Application of a local attractor dimension to reduced space strongly coupled data assimilation for chaotic multiscale systems

The basis and challenge of strongly coupled data assimilation (CDA) is the accurate representation of cross-domain covariances between various coupled subsystems with disparate spatio-temporal scales, where often one or more subsystems are unobserved. In this study, we explore strong CDA using ensemble Kalman filtering methods applied to a conceptual multiscale chaotic model consisting of three coupled Lorenz attractors. We introduce the use of the local attractor dimension (i.e. the Kaplan–Yorke dimension, dimKY) to prescribe the rank of the background covariance matrix which we construct using a variable number of weighted covariant Lyapunov vectors (CLVs). Specifically, we consider the ability to track the nonlinear trajectory of each of the subsystems with different variants of sparse observations, relying only on the cross-domain covariance to determine an accurate analysis for tracking the trajectory of the unobserved subdomain. We find that spanning the global unstable and neutral subspaces is not sufficient at times where the nonlinear dynamics and intermittent linear error growth along a stable direction combine. At such times a subset of the local stable subspace is also needed to be represented in the ensemble. In this regard the local dimKY provides an accurate estimate of the required rank. Additionally, we show that spanning the full space does not improve performance significantly relative to spanning only the subspace determined by the local dimension. Where weak coupling between subsystems leads to covariance collapse in one or more of the unobserved subsystems, we apply a novel modified Kalman gain where the background covariances are scaled by their Frobenius norm. This modified gain increases the magnitude of the innovations and the effective dimension of the unobserved domains relative to the strength of the coupling and timescale separation. We conclude with a discussion on the implications for higher-dimensional systems.

Stability Within $$T^2$$-Symmetric Expanding Spacetimes

We prove a nonpolarised analogue of the asymptotic characterisation of $$T^2$$T2-symmetric Einstein flow solutions completed recently by LeFloch and Smulevici. In this work, we impose a condition weaker than polarisation and so our result applies to a larger class. We obtain similar rates of decay for the normalised energy and associated quantities for this class. We describe numerical simulations which indicate that there is a locally attractive set for $$T^2$$T2-symmetric solutions not covered by our main theorem. This local attractor is distinct from the local attractor in our main theorem, thereby indicating that the polarised asymptotics are unstable.

Application of local attractor dimension to reduced space strongly coupled data assimilation for chaotic multiscale systems

The basis and challenge of strongly coupled data assimilation (CDA) is the accurate representation of cross-domain covariances between various coupled subsystems with disparate spatio-temporal scales, where often one or more subsystems are unobserved. In this study, we explore strong CDA using ensemble Kalman filtering methods applied to a conceptual multiscale chaotic model consisting of three coupled Lorenz attractors. We introduce the use of the local attractor dimension (i.e. the Kaplan-Yorke dimension, dimKY) to determine the rank of the background covariance matrix which we construct using a variable number of weighted covariant Lyapunov vectors (CLVs). Specifically, we consider the ability to track the nonlinear trajectory of each of the subsystems with different variants of sparse observations, relying only on the cross-domain covariance to determine an accurate analysis for tracking the trajectory of the unobserved subdomain. We find that spanning the global unstable and neutral subspaces is not sufficient at times where the nonlinear dynamics and intermittent linear error growth along a stable direction combine. At such times a subset of the local stable subspace is also needed to be represented in the ensemble. In this regard the local dimKY provides an accurate estimate of the required rank. Additionally, we show that spanning the full space does not improve performance significantly relative to spanning only the subspace determined by the local dimension. Where weak coupling between subsystems leads to covariance collapse in one or more of the unobserved subsystems, we apply a novel modified Kalman gain where the background covariances are scaled by their Frobenius norm. This modified gain increases the magnitude of the innovations and the effective dimension of the unobserved domains relative to the strength of the coupling and time-scale separation. We conclude with a discussion on the implications for higher dimensional systems.

Quantum-Behaved Particle Swarm Optimization with Weighted Mean Personal Best Position and Adaptive Local Attractor

Motivated by concepts in quantum mechanics and particle swarm optimization (PSO), quantum-behaved particle swarm optimization (QPSO) was proposed as a variant of PSO with better global search ability. In this paper, a QPSO with weighted mean personal best position and adaptive local attractor (ALA-QPSO) is proposed to simultaneously enhance the search performance of QPSO and acquire good global optimal ability. In ALA-QPSO, the weighted mean personal best position is obtained by distinguishing the difference of the effect of the particles with different fitness, and the adaptive local attractor is calculated using the sum of squares of deviations of the particles’ fitness values as the coefficient of the linear combination of the particle best known position and the entire swarm’s best known position. The proposed ALA-QPSO algorithm is tested on twelve benchmark functions, and compared with the basic Artificial Bee Colony and the other four QPSO variants. Experimental results show that ALA-QPSO performs better than those compared method in all of the benchmark functions in terms of better global search capability and faster convergence rate.

Initial conditions for inflation — A short review

I give a brief review of the status of research on the nature of initial conditions required to obtain a period of cosmological inflation. It is shown that there is good evidence that in the case of large field models, the inflationary slow-roll trajectory is a local attractor in initial condition space, whereas it is not in the case of small field models.