Large Dimensional Systems Research Articles

Forecasting Lattice and Point Spatial Data: Comparison of Unilateral and Multilateral SAR Models

Spatial auto-regressive (SAR) models are widely used in geosciences for data analysis; their main feature is the presence of weight (W) matrices, which define the neighboring relationships between the spatial units. The statistical properties of parameter and forecast estimates strongly depend on the structure of such matrices. The least squares (LS) method is the most flexible and can estimate systems of large dimensions; however, it is biased in the presence of multilateral (sparse) matrices. Instead, the unilateral specification of SAR models provides triangular weight matrices that allow consistent LS estimates and sequential prediction functions. These two properties are strictly related and depend on the linear and recursive nature of the system. In this paper, we show the better performance in out-of-sample forecasting of unilateral SAR (estimated with LS), compared to multilateral SAR (estimated with maximum likelihood, ML). This conclusion is supported by numerical simulations and applications to real geological data, both on regular lattices and irregularly distributed points.

Tam Bulanık Lineer Denklem Sistemleri için Bulanık Sayısal Simülasyon Tabanlı Sezgisel Bir Yöntem

In this paper, a new method is proposed to find the approximate solutions to fully fuzzy systems of linear equations (FFSLEs). The technique integrates a bisection method with Fuzzy Numerical Simulation (FNS). The procedure starts with generating single values of fuzzy parameters and solving the resulting crisp problems repeatedly to determine the lower and upper bounds of the solutions. After computing the mean lower and upper bound values, the obtained supremum and infimum values are considered to be the lower and upper bounds of the solutions, respectively. It is attempted to improve solutions by considering an error function related to the sum of the absolute differences between the corresponding lower and upper bounds of the left and right sides of the equalities. When very large intervals are obtained for the solutions, the bisection algorithm is applied to reduce the error value. The method intends to solve square systems of large dimensions for arbitrary fuzzy numbers (FNs) by removing non-negativity confinements of the variables and/or coefficients to be more realistic. After the computational method is presented thoroughly, some benchmark examples are finally provided.

Randomized vector iterative linear solvers of high precision for large dense system

Abstract In this paper we suggest randomized linear solvers with a focus on refinement issue to achieve a high precision while maintaining all the advantages of the Monte Carlo method for solving systems of large dimension with dense matrices. It is shown that each iterative refinement step reduces the error by one order of magnitude. The crucial point of the suggested method is, in contrast to the standard Monte Carlo method, that the randomized vector algorithm computes the entire solution column at once, rather than a single component. This makes it possible to efficiently construct the iterative refinement method. We apply the developed method for solving a system of elasticity equations.

CEBoosting: Online sparse identification of dynamical systems with regime switching by causation entropy boosting.

Regime switching is ubiquitous in many complex dynamical systems with multiscale features, chaotic behavior, and extreme events. In this paper, a causation entropy boosting (CEBoosting) strategy is developed to facilitate the detection of regime switching and the discovery of the dynamics associated with the new regime via online model identification. The causation entropy, which can be efficiently calculated, provides a logic value of each candidate function in a pre-determined library. The reversal of one or a few such causation entropy indicators associated with the model calibrated for the current regime implies the detection of regime switching. Despite the short length of each batch formed by the sequential data, the accumulated value of causation entropy corresponding to a sequence of data batches leads to a robust indicator. With the detected rectification of the model structure, the subsequent parameter estimation becomes a quadratic optimization problem, which is solved using closed analytic formulas. Using the Lorenz 96 model, it is shown that the causation entropy indicator can be efficiently calculated, and the method applies to moderately large dimensional systems. The CEBoosting algorithm is also adaptive to the situation with partial observations. It is shown via a stochastic parameterized model that the CEBoosting strategy can be combined with data assimilation to identify regime switching triggered by the unobserved latent processes. In addition, the CEBoosting method is applied to a nonlinear paradigm model for topographic mean flow interaction, demonstrating the online detection of regime switching in the presence of strong intermittency and extreme events.

We modeled long memory with just one lag!

