Linear Recursive Equations Research Articles

Stochastic optimal control problems of discrete‐time Markov jump systems

AbstractIn this paper, we consider the indefinite stochastic optimal control problems of discrete‐time Markov jump linear systems. Firstly, we establish the new stochastic maximum principle, and by solving the forward‐backward stochastic difference equations with Markov jump (FBSDEs‐MJ), we derive the necessary and sufficient solvability condition of the indefinite control problem with non‐discounted cost, which is in an explicit analytical expression. Then, the optimal control is designed by a series of coupled generalized Riccati difference equations with Markov jump (GRDEs‐MJ) and linear recursive equations with Markov jump (LREs‐MJ). Moreover, based on the non‐discounted cost case, we deduce the optimal control problem with discounted cost. Finally, a numerical example for defined‐benefit (DB) pension fund with regime switching is exploited to illustrate the validity of the obtained results.

Secure and Efficient Smart Healthcare System Based on Federated Learning

The rapid development of smart healthcare system in the Internet of Things (IoT) has made the early detection of many chronic diseases more convenient, quick, and economical. However, when healthcare organizations collect users’ health data through deployed IoT devices, there are issues of compromising users’ privacy. In view of this situation, this paper introduces federated learning technology to solve the problem of data security. In this paper, we consider the two main problems of federated learning applications in IoT smart healthcare system: (1) how to reduce the time overhead of system running and (2) how to authenticate that the user device uploading data is deployed by the system itself. To solve the above problems, we propose the first federated learning scheme based on full dynamic secret sharing. First, we use a two-mask protocol to keep the user’s local model parameters confidential during federated learning. Then, based on homogeneous linear recursive equation, homomorphic hash function, and elliptic curve cryptosystem, the full dynamic secret sharing and user identity authentication are realized. In addition, our scheme allows users to join or quit during training. Finally, we have carried out simulation test on this scheme. The experimental results show that the efficiency of our scheme is improved by about 60% on average in the case of no user dropping and by about 30% in the case of some users dropping.

Electrical characterization of the 2 × 6 × n cobweb cascaded resistor network model by the improved recursion-transform method

Since the cobweb model combines the properties of several network topologies, the cobweb resistor network based on the cobweb model has aroused the interest of scholars. However, owing to the complicated structure of the cobweb cascaded resistor network, it is very challenging to solve with conventional approaches. To address this issue, an improved recursion-transform approach was employed. The electrical characterization of the 2 × 6 × n cobweb cascade resistor network model was studied in this paper. Firstly, the equivalent circuit of the resistor network to be solved was found. Secondly, the recursive equation of the equivalent circuit was constructed. Then, the recursive equation was linearized. Finally, the equivalent resistance of the resistor network was obtained by solving the linear recursive equation. This approach significantly reduces the computation procedure because it adopts the scheme of simplifying the circuit first, then establishing the equation and solving it. The findings indicate that the equivalent resistances between the two nodes of this resistor network change with the change of the order n. When n tends to infinity, these equivalent resistances will have definite convergence values. The calculation results show that these equivalent resistances are already very close to their convergence values, respectively, when n takes 7 or 8.

A fully dynamic secret sharing scheme

Based on the homogeneous linear recursive equation and the Elliptic curve cryptography (ECC), a fully dynamic secret sharing scheme is proposed for the general access structure of the participants. The participants choose their own shadows by themselves. When the shared secrets are updated, the access structure is changed, and other participants are added to or deleted from the existed group, but the shadow of each participant remains unchanged. Each participant should only maintain a shadow to share the multiple secrets. Compared with the existing popular dynamic schemes, the key construction of the proposed scheme is simpler and the proposed scheme has a completely dynamic characteristic, so that it has the potential to extend its practical application scenarios. That is, the proposed scheme can improve the performances of the key management and the distributed system. The security of the scheme is based on the Shamir threshold scheme, the computational Diffie–Hellman problem (CDHP) and the discrete logarithm problem (DLP).

