Factor Of Square Root Research Articles

Forecasting-Aided State Estimation in Power Systems During Normal Load Variations Using Iterated Square-Root Cubature Kalman Filter

The main aim is to estimating the voltage profile at all the buses in the system before the arrival of next set of hybrid measurements from field. The effectiveness of the algorithm ISCKF during load variations is evaluated with respect to already implemented Kalman filter approaches for this application. This includes the sudden changes in loads which occurring in practical power systems. The utilization of an iterated square-root cubature Kalman filter (ISCKF) for power system forecasting-aided state estimation (FASE) is being studied during normal load variations. Its implementation involves "Newton-Gauss iterative method being embedded into the square-root cubature Kalman filter (SCKF)" at the measurement update step of Kalman filter. The square-root factor of error covariance matrices is calculated by utilizing QR decomposition to avoid losing of positive definiteness of the matrix. The estimation is carried out utilizing hybrid measurements from remote terminal units and phasor measurement units. The state vector is forecasted using the proposed method in the interval period between two measurement arrivals from the devices. Thereby, caters to state estimation of the voltages at buses in the system even when the measurements are unavailable. The efficacy of the proposed algorithm to FASE is evaluated for IEEE 30-bus system and Northern Region Power Grid (NRPG) 246-bus system. The simulation results show that the proposed ISCKF outperforms the CKF by significant improvement in accuracy of forecasting-aided state estimation. ISCKF will be able to give results before the next set of hybrid data arrives (expected from an estimation algorithm). Therefore, the proposed estimation algorithm is applicable for real-time practical application, with respect to large power systems as well.

Robust optimal asset-liability management with mispricing and stochastic factor market dynamics

This paper investigates a robust optimal asset-liability management problem under an expected utility maximization criterion. More specifically, the manager is concerned about the potential model uncertainty and aims to seek the robust optimal investment strategies. We incorporate an uncontrollable random liability described by a generalized drifted Brownian motion. Also, the manager has access to an incomplete financial market consisting of a risk-free asset, a market index with potentially path-dependent, time-varying risk premium and volatility, and a pair of mispriced stocks. The market dynamics are assumed to rely on an affine-form, square-root factor process and the price error is modeled by a co-integrated system. We adopt a backward stochastic differential equation approach hinging on the martingale optimality principle to solve this non-Markovian robust control problem. Closed-form expressions for the robust optimal investment strategies, the probability perturbation process under the well-defined worst-case scenario and the corresponding value function are derived. The admissibility of the robust optimal controls is verified under some technical conditions. Finally, we perform some numerical examples to illustrate the effects of model parameters on the robust investment strategies and draw some economic interpretations from these results.

Robust optimal asset-liability management under square-root factor processes and model ambiguity: a BSDE approach

This article studies robust optimal asset-liability management problems for an ambiguity-averse manager in a possibly non-Markovian environment with stochastic investment opportunities. The manager has access to one risk-free asset and one risky asset in a financial market. The market price of risk relies on a stochastic factor process satisfying an affine-form, square-root, Markovian model, whereas the risky asset’s return rate and volatility are potentially given by general non-Markovian, unbounded stochastic processes. This financial framework includes, but is not limited to, the constant elasticity of variance (CEV) model, the family of 4/2 stochastic volatility models, and some path-dependent non-Markovian models, as exceptional cases. As opposed to most of the papers using the Hamilton-Jacobi-Bellman-Issacs (HJBI) equation to deal with model ambiguity in the Markovian cases, we address the non-Markovian case by proposing a backward stochastic differential equation (BSDE) approach. By solving the associated BSDEs explicitly, we derive, in closed form, the robust optimal controls and robust optimal value functions for power and exponential utility, respectively. In addition, analytical solutions to some particular cases of our model are provided. Finally, the effects of model ambiguity and market parameters on the robust optimal investment strategies are illustrated under the CEV model and 4/2 model with numerical examples.