Two recent contributions have found conditions for large dimensional networks or systems to generate long memory in their individual components. We build on these and provide a multivariate methodology for modeling and forecasting series displaying long range dependence. We model long memory properties within a vector autoregressive system of order 1 and consider Bayesian estimation or ridge regression. For these, we derive a theory-driven parametric setting that informs a prior distribution or a shrinkage target. Our proposal significantly outperforms univariate time series long-memory models when forecasting a daily volatility measure for 250 U.S. company stocks over twelve years. This provides an empirical validation of the theoretical results showing long memory can be sourced to marginalization within a large dimensional system.

Optimal quantum control via genetic algorithms for quantum state engineering in driven-resonator mediated networks

We employ a machine learning-enabled approach to quantum state engineering based on evolutionary algorithms. In particular, we focus on superconducting platforms and consider a network of qubits—encoded in the states of artificial atoms with no direct coupling—interacting via a common single-mode driven microwave resonator. The qubit-resonator couplings are assumed to be in the resonant regime and tunable in time. A genetic algorithm is used in order to find the functional time-dependence of the couplings that optimise the fidelity between the evolved state and a variety of targets, including three-qubit GHZ and Dicke states and four-qubit graph states. We observe high quantum fidelities (above 0.96 in the worst case setting of a system of effective dimension 96), fast preparation times, and resilience to noise, despite the algorithm being trained in the ideal noise-free setting. These results show that the genetic algorithms represent an effective approach to control quantum systems of large dimensions.

SIMPLE AND EFFICIENT FIFTH ORDER SOLVERS FOR SYSTEMS OF NONLINEAR PROBLEMS

In this study, two multi-step iterative techniques of fifth order convergence are explored to solve nonlinear equations. The techniques are designed with the prime objective of keeping the computational cost as low as possible. To claim this objective, the efficiency indices are determined and compared with the efficiencies of the existing techniques of same order. The outcome of comparison analysis is remarkable from the view of high computational efficiency of new methods. Performance and stability are illustrated by executing the numerical tests on some nonlinear problems of diverse nature. The entire analysis significantly favors the new techniques compared to their existing counterparts, especially for the case of large dimensional systems.

New Approach to Split Variational Inclusion Issues through a Three-Step Iterative Process

Split variational inclusions are revealed as a large class of problems that includes several other pre-existing split-type issues: split feasibility, split zeroes problems, split variational inequalities and so on. This makes them not only a rich direction of theoretical study but also one with important and varied practical applications: large dimensional linear systems, optimization, signal reconstruction, boundary value problems and others. In this paper, the existing algorithmic tools are complemented by a new procedure based on a three-step iterative process. The resulting approximating sequence is proved to be weakly convergent toward a solution. The operation mode of the new algorithm is tracked in connection with mixed optimization–feasibility and mixed linear–feasibility systems. Standard polynomiographic techniques are applied for a comparative visual analysis of the convergence behavior.

Quantum Anharmonic Calculations of Vibrational Spectra for Water Adsorbed on Titania Anatase(101) Surface: Dissociative versus Molecular Adsorption.

The interaction of water molecules and hydroxyl groups with titanium dioxide (TiO2) surfaces is ubiquitous and very important in anatase nanoparticle photocatalytic processes. Infrared spectroscopy, assisted by ab initio calculations of vibrational frequencies, can be a powerful tool to elucidate the mechanisms behind water adsorption. However, a straightforward comparison between measurements and calculations remains a challenging task because of the complexity of the physical phenomena occurring on nanoparticle surfaces. Consequently, severe computational approximations, such as harmonic vibrational ones, are usually employed. In the present work we partially address this complexity issue by overcoming some of the standard approximations used in theoretical simulations and employ the Divide and Conquer Semiclassical Initial Value Representation (DC-SCIVR) molecular dynamics. This method allows to perform simulations of vibrational spectra of large dimensional systems accounting not only for anharmonicities, but also for nuclear quantum effects. We apply this computational method to water and deuterated water adsorbed on the ideal TiO2 anatase(101) surface, contemplating both the molecular and the dissociated adsorption processes. The results highlight not only the presence of an anharmonic shift of the frequencies in agreement with the experiments, but also complex quantum mechanical spectral signatures induced by the coupling of molecular vibrational modes with the surface ones, which are different in the hydrogenated case from the deuterated one. These couplings are further analyzed by exploiting the mode subdivision performed during the divide and conquer procedure.

