A two-sided Arnoldi algorithm with stopping criterion and MIMO selection procedure

In this paper we introduce a two-sided Arnoldi method for the reduction of high order linear systems and we propose useful extensions, first of all a stopping criterion to find a suitable order for the reduced model and secondly, a selection procedure to significantly improve the performance in the multi-input multi-output (MIMO) case. One application is in micro-electro-mechanical systems (MEMS). We consider a thermo-electric micro thruster model, and a comparison between the commonly used Arnoldi algorithm and the two-sided Arnoldi is performed.     

Two-sided model order reduction for two-directional difference system

This paper defines a two-directional difference system and constructs the projection matrix. Then the original system is projected into the smaller system, and we discuss its moment-matching properties. Next we define the dual system, and discuss the dual relation between the dual system and the original system. Then we can construct the projection matrix with the above mentioned dual relation, and project the dual system into the respectively smaller system, hence derive the moment-matching properties. Finally synthesizing the above two moment-matching properties we obtain the main results that the number of moments matched is twice as much as the number of the generating terms of the constructed projection subspace. We apply this result to the two-sided model order reduction for parameter time delay system, and obtain the result that the reduced system can preserve twice moments as the number of the generating terms of the constructed projection subspace. Finally we derive an algorithm to compute the basis of the subspace involved in the reduction process.     

Model order reduction and error estimation with an application to the parameter-dependent eddy current equation

In product development, engineers simulate the underlying partial differential equation many times with commercial tools for different geometries. Since the available computation time is limited, we look for reduced models with an error estimator that guarantees the accuracy of the reduced model. Using commercial tools the theoretical methods proposed by G. Rozza, D.B.P. Huynh and A.T. Patera [Reduced basis approximation and a posteriori error estimation for affinely parameterized elliptic coercive partial differential equations, Arch. Comput. Methods Eng. 15 (2008), pp. 229–275] lead to technical difficulties. We present how to overcome these challenges and validate the error estimator by applying it to a simple model of a solenoid actuator that is a part of a valve.     

Invasive weed optimization for model order reduction of linear MIMO systems

In this work, a model order reduction (MOR) technique for a linear multivariable system is proposed using invasive weed optimization (IWO). This technique is applied with the combined advantages of retaining the dominant poles and the error minimization. The state space matrices of the reduced order system are chosen such that the dominant eigenvalues of the full order system are unchanged. The other system parameters are chosen using the invasive weed optimization with objective function to minimize the mean squared errors between the outputs of the full order system and the outputs of the reduced order model when the inputs are unit step. The proposed algorithm has been applied successfully, a 10th order Multiple-Input–Multiple-Output (MIMO) linear model for a practical power system was reduced to a 3rd order and compared with recently published work.     

A MODIFIED GMRES METHOD FOR SOLVING LARGE NONSYMMETRIC LINEAR SYSTEMS

A modified GMRES method is proposed in this paper, the method replaces the approximation xm obtained by the GMRES method with a new approximation xm which is a linear combination of xm and the wasted basis vector vm 1. The residual norm of the new approximation satisfies a small one-dimensional minimization problem. Relationships between the residual norms of xm and xm are given. We show that the resulting m-step modified GMRES method is better than the original m-step GMRES method in theory and is consi...     

On dominant poles and model reduction of second order time-delay systems

The method known as the dominant pole algorithm (DPA) has previously been successfully used in combination with model order reduction techniques to approximate standard linear time-invariant dynamical systems and second order dynamical systems. In this paper, we show how this approach can be adapted to a class of second order delay systems, which are large scale nonlinear problems whose transfer functions have an infinite number of simple poles. Deflation is a very important ingredient for this type of methods. Because of the nonlinearity, many deflation approaches for linear systems are not applicable. We therefore propose an alternative technique that essentially removes computed poles from the system?s input and output vectors. In general, this technique changes the residues, and hence, modifies the order of dominance of the poles, but we prove that, under certain conditions, the residues stay near the original residues. The new algorithm is illustrated by numerical examples.     