Weighted moments of solutions of the stochastic linear recursive distributional equation and applications

Weighted moments of solutions of the stochastic linear recursive distributional equation and applications

Parameter identifiability-based optimal observation remedy for biological networks

BackgroundTo systematically understand the interactions between numerous biological components, a variety of biological networks on different levels and scales have been constructed and made available in public databases or knowledge repositories. Graphical models such as structural equation models have long been used to describe biological networks for various quantitative analysis tasks, especially key biological parameter estimation. However, limited by resources or technical capacities, partial observation is a common problem in experimental observations of biological networks, and it thus becomes an important problem how to select unobserved nodes for additional measurements such that all unknown model parameters become identifiable. To the best knowledge of our authors, a solution to this problem does not exist until this study.ResultsThe identifiability-based observation problem for biological networks is mathematically formulated for the first time based on linear recursive structural equation models, and then a dynamic programming strategy is developed to obtain the optimal observation strategies. The efficiency of the dynamic programming algorithm is achieved by avoiding both symbolic computation and matrix operations as used in other studies. We also provided necessary theoretical justifications to the proposed method. Finally, we verified the algorithm using synthetic network structures and illustrated the application of the proposed method in practice using a real biological network related to influenza A virus infection.ConclusionsThe proposed approach is the first solution to the structural identifiability-based optimal observation remedy problem. It is applicable to an arbitrary directed acyclic biological network (recursive SEMs) without bidirectional edges, and it is a computerizable method. Observation remedy is an important issue in experiment design for biological networks, and we believe that this study provides a solid basis for dealing with more challenging design issues (e.g., feedback loops, dynamic or nonlinear networks) in the future. We implemented our method in R, which is freely accessible at https://github.com/Hongyu-Miao/SIOOR.

Direct solutions of linear non-homogeneous difference equations

In this paper, the authors develop a direct method used to solve the initial value problems of a linear non-homogeneous time-invariant difference equation. In this method, the obtained general term of the solution sequence has an explicit formula, which includes coefficients, initial values, and right-side terms of the solved equation only. Furthermore, the authors find that when the solution sequence has a nonzero first term, it satisfies two adjoint linear recursive equations; this usually shows several new features of the solution sequence.

Effect of yerba mate (Ilex paraguariensis) on bone tissue

In the article, we propose an incremental algorithm for calculating the inversion of the confluent Vandermonde matrix and triangular factorization of this inversion. We implemented all the incremental operations, i.e. adding, deleting and changing the single matrix parameter to avoid repeating the same calculations again and again. Besides, contrary to other works in this field, the article derives an explicit analytic formula for the calculation of the triangular factorization of the confluent Vandermonde matrix inversion. Additionally, we propose a solution to these two problems with the use of a system of linear recursive equations. The results of this article do not require any symbolic calculations. Therefore they can be performed by a numerical algorithm implemented in any general-purpose programming language.

Incremental numerical recipes for the high efficient inversion of the confluent Vandermonde matrices

In the article, we propose an incremental algorithm for calculating the inversion of the confluent Vandermonde matrix and triangular factorization of this inversion. We implemented all the incremental operations, i.e. adding, deleting and changing the single matrix parameter to avoid repeating the same calculations again and again. Besides, contrary to other works in this field, the article derives an explicit analytic formula for the calculation of the triangular factorization of the confluent Vandermonde matrix inversion. Additionally, we propose a solution to these two problems with the use of a system of linear recursive equations. The results of this article do not require any symbolic calculations. Therefore they can be performed by a numerical algorithm implemented in any general-purpose programming language.