A Second Order Arnoldi Method with Stopping Criterion and Reduced Order Selection for Reducing Second Order Systems

This paper introduces a new algorithm for reducing large dimensional second-order dynamic systems through the Second Order Arnold Reduction (SOAR) procedure, with a stopping criterion to select an acceptable good order for the reduced model based on a new coefficient called the Numerical-Rank Performance Coefficient (NRPC), for efficient early termination and automatic optimal order selection of the reduced model. The key idea of this method is to calculate the NRPC coefficient for each iteration of the SOAR algorithm and measure the dynamic evolution information of the original system, which is added to each vector of the Krylov subspace generated by the SOAR algorithm. When the dynamical tolerance condition is verified, the iterative procedure of the algorithm stops. Three benchmark models were used as numerical examples to check the effectiveness and simplicity of the proposed algorithm.

A machine learning-based process operability framework using Gaussian processes

The objective in this work is to develop a machine learning-based framework for process operability using surrogate responses based on Kriging (also known as Gaussian Process Regression). Currently, the available operability approaches for nonlinear systems are limited by the problem dimensionality that they can address, not being computationally tractable for high-dimensional systems. The proposed approach will use Kriging-based models to substitute the developed first-principles or process simulation-based models. The built surrogate models can generate responses that are comparable to the first-principles nonlinear models in terms of accuracy, while reducing the computational effort. To achieve this goal, a framework for the systematic analysis of highly nonlinear, large-dimensional systems at steady state is developed. The proposed approach is benchmarked against current operability methods and provides a new direction in the process operability field employing Kriging models. Two case studies associated with natural/shale gas conversion are addressed to illustrate the effectiveness of the proposed methods, namely a membrane reactor for direct methane conversion to fuels and chemicals and a natural gas combined cycle power plant. It is shown that the computational time for operability calculations is significantly decreased when using the developed approach, with reductions of up to four orders of magnitude, while the relative errors with respect to the output responses is below 0.3% for the worst-case scenario considering all cases. This work thus contributes to machine learning formulations and algorithms for process operability to enable the improved design, operations and manufacturing of chemical and energy systems.

High-dimensional VARs with common factors

This paper studies high-dimensional vector autoregressions (VARs) augmented with common factors that allow for strong cross-sectional dependence. Models of this type provide a convenient mechanism for accommodating the interconnectedness and temporal co-variability that are often present in large dimensional systems. We propose an ℓ1-nuclear-norm regularized estimator and derive the non-asymptotic upper bounds for the estimation errors as well as large sample asymptotics for the estimates. A singular value thresholding procedure is used to determine the correct number of factors with probability approaching one. Both the LASSO estimator and the conservative LASSO estimator are employed to improve estimation precision. The conservative LASSO estimates of the non-zero coefficients are shown to be asymptotically equivalent to the oracle least squares estimates. Simulations demonstrate that our estimators perform reasonably well in finite samples given the complex high-dimensional nature of the model. In an empirical illustration we apply the methodology to explore dynamic connectedness in the volatilities of financial asset prices and the transmission of ‘investor fear’. The findings reveal that a large proportion of connectedness is due to the common factors. Conditional on the presence of these common factors, the results still document remarkable connectedness due to the interactions between the individual variables, thereby supporting a common factor augmented VAR specification.

Gramian‐based vulnerability analysis of dynamic networks

In this paper, the vulnerability of large-dimensional dynamic networks to false data injections is analysed. The malicious data can manipulate input injection at the control nodes and affect the outputs of the network. The objective is to analyse and quantify the potential vulnerability of the dynamics by such adversarial inputs when the opponents try to avoid being detected as much as possible. A joint set of most effective actuation nodes and most vulnerable target nodes are introduced with minimal detectability by the monitoring system. Detection of this joint set of actuation-target nodes is carried out by introducing a Gramian-based measure and reformulating the vulnerability problem as a standard optimisation problem. However, the underlying optimisation problem appears to be a binary program and intractable in large-dimensional systems. Balanced realisation of the system dynamics, combined with a Gauss–Newton approach is employed to provide a two-stage algorithm for the selection of the most effective joint sets. The proposed approach relies on QR decomposition and QR pivoting, does not require iterative computation of Gramian matrices, and scales well with the system dimensions. The proposed approach is evaluated on a set of synthetic and real-life examples including IEEE 118 bus power network.