Robust equilibrium investment-reinsurance strategy for n competitive insurers with square-root factor process

We investigate a robust equilibrium investment-reinsurance problem for n ambiguity-averse competitive insurers, . Each insurer is allowed to purchase proportional reinsurance and invest in a risk-free asset and a risky asset. Each insurer aims to maximize the expected utility of a weighted relative terminal wealth with respect to the other competitors. In this article, the risky asset is assumed to follow a general and flexible model: the square root factor process. Following the game theory approach, we derive the closed solutions of the robust equilibrium investment-reinsurance strategies. Moreover, the verification theorem is provided in this article. Finally, we demonstrate some numerical analyses and give the economic explanations as well.

Perfect optomechanically induced transparency and slow light in an Rydberg atom-assisted optomechanical system

Rydberg atoms have attracted considerable attention due to their several singular properties, such as strong long-range interactions, extremely long lifetimes, and very large polarizability. These characteristics make Rydberg atoms a good candidates for cavity quantum electrodynamics, quantum information, and many-body physics. Recently, the study of optomechanics has become a rapidly developing field due to its important applications in cooling of mechanical resonators, implementing precision measurements, slowing light, and preparing quantum entanglement. Obviously, optomechanical systems pave the way for quantum information processing and quantum communication. Specially, there is significant interest in quantum optomechanics due to its high compatibility that can be combined, to build hybrid systems for certain purposes, with the ultracold atoms, a superconducting single electron transistor, a magnetostrictive actuation, a charged oscillator resonator, etc. In this work, we investigate the optomechanically induced transparency (OMIT) and the resulting effect of slow light in a hybrid system composed of a Rydberg atomic ensemble embedded inside a simple optomechanical cavity. As a typical effect of destructive quantum interference, OMIT is widely used in quantum optics and quantum information processing. Based on the Rydberg blockade effect, a Rydberg atomic ensemble in the same blockade region embedded inside an optomechanical cavity can be regarded as a superatom that contains only a single Rydberg excitation. Therefore, the problem of exponentially increasing system size with the number of atoms increasing can be circumvented easily. The hybrid system becomes a coupling between a Rydberg superatom and an optomechanical cavity and the coupling strength is enhanced by a factor of square root of the number of atoms in the ensemble. In this system, the perfect OMIT, namely, an ideal OMIT dip with a very narrow window, can be attained when an effect of non-rotating wave approximation (NRWA) is considered. Further, we demonstrate that the term of NRWA plays a key important role in achieving perfect OMIT by comparing the optomechanical spectra obtained with and without NRWA effects. Our results show that in the resolved sideband regime the higher the quality factor of cavity is, the stronger the slow light effect becomes in the window of the perfect OMIT. Particularly, in achieving the ultraslow light, the long lifetime of the Rydberg atom shows its superiority.

Efficient Construction of Canonical Polyadic Approximations of Tensor Networks.

We consider the problem of constructing a canonical polyadic (CP) decomposition for a tensor network, rather than a single tensor. We illustrate how it is possible to reduce the complexity of constructing an approximate CP representation of the network by leveraging its structure in the course of the CP factor optimization. The utility of this technique is demonstrated for the order-4 Coulomb interaction tensor approximated by two order-3 tensors via an approximate generalized square-root (SQ) factorization, such as density fitting or (pivoted) Cholesky. The complexity of constructing a four-way CP decomposition is reduced from (for the nonapproximated Coulomb tensor) to (for the SQ-factorized Coulomb tensor), where n and RCP are the basis and CP ranks, respectively. This reduces the cost of constructing the CP approximation of two-body interaction tensors of relevance to accurate many-body electronic structure by up to 2 orders of magnitude for systems with up to 36 atoms studied here. The full four-way CP approximation of the Coulomb interaction tensor is shown to be more accurate than the known approaches which utilize CP-factorizations of the SQ factors (which are also constructed with an cost), such as the algebraic pseudospectral and tensor hypercontraction approaches. The CP-decomposed SQ factors can also serve as a robust initial guess for the four-way CP factors.