Indefinite Mean-Field Stochastic Linear-Quadratic Optimal Control

This paper is concerned with the discrete-time indefinite mean-field linear-quadratic optimal control problem. The so-called mean-field type stochastic control problems refer to the problem of incorporating the means of the state variables into the state equations and cost functionals, such as the mean-variance portfolio selection problems. A dynamic optimization problem is called to be nonseparable in the sense of dynamic programming if it is not decomposable by a stage-wise backward recursion. The classical dynamic-programming-based optimal stochastic control methods would fail in such nonseparable situations as the principle of optimality no longer applies. In this paper, we show that both the well-posedness and the solvability of the indefinite mean-field linear-quadratic problem are equivalent to the solvability of two coupled constrained generalized difference Riccati equations and a constrained linear recursive equation. We characterize the optimal control set completely, and obtain a set of necessary and sufficient conditions on the mean-variance portfolio selection problem. The results established in this paper offer a more accurate solution scheme in tackling directly the issue of nonseparability and deriving the optimal policies analytically for the mean-variance-type portfolio selection problems.

Optimal tensegrity structures in bending: The discrete Michell truss

This paper provides the closed form analytical solution to the problem of minimizing the material volume required to support a given set of bending loads with a given number of discrete structural members, subject to material yield constraints. The solution is expressed in terms of two variables, the aspect ratio, ρ-1, and complexity of the structure, q (the total number of members of the structure is equal to q(q+1)). The minimal material volume (normalized) is also given in closed form by a simple function of ρ and q, namely, V=q(ρ-1/q-ρ1/q). The forces for this nonlinear problem are shown to satisfy a linear recursive equation, from node-to-node of the structure. All member lengths are specified by a linear recursive equation, dependent only on the initial conditions involving a user specified length of the structure. The final optimal design is a class 2 tensegrity structure. Our results generate the 1904 results of Michell in the special case when the selected complexity q approaches infinity. Providing the optimum in terms of a given complexity has the obvious advantage of relating complexity q to other criteria, such as costs, fabrication issues, and control. If the structure is manufactured with perfect joints (no glue, welding material, etc.), the minimal mass complexity is infinite. But in the presence of any joint mass, the optimal structural complexity is finite, and indeed quite small. Hence, only simple structures (low complexity q) are needed for practical design.

Calibration of a water and energy balance model: Recursive parameter estimation versus particle swarm optimization

It is well known that one of the major problems in the application of land surface models is the determination of the various model parameters. In most cases, only one or a limited number of variables are used to estimate these parameters. This study evaluates the use of two fundamentally different global optimization methods, multistart weight‐adaptive recursive parameter estimation (MWARPE) and particle swarm optimization (PSO), for the estimation of hydrologic model parameters on the basis of data for multiple variables. MWARPE iteratively uses the linear recursive filter equations in a Monte Carlo setting and therefore does not rely on the explicit minimization of an objective function. However, a major drawback of the MWARPE method is the high dimensionality, determined by the number of observations, of the matrix to be inverted. On the other hand, PSO is a stochastic optimization method based on the collective strength of a population of individuals with flocking or herding behavior, as observed in a wide number of biological systems. In situ observations of net radiation; latent, sensible, and ground heat fluxes; and the soil moisture profile are used to determine the parameters of a simplified water and energy balance model. Both optimization methods are analyzed in terms of model performance and computational efficiency. Comparable results, expressed in terms of the root mean square error values, were obtained for both methods. However, it was found that MWARPE tends to slightly overfit the data.

An algorithm for radial distribution power flow in Complex mode including voltage controlled buses

This paper presents an efficient algorithm to solve the radial distribution power flow problem in complex mode. The relationship between the complex branch powers and complex bus powers is derived as a non singular square matrix known as element incidence matrix. The power flow equations are rewritten in terms of a new variable as linear recursive equations. The linear equations are solved to determine the bus voltages and branch currents in terms of new variable as complex numbers. The advantage of this algorithm is that it does not need any initial value and easier to develop the code since all the equations are expressed in matrix format. It is tested on the distribution systems available in the literature. This proposed method could be applied to distribution systems having voltage-controlled buses also. The results prove the efficiency of the proposed method. Keywords: radial distribution power flow, element incidence matrix, transmission loss, linear recursive equations.