Numerical Solutions for Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equations

Analyzing the stability of many control systems required solving a couple of crisp Sylvester matrix equations (CSMEs) simultaneously. However, there are some situations in which the crisp Sylvester matrix equations are not well equipped to deal with the uncertainty problem during the stability analysis of control systems. This paper constructs analytical and numerical methods for solving a couple of trapezoidal fully fuzzy Sylvester matrix equations (CTrFFSMEs) to overcome the drawbacks of the existing crisp methods. In developing these new methods, fuzzy arithmetic multiplication is applied on the CTrFFSME to transform it into an equivalent system of four CSMEs. Then, the fuzzy solution is obtained analytically by the fuzzy matrix vectorization method and numerically by gradient and least square methods. The analytical method can obtain the exact solution; however, it is limited to small-sized systems while the numerical methods can approximate the solution for large dimensional systems up to 100 × 100 with a very small error bound for any initial value. In addition, the proposed methods are applied to other fuzzy systems such as Sylvester and Lyapunov matrix equations. The proposed methods are illustrated by solving numerical examples with different size systems.

A strongly coupled immersed boundary method for fluid-structure interaction that mimics the efficiency of stationary body methods

Strongly coupled immersed boundary (IB) methods solve the nonlinear fluid and structural equations of motion simultaneously for strongly enforcing the no-slip constraint on the body. Handling this constraint requires solving several large dimensional systems that scale by the number of grid points in the flow domain even though the nonlinear constraints scale only by the small number of points used to represent the fluid-structure interface. These costly large scale operations for determining only a small number of unknowns at the interface creates a bottleneck to efficiently time-advancing strongly coupled IB methods. In this manuscript, we present a remedy for this bottleneck that is motivated by the efficient strategy employed in stationary-body IB methods while preserving the favorable stability properties of strongly coupled algorithms—we precompute a matrix that encapsulates the large dimensional system so that the prohibitive large scale operations need not be performed at every time step. This precomputation process yields a modified system of small-dimensional constraint equations that is solved at minimal computational cost while time advancing the equations. We also present a parallel implementation that scales favorably across multiple processors. The accuracy, computational efficiency and scalability of our approach are demonstrated on several two dimensional flow problems. Although the demonstration problems consist of a combination of rigid and torsionally mounted bodies, the formulation is derived in a more general setting involving an arbitrary number of rigid, torsionally mounted, and continuously deformable bodies.

Common Bubble Detection in Large Dimensional Financial Systems

Abstract Price bubbles in multiple assets are sometimes nearly coincident in occurrence. Such near-coincidence is strongly suggestive of co-movement in the associated asset prices and is likely driven by certain factors that are latent in the financial or economic system with common effects across several markets. Can we detect the presence of such common factors at the early stages of their emergence? To answer this question, we build a factor model that includes I(1), mildly explosive, and stationary factors to capture normal, exuberant, and collapsing phases in such phenomena. The I(1) factor models the primary driving force of market fundamentals. The explosive and stationary factors model latent forces that underlie the formation and destruction of asset price bubbles, which typically exist only for subperiods of the sample. The article provides an algorithm for testing the presence of and date-stamping the origination and termination of price bubbles determined by latent factors in a large-dimensional system embodying many markets. Asymptotics of the bubble test statistic are given under the null of no common bubbles and the alternative of a common bubble across these markets. We prove the consistency of a factor bubble detection process for the origination and termination dates of the common bubble. Simulations show good finite sample performance of the testing algorithm in terms of its successful detection rates. Our methods are applied to real estate markets covering eighty-nine major cities in China over the period January 2005 to December 2008. Results suggest the presence of a common bubble episode in what are known as China’s Tier 1 and Tier 2 cities from June 2007 to February 2008. There is also a common bubble episode in Tier 3 cities but of shorter duration.