Continuous–discrete unscented Kalman filtering framework by MATLAB ODE solvers and square-root methods

This paper addresses the problem of designing the continuous–discrete unscented Kalman filter (UKF) implementation methods. More precisely, the aim is to propose the MATLAB-based UKF algorithms for accurate and robust state estimation of stochastic dynamic systems. The accuracy of the continuous–discrete nonlinear filters heavily depends on how the implementation method manages the discretization error arisen at the filter prediction step. We suggest the elegant and accurate implementation framework for tracking the hidden states by utilizing the MATLAB built-in numerical integration schemes developed for solving ordinary differential equations (ODEs). The accuracy is boosted by the discretization error control involved in all MATLAB ODE solvers. This keeps the discretization error below the tolerance value provided by users, automatically. Meanwhile, the robustness of the UKF filtering methods is examined in terms of the stability to roundoff. In contrast to the pseudo-square-root UKF implementations established in engineering literature, which are based on the one-rank Cholesky updates, we derive the stable square-root methods by utilizing the J-orthogonal transformations for calculating the Cholesky square-root factors.

Optimal DC pension investment with square-root factor processes under stochastic income and inflation risks

ABSTRACT This paper studies optimally defined contribution (DC) pension investment problems under the expected utility maximization framework with stochastic income and inflation risks. The member has access to a financial market consisting of a risk-free asset (money account), an inflation-indexed bond, and a stock. The market price of volatility risk is assumed to depend on an affine-form, Markovian, square-root factor process, while the return rate and the volatility of the stock are possibly given by general non-Markovian, unbounded stochastic processes. This financial framework recovers the Black–Scholes model, constant elasticity of variance (CEV) model, Heston model, 3/2 model, 4/2 model, and some non-Markovian models as exceptional cases. To tackle the potentially non-Markovian structures, we adopt a backward stochastic differential equation (BSDE) approach. By solving the associated BSDEs explicitly, closed-form expressions for the optimal investment strategies and optimal value functions are obtained for the power, logarithmic, and exponential utility functions. Moreover, explicit solutions to some special cases of our portfolio model are provided. Finally, numerical examples are provided to illustrate the effects of model parameters on the optimal investment strategies under the 4/2 model.

Robust Optimal Asset-Liability Management Under Square-Root Factor Processes and Model Ambiguity: A BSDE Approach

Robust Optimal Asset-Liability Management Under Square-Root Factor Processes and Model Ambiguity: A BSDE Approach

A Square-Root Factor-Based Multi-Population Extension of the Mortality Laws

In this paper, we present and calibrate a multi-population stochastic mortality model based on latent square-root affine factors of the Cox-Ingersoll and Ross type. The model considers a generalization of the traditional actuarial mortality laws to a stochastic, multi-population and time-varying setting. We calibrate the model to fit the mortality dynamics of UK males and females over the last 50 years. We estimate the optimal states and model parameters using quasi-maximum likelihood techniques.

ISM-Assisted Tomographic STED Microscopy Allows for Sample-Gentle Superresolution Imaging

Fluorescence nanoscopy allows to non-invasively resolve three-dimensional cell structures beyond the diffraction limit. One of these high resolution imaging techniques is stimulated emission depletion (STED) microscopy. However, the practically achievable resolution of a STED microscope is often limited by photobleaching. One method to overcome this limitation is tomographic STED (tomoSTED) microscopy. In tomoSTED microscopy, excited fluorophores in the sample plane are depleted by a 0-π phase step modified and subsequently focused STED beam, which leads to an effective narrowing of the fluorescence-allowed area in a single direction. A full highly resolved spatial information of the object in two dimensions is then obtained by rotating the depletion pattern. The number of required pattern rotations depends on the ratio of the resolutions in the depleted and non-depleted directions. Since the resolution per pattern orientation is only increased along a single direction, imaging along the other direction is still diffraction-limited. Therefore, the resolution along this direction can be increased by utilizing the concept of image scanning microscopy (ISM). ISM allows to improve the resolution of a scanning microscope by a factor of square root of two. By combining tomoSTED with ISM, the number of required pattern orientations can therefore be reduced leading to a lower overall acquisition time which translates directly into a lower light dose. On simulated as well as on experimental data, we show that approximately a factor of square root of two lower number of orientations are required to achieve the same resolution and image quality as in tomoSTED microscopy without ISM assistance. Under typical recording conditions, this corresponds to a six-fold reduction of the required light dose as compared to a classical STED microscope.