Indefinite quadratic with linear costs optimal control of Markov jump with multiplicative noise systems

In this paper we consider the stochastic optimal control problem of discrete-time Markov jump with multiplicative noise linear systems. The performance criterion is assumed to be formed by a linear combination of a quadratic part and a linear part in the state and control variables. The weighting matrices of the state and control for the quadratic part are allowed to be indefinite. We present a necessary and sufficient condition under which the problem is well posed and a state feedback solution can be derived from a set of coupled generalized Riccati difference equations interconnected with a set of coupled linear recursive equations. For the case in which the quadratic-term matrices are non-negative, this necessary and sufficient condition can be written in a more explicit way. The results are applied to a problem of portfolio optimization.

DYNAMIC EQUIVALENCE PRINCIPLE IN LINEAR RATIONAL EXPECTATIONS MODELS

Linear models with infinite horizon generally admit infinitely many rational expectations solutions. Consequently, some additional selection devices are needed to narrow the set of relevant solutions. The viewpoint of this paper is that a solution will be more likely to arise if it is locally determinate (i.e., locally isolated), locally immune to sunspots, and locally stable under learning. These three criteria are applied to solutions of linear univariate models along which the level of the state variable evolves through time. In such models the equilibrium behavior of the level of the state variable is described by a linear recursive equation characterized by the set of its coefficients. The main innovation of this paper is to define new perfect-foresight dynamics whose fixed points are these sets of coefficients, thus allowing us to study the property of determinacy of these sets, or, equivalently, of the associated solutions. It is shown that only one solution is locally determinate in the new dynamics. It is also locally immune to sunspots and locally stable under myopic learning. This solution corresponds to the saddle path in the saddle-point case.

Determinants of Triangular Matrices and Trajectories on Ferre Diagrams

Some functions defined on the set of integer triangular matrices and their modifications are considered. These functions are analogs of the classical determinant and permanent functions and provide a powerful apparatus for studying sequences generated by linear recursive equations. Modifications of these functions are used to solve the problem about trajectories on Ferre diagrams and to establish new combinatorial identities.

Perturbation analysis of linear programming problems with random parameters

This paper presents a methodology for analyzing large scale engineering problems with uncertain parameters. The coefficients of the objective function, the coefficients of the decision variables and the right hand side of the constraints are functions of random variables. The optimization problem is solved using the traditional simplex method. The uncertain parameters in the equations are expanded in the Taylor series. It is shown that the resulting equations are actually a set of linear programming recursive equations. Upon solving these equations, the required probabilistic statements can be easily established. An engineering example is provided to demonstrate the use of this methodology.

Laguerre series approach to the analysis of a linear control system incorporating observers

Application of the Laguerre polynomial expansion to the analysis of a linear optimal control system incorporating observers is studied. The method simplifies the system of equations into the successive solution of a set of linear algebraic recursive equations. Compared with other orthogonal polynomials, the present method is simpler in form and more convenient for use with a digital computer. A numerical example is given to illustrate the use of this method.

Functional programming with streams —Part II—

This paper complements our previous paper “Functional programming with streams.” The purpose of this paper is two-fold: to further develop the concept of a stream, and to present an implementation aspect of stream programming. Stream programming is decomposed into three phases, i.e. stream generation, stream transformation and stream reduction and for each phase we have (stream) generators, transformers and reducers, respectively. A linear recursive function equation, for example, is described as a composition of a stream generator and a reducer. We also give a listing of implemented stream processing functions in this paper.

The block-pulse functions method of the two-dimensional Laplace transform

This paper extends the recently developed method of one-dimensional (1D) operational calculus via block-pulse functions method. It was shown that the inverse Laplace transform of a two-dimensional (2D) rational transfer function can easily be obtained in terms of block-pulse functions as a set of linear recursive equations.