Observation error covariance specification in dynamical systems for data assimilation using recurrent neural networks

Data assimilation techniques are widely used to predict complex dynamical systems with uncertainties, based on time-series observation data. Error covariance matrices modeling is an important element in data assimilation algorithms which can considerably impact the forecasting accuracy. The estimation of these covariances, which usually relies on empirical assumptions and physical constraints, is often imprecise and computationally expensive, especially for systems of large dimensions. In this work, we propose a data-driven approach based on long short term memory (LSTM) recurrent neural networks (RNN) to improve both the accuracy and the efficiency of observation covariance specification in data assimilation for dynamical systems. Learning the covariance matrix from observed/simulated time-series data, the proposed approach does not require any knowledge or assumption about prior error distribution, unlike classical posterior tuning methods. We have compared the novel approach with two state-of-the-art covariance tuning algorithms, namely DI01 and D05, first in a Lorenz dynamical system and then in a 2D shallow water twin experiments framework with different covariance parameterization using ensemble assimilation. This novel method shows significant advantages in observation covariance specification, assimilation accuracy, and computational efficiency.

On-the-fly adiabatically switched semiclassical initial value representation molecular dynamics for vibrational spectroscopy of biomolecules.

Semiclassical (SC) vibrational spectroscopy is a technique capable of reproducing quantum effects (such as zero-pointenergies, quantum resonances, and anharmonic overtones) from classical dynamics runs even in the case of very large dimensional systems. In a previous study [Conte et al. J. Chem. Phys. 151, 214107 (2019)], a preliminary sampling based on adiabatic switching has been shown to be able to improve the precision and accuracy of semiclassical results for challenging model potentials and small molecular systems. In this paper, we investigate the possibility to extend the technique to larger (bio)molecular systems whose dynamics must be integrated by means of ab initio "on-the-fly" calculations. After some preliminary tests on small molecules, we obtain the vibrational frequencies of glycine improving on pre-existing SC calculations. Finally, the new approach is applied to 17-atom proline, an amino acid characterized by a strong intramolecular hydrogen bond.

Mathematical Modeling of the Phytoplankton Populations Geographic Dynamics for Possible Scenarios of Changes in the Azov Sea Hydrological Regime

Increased influence of abiotic and anthropogenic factors on the ecological state of coastal systems leads to uncontrollable changes in the overall ecosystem. This paper considers the crucial problem of studying the effect of an increase in the water’s salinity in the Azov Sea and the Taganrog Bay on hydrobiological processes. The main aim of the research is the diagnostic and predictive modeling of the geographic dynamics of the general phytoplankton populations. A mathematical model that describes the dynamics of three types of phytoplankton is proposed, considering the influence of salinity and nutrients on algae development. Discretization is carried out based on a linear combination of Upwind Leapfrog difference schemes and a central difference scheme, which makes it possible to increase the accuracy of solving the biological kinetics problem at large values of the grid Péclet number (Peh > 2). A software package has been developed that implements interrelated models of hydrodynamics and biogeochemical cycles. A modified alternating-triangular method was used to solve large-dimensional systems of linear algebraic equations (SLAE). Based on the scenario approach, several numerical experiments were carried out to simulate the dynamics of the main species of phytoplankton populations at different levels of water salinity in coastal systems. It is shown that with an increase in the salinity of waters, the habitats of phytoplankton populations shift, and marine species invasively replace freshwater species of algae.

Benchmarking quantum tomography completeness and fidelity with machine learning

We train convolutional neural networks to predict whether or not a set of measurements is informationally complete to uniquely reconstruct any given quantum state with no prior information. In addition, we perform fidelity benchmarking based on this measurement set without explicitly carrying out state tomography. The networks are trained to recognize the fidelity and a reliable measure for informational completeness. By gradually accumulating measurements and data, these trained convolutional networks can efficiently establish a compressive quantum-state characterization scheme by accelerating runtime computation and greatly reducing systematic drifts in experiments. We confirm the potential of this machine-learning approach by presenting experimental results for both spatial-mode and multiphoton systems of large dimensions. These predictions are further shown to improve when the networks are trained with additional bootstrapped training sets from real experimental data. Using a realistic beam-profile displacement error model for Hermite–Gaussian sources, we further demonstrate numerically that the orders-of-magnitude reduction in certification time with trained networks greatly increases the computation yield of a large-scale quantum processor using these sources, before state fidelity deteriorates significantly.