Probing Many-Body Systems Near Spectral Degeneracies

The diagonal elements of the time correlation matrix are used to probe closed quantum systems that are measured at random times. This enables us to extract two distinct parts of the quantum evolution, a recurrent part and an exponentially decaying part. This separation is strongly affected when spectral degeneracies occur, for instance, in the presence of spontaneous symmetry breaking. Moreover, the slowest decay rate is determined by the smallest energy level spacing, and this decay rate diverges at the spectral degeneracies. Probing the quantum evolution with the diagonal elements of the time correlation matrix is discussed as a general concept and tested in the case of a bosonic Josephson junction. It reveals for the latter characteristic properties at the transition to Hilbert-space localization.

Criteria for the description of the financial behavior model of households in the digitalization conditions

The crisis caused by the COVID-19 pandemic has highlighted the need to update the criteria for describing the financial behavior of households. The purpose of the article is to substantiate the criteria for describing the financial behavior model of households in a specific period and in terms of the economic space localization of the regions. It is emphasized in the article that a set of criteria of the economic, psychological, demographic, sociological, and social nature should be used to describe the financial behavior model of households. It is substantiated that in the conditions of the COVID-19 virus spread, the criteria of the digitalization and localization of economic space of the regions should be added to the set of criteria for this model description. Prospects for further research determine the application of the criteria presented in the article in the economic-mathematical model of the description of the financial behavior transformation of households in terms of digitalization and spread of the COVID-19 virus.

The Maximum Likelihood Ensemble Filter with State Space Localization

New method for ensemble data assimilation that incorporates state space covariance localization, global numerical optimization, and implied Bayesian inference, is presented. The method is referred to as the MLEF with State Space Localization (MLEF-SSL) due to its similarity with the Maximum Likelihood Ensemble Filter (MLEF). One of the novelties introduced in MLEF-SSL is the calculation of a reduced-rank localized forecast error covariance using random projection. The Hessian preconditioning is accomplished via Cholesky decomposition of the Hessian matrix, accompanied with solving triangular system of equations instead of directly inverting matrices. For ensemble update the MLEF-SSL system employs resampling of posterior perturbations. The MLEF-SSL was applied to Lorenz model II and compared to Ensemble Kalman Filter with state space localization and to MLEF with observation space localization. The observations include linear and nonlinear observation operators, each applied to integrated and point observations. Results indicate improved performance of MLEF-SSL, particularly in assimilation of integrated nonlinear observations. Resampling of posterior perturbations for ensemble update also indicates a satisfactory performance. Additional experiments were conducted to examine the sensitivity of the method to the rank of random matrix and to compare it to truncated eigenvectors of the localization matrix. The two methods are comparable in application to low-dimensional Lorenz model, except that the new method outperforms the truncated eigenvector method in case of severe rank reduction. The random basis method is simple to implement and may be more promising for realistic high-dimensional applications.

A Multiscale Local Gain Form Ensemble Transform Kalman Filter (MLGETKF)

A new multiscale, ensemble-based data assimilation (DA) method, MLGETKF (Multiscale Local Gain Form Ensemble Transform Kalman Filter), is introduced. MLGETKF allows simultaneous update of multiple scales for both the mean and ensemble perturbations through assimilating all observations at once. MLGETKF performs DA in independent local volumes, which lends the algorithm a high degree of computational scalability. The multiscale analysis is enabled through the rapid creation of many pseudo ensemble perturbations via a multiscale ensemble modulation procedure. The Kalman gain that is used to update the raw background ensemble mean and perturbations is based on this modulated ensemble, which intrinsically includes multi-scale model space localization.Experiments with a non-cycled statistical model show that the full background covariance estimated by MLGETKF more accurately resembles the shape of the true covariance than a scale-unaware localization. The mean analysis from the best-performing MLGETKF is statistically significantly more accurate than the best performing scale unaware LGETKF. The accuracy of the MLGETKF analysis is more sensitive to small-scale band localization radius than large-scale band. MLGETKF is further examined in a cycling DA context with a Surface Quasi-Geostrophic model. The root-mean-square potential temperature analysis error of the best performing MLGETKF is 17.2% lower than that of the best-performing LGETKF. MLGETKF reduces analysis errors measured in kinetic energy spectra space by 30-80% relative to LGETKF with the largest improvement at large scales. MLGETKF deterministic and ensemble mean forecasts are more accurate than LGETKF for full and large scales up to 5-6 day lead-time and for small scales up to 3-4 day lead-time, gaining 12-hour ~ 1-day of predictability.

Development of a nonlinear ensemble data assimilation method with global state-space covariance localization

<p>High-dimensional ensemble data assimilation applications require error covariance localization in order to address the problem of insufficient degrees of freedom, typically accomplished using the observation-space covariance localization. However, this creates a challenge for vertically integrated observations, such as satellite radiances, aerosol optical depth, etc., since the exact observation location in vertical does not exist. For nonlinear problems, there is an implied inconsistency in iterative minimization due to using observation-space localization which effectively prevents finding the optimal global minimizing solution. Using state-space localization, however, in principal resolves both issues associated with observation space localization.</p><p> </p><p>In this work we present a new nonlinear ensemble data assimilation method that employs covariance localization in state space and finds an optimal analysis solution. The new method resembles “modified ensembles” in the sense that ensemble size is increased in the analysis, but it differs in methodology used to create ensemble modifications, calculate the analysis error covariance, and define the initial ensemble perturbations for data assimilation cycling. From a practical point of view, the new method is considerably more efficient and potentially applicable to realistic high-dimensional data assimilation problems. A distinct characteristic of the new algorithm is that the localized error covariance and minimization are global, i.e. explicitly defined over all state points. The presentation will focus on examining feasible options for estimating the analysis error covariance and for defining the initial ensemble perturbations.</p>