depending on the trade-off between precision and computational load.

Dynamic perceptual grouping in schizophrenia: Effects of presentation context and learning

Noninvasive transmural electrophysiological imaging (TEPI) combines body-surface electrocardiograms and image-derived anatomic data to compute subject-specific electrical activity and the relevant diseased substrates deep into the ventricular myocardium. Based on the Bayesian estimation where the priors come from probabilistic simulations of high dimensional EP models, TEPI engages intensive computation that hinders its clinical translation. We present a reduced-rank square-root (RRSR) algorithm for TEPI that reduces computational time by neglecting minor components of estimation uncertainty and improves numerical stability by the square-root structure. Phantom and real-data experiments demonstrate the ability of RRSR-TEPI to bring notable computational reduction without significant sacrifice of diagnostic efficacy, particularly in imaging and quantifying post-infarct substrates.

Application of advanced data assimilation methods for the initialisation of storm surge models

A sequential data assimilation technique (utilising a reduced-rank square-root filter) has been incorporated into the complete non-linear equations of a hydrodynamical module of a standard two dimensional hydrodynamic modelling tool. The deterministic model output is corrected using the available data and the best model initial conditions are generated. The deterministic model is then used to generate the water level forecast. A description of the general technique that is applied and its adaptation to this particular problem are provided. The application to a real storm that occurred during February 1993 in the North sea is presented.