Equilibrium excess-of-loss reinsurance and investment strategies for an insurer and a reinsurer

In this paper, we consider the equilibrium excess-of-loss reinsurance and investment problem for both an insurer and a reinsurer. The risk process of the insurer is described by a classical Cramér-Lundberg (C-L) risk model and the insurer can purchase excess-of-loss reinsurance from the reinsurer. Both the insurer and the reinsurer are allowed to invest in a financial market consisting of a risk-free asset and two risky assets. The market price of risk depends on a Markovian, affine-form and square-root stochastic factor process. Dynamic mean-variance criterion is considered in this paper. We aim to maximize the weighted sum of the insurer’s and the reinsurer’s objectives with different risk averse coefficients. By solving the corresponding extended Hamilton-Jacobi-Bellman (HJB) equations, we derive the equilibrium reinsurance and investment strategies and the corresponding equilibrium value function. Finally, the economic implications of our findings are illustrated.

Modular framework for implementation and analysis of recursive filters with considered and neglected parameters

This paper develops novel covariance and square-root factor formulations of a consider-neglect Kalman filter for navigation applications. The proposed filter partitions system parameters into three distinct categories: those to be estimated by the filter, those whose contribution to the system are considered without being explicitly estimated, and those with sufficiently low effect on the system such that their contribution can be neglected altogether. Discussion on appropriate selection of parameters to be considered and neglected is provided with specific attention given to descent-to-landing navigation. Monte Carlo simulations and analysis are performed to assess the performance of the developed square-root consider-neglect filter in a descent-to-landing navigation scenario.

An accelerated directional derivative method for smooth stochastic convex optimization

We consider smooth stochastic convex optimization problems in the context of algorithms which are based on directional derivatives of the objective function. This context can be considered as an intermediate one between derivative-free optimization and gradient-based optimization. We assume that at any given point and for any given direction, a stochastic approximation for the directional derivative of the objective function at this point and in this direction is available with some additive noise. The noise is assumed to be of an unknown nature, but bounded in the absolute value. We underline that we consider directional derivatives in any direction, as opposed to coordinate descent methods which use only derivatives in coordinate directions. For this setting, we propose a non-accelerated and an accelerated directional derivative method and provide their complexity bounds. Our non-accelerated algorithm has a complexity bound which is similar to the gradient-based algorithm, that is, without any dimension-dependent factor. Our accelerated algorithm has a complexity bound which coincides with the complexity bound of the accelerated gradient-based algorithm up to a factor of square root of the problem dimension. We extend these results to strongly convex problems.

BSDE approach to utility maximization with square-root factor processes

We consider the utility-based portfolio selection problem in a continuous-time setting. We assume the market price of risk depends on a stochastic factor that satisfies an affine-form, square-root, Markovian model. This financial market framework includes the classical geometric Brownian motion, CEV model, and Heston’s model as special cases. Adopting the BSDE approach, we obtain closed-form solutions for the optimal portfolio strategies and value functions for the logarithmic, power, and exponential utility functions.

On the Stable Cholesky Factorization-based Method for the Maximum Correntropy Criterion Kalman Filtering

This paper continues the research devoted to the design of numerically stable square-root implementations for the maximum correntropy criterion Kalman filtering (MCC-KF). In contrast to the previously obtained results, here we reveal the first robust (with respect to roundoff errors) method within the Cholesky factorization-based approach. The method is formulated in terms of square-root factors of the covariance matrices, i.e. it belongs to the covariance-type filtering methodology. Additionally, a numerically stable orthogonal transformation is utilized at each iterate of the algorithm for accurate propagation of the Cholesky factors involved. The results of numerical experiments illustrate a superior performance of the novel MCC-KF implementation compared to both the conventional algorithm and its previously published Cholesky-based variant.

Fusion estimation for nonlinear multi-sensor networked systems with packet loss compensation and correlated noises

PurposeThis paper aims to give the centralized and distributed fusion estimator for nonlinear multi-sensor networked systems with packet loss compensation and correlated noises and give the corresponding square-root cubature Kalman filter.Design/methodology/approachBased on the Gaussian approximation recursive filter framework, the authors derive the centralized fusion filter and using the projection theorem, the authors derive the centralized fusion smoother. Then, based on the fast batch covariance intersection fusion algorithm, the authors give the corresponding results for distributed fusion estimators.FindingsDesigning the fusion estimators for nonlinear multi-sensor networked systems with packet loss compensation and correlated noises is necessary. It is useful for general nonlinear systems.Originality/valueThroughout the whole study, the main highlights of this paper are as follows: packet loss compensation and correlated noises are considered in nonlinear multi-sensor networked systems. There are no relevant conclusions in the existing literature; centralized and distributed fusion estimators are derived based on the above system; for the posterior covariance with compensation factor and correlated noises, a new square-root factor of the error covariance is derived; and the new square-root factor of the error covariance is used to replace the numerical implementation of the covariance in cubature Kalman filter (CKF), which simplified the problem in calculating the posterior covariance in CKF.

An improved numerically-stable equivalent static loads (ESLs) algorithm based on energy-scaling ratio for stiffness topology optimization under crash loads

The standard equivalent static loads (ESLs) method for stiffness topology optimization under crash condition may lead to exaggerated equivalent loads, which is not appropriate to be incorporated into the linear static topology optimization and whereby hinder the optimization process. To overcome this disadvantage, an improved ESLs algorithm based on energy-scaling ratio is proposed to guarantee the numerical stability, especially for the first several cycles with relatively larger differences of strain energy between the original crash simulation and equivalent static analysis. At each cycle, the equivalent external static forces are calculated by multiplying the stiffness matrix and the displacement vector at the time with maximal strain energy during the crash simulation. A further adaptive energy-scaling operation for those forces are performed by a weighting factor of square root of the energy ratio to the standard equivalent static loads for the crash problem based on the judging criterion. The newly equivalent loads are incorporated into the static topology optimization, and topology results are filtered into a black-white design for the crash simulation to avoid the numerical issues due to existing of low-density elements. The process is repeated until the convergence criteria is satisfied. The effectiveness of the proposed method is demonstrated by investing two crash design problems.

Gradient-Based Parameter Estimation in Pairwise Linear Gaussian System

This technical note addresses the gradient-based parameter estimation problem for pairwise linear Gaussian systems. The new adaptive filtering scheme is based on gradient-based optimization methods for estimating the uncertain system parameters and a robust square-root variant of the pairwise Kalman filter used for estimating the unknown states of the pairwise linear Gaussian systems. Hence, in the adaptive filtering techniques, the dynamic state and system parameters are estimated simultaneously. The method of the filter sensitivities computation required in gradient evaluation is derived in terms of square-root factors of covariance matrices.

Supercritical entanglement in local systems: Counterexample to the area law for quantum matter

Quantum entanglement is the most surprising feature of quantum mechanics. Entanglement is simultaneously responsible for the difficulty of simulating quantum matter on a classical computer and the exponential speedups afforded by quantum computers. Ground states of quantum many-body systems typically satisfy an "area law": The amount of entanglement between a subsystem and the rest of the system is proportional to the area of the boundary. A system that obeys an area law has less entanglement and can be simulated more efficiently than a generic quantum state whose entanglement could be proportional to the total system's size. Moreover, an area law provides useful information about the low-energy physics of the system. It is widely believed that for physically reasonable quantum systems, the area law cannot be violated by more than a logarithmic factor in the system's size. We introduce a class of exactly solvable one-dimensional physical models which we can prove have exponentially more entanglement than suggested by the area law, and violate the area law by a square-root factor. This work suggests that simple quantum matter is richer and can provide much more quantum resources (i.e., entanglement) than expected. In addition to using recent advances in quantum information and condensed matter theory, we have drawn upon various branches of mathematics such as combinatorics of random walks, Brownian excursions, and fractional matching theory. We hope that the techniques developed herein may be useful for other problems in physics as well.