Model reduction using the Vorobyev moment problem

Given a nonsingular complex matrix and complex vectors v and w of length N, one may wish to estimate the quadratic form w ^* A ^− 1 v, where w ^* denotes the conjugate transpose of w. This problem appears in many applications, and Gene Golub was the key figure in its investigations for decades. He focused mainly on the case A Hermitian positive definite (HPD) and emphasized the relationship of the algebraically formulated problems with classical topics in analysis - moments, orthogonal polynomials and quadrature. The essence of his view can be found in his contribution Matrix Computations and the Theory of Moments, given at the International Congress of Mathematicians in Zürich in 1994. As in many other areas, Gene Golub has inspired a long list of coauthors for work on the problem, and our contribution can also be seen as a consequence of his lasting inspiration. In this paper we will consider a general mathematical concept of matching moments model reduction, which as well as its use in many other applications, is the basis for the development of various approaches for estimation of the quadratic form above. The idea of model reduction via matching moments is well known and widely used in approximation of dynamical systems, but it goes back to Stieltjes, with some preceding work done by Chebyshev and Heine. The algebraic moment matching problem can for A HPD be formulated as a variant of the Stieltjes moment problem, and can be solved using Gauss-Christoffel quadrature. Using the operator moment problem suggested by Vorobyev, we will generalize model reduction based on matching moments to the non-Hermitian case in a straightforward way. Unlike in the model reduction literature, the presented proofs follow directly from the construction of the Vorobyev moment problem. The work was supported by the GAAS grant IAA100300802 and by the Institutional Research Plan AV0Z10300504.     

A connection between time domain model order reduction and moment matching for LTI systems

We investigate the time domain model order reduction (MOR) framework using general orthogonal polynomials by Jiang and Chen [1 Y.L. Jiang and H.B. Chen, Time domain model order reduction of general orthogonal polynomials for linear input-output systems, IEEE Trans. Autom. Control 57 (2012), pp. 330–343. doi:10.1109/TAC.2011.2161839[Crossref], [Web of Science ®] , [Google Scholar]] and extend their idea by exploiting the structure of the corresponding linear system of equations. Identifying an equivalent Sylvester equation, we show a connection to a rational Krylov subspace, and thus to moment matching. This theoretical link between the MOR techniques is illustrated by three numerical examples. For linear time-invariant systems, the link also motivates that the time-domain approach can be at best as accurate as moment matching, since the expansion points are fixed by the choice of the polynomial basis, while in moment matching they can be adapted to the system.     

A matrix rational Lanczos method for model reduction in large‐scale first‐ and second‐order dynamical systems

In the present paper, we describe an adaptive modified rational global Lanczos algorithm for model‐order reduction problems using multipoint moment matching‐based methods. The major problem of these methods is the selection of some interpolation points. We first propose a modified rational global Lanczos process and then we derive Lanczos‐like equations for the global case. Next, we propose adaptive techniques for choosing the interpolation points. Second‐order dynamical systems are also considered in this paper, and the adaptive modified rational global Lanczos algorithm is applied to an equivalent state space model. Finally, some numerical examples will be given.     

Characterization of chaotic order and its application to furuta inequality

In this note, we give a simple characterization of the chaotic order among positive invertible operators on a Hilbert space. As an application, we discuss Furuta's type operator inequality.

     

A tactical model for resource allocation and its application to advertising budgeting

Models for optimal product positioning have received considerable attention by marketing researchers and marketing scientists over the past decade. Typically, optimizing models take the viewpoint that the manager wishes to find a specific vector of product attribute levels that, in the face of competitors’ product profiles, maximizes the firm’s market share (or, perhaps, return) over some designated planning horizon. This class of models emphasizes long run strategic modeling.In contrast, the authors introduce a tactical, short-term model, called SALIENCE, whose purpose is to allocate sales efforts in such a way as to increase the relative importance of attributes for which the sponsoring firm’s current product has a (possibly temporary) differential advantage. In this case emphasis is on short-run, tactical decision making.We describe the SALIENCE model, both informally and mathematically. The model is applied, illustratively, to a real (disguised) study of overnight air shipment delivery.     

Cognitive learning in differential evolution and its application to model order reduction problem for single-input single-output systems

Differential evolution (DE) is a well known and simple population based probabilistic approach for global optimization over continuous spaces. It has reportedly outperformed a few evolutionary algorithms and other search heuristics like the particle swarm optimization when tested over both benchmark and real world problems. DE, like other probabilistic optimization algorithms, has inherent drawback of premature convergence and stagnation. Therefore, in order to find a trade-off between exploration and exploitation capability of DE algorithm, a new parameter namely, cognitive learning factor (CLF) is introduced in the mutation process. Cognitive learning is a powerful mechanism that adjust the current position of individuals by the means of some specified knowledge (previous experience of individuals). The proposed strategy is named as cognitive learning in differential evolution (CLDE). To prove the efficiency of various approaches of CLF in DE,?CLDE is tested over 25 benchmark problems. Further, to establish the wide applicability of CLF,?CLDE is applied to two advanced DE variants. CLDE is also applied to solve a well known electrical engineering problem called model order reduction problem for single input single output systems.     

Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication

This paper presents a new fractional-order hyperchaotic system. The chaotic behaviors of this system in phase portraits are analyzed by the fractional calculus theory and computer simulations. Numerical results have revealed that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4 and the lowest order to have hyperchaos in this system is 3.664. The existence of two positive Lyapunov exponents further verifies our results. Furthermore, a novel modified generalized projective synchronization (MGPS) for the fractional-order chaotic systems is proposed based on the stability theory of the fractional-order system, where the states of the drive and response systems are asymptotically synchronized up to a desired scaling matrix. The unpredictability of the scaling factors in projective synchronization can additionally enhance the security of communication. Thus MGPS of the new fractional-order hyperchaotic system is applied to secure communication. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient.     

A self‐adaptive intelligence gray prediction model with the optimal fractional order accumulating operator and its application

The self‐adaptive intelligence gray predictive model (SAIGM) has an alterable‐flexible model structure, and it can build a dynamic structure to fit different external environments by adjusting the parameter values of SAIGM. However, the order number of the raw SAIGM model is not optimal, which is an integer. For this, a new SAIGM model with the fractional order accumulating operator (SAIGM_FO) was proposed in this paper. Specifically, the final restored expression of SAIGM_FO was deduced in detail, and the parameter estimation method of SAIGM_FO was studied. After that, the Particle Swarm Optimization algorithm was used to optimize the order number of SAIGM_FO, and some steps were provided. Finally, the SAIGM_FO model was applied to simulate China's electricity consumption from 2001 to 2008 and forecast it during 2009 to 2015, and the mean relative simulation and prediction percentage errors of the new model were only 0.860% and 2.661%, in comparison with the ones obtained from the raw SAIGM model, the GM(1, 1) model with the optimal fractional order accumulating operator and the GM(1, 1) model, which were (1.201%, 5.321%), (1.356%, 3.324%), and (2.013%, 23.944%), respectively. The findings showed both the simulation and the prediction performance of the proposed SAIGM_FO model were the best among the 4 models.     

A refinement of an optimality criterion and its application to parametric programming

We consider semi-infinite linear minimization problems and prove a refinement of an optimality condition proved earlier by the author. This refinement is used to derive a sufficient condition for strong uniqueness of a minimal point. As an application, we show that these strongly unique minimal points depend (pointwise) Lipschitz-continuously on the parameter of the minimization problem. Finally, we consider numerical algorithms for semi-infinite optimization problems and we apply the above results to derive error estimates for these algorithms.     

A generalized Gronwall inequality and its application to a fractional differential equation

This paper presents a generalized Gronwall inequality with singularity. Using the inequality, we study the dependence of the solution on the order and the initial condition of a fractional differential equation.     

A Bayesian nonparametric model and its application in insurance loss prediction

Predicting insurance losses is an eternal focus of actuarial science in the insurance sector. Due to the existence of complicated features such as skewness, heavy tail, and multi-modality, traditional parametric models are often inadequate to describe the distribution of losses, calling for a mature application of Bayesian methods. In this study we explore a Gaussian mixture model based on Dirichlet process priors. Using three automobile insurance datasets, we employ the probit stick-breaking method to incorporate the effect of covariates into the weight of the mixture component, improve its hierarchical structure, and propose a Bayesian nonparametric model that can identify the unique regression pattern of different samples. Moreover, an advanced updating algorithm of slice sampling is integrated to apply an improved approximation to the infinite mixture model. We compare our framework with four common regression techniques: three generalized linear models and a dependent Dirichlet process ANOVA model. The empirical results show that the proposed framework flexibly characterizes the actual loss distribution in the insurance datasets and demonstrates superior performance in the accuracy of data fitting and extrapolating predictions, thus greatly extending the application of Bayesian methods in the insurance sector.     

A study of the fractional‐order mathematical model of diabetes and its resulting complications

Diabetes is a worldwide problem that affects one of every 11 persons nowadays. The IDF Diabetes Atlas (Eighth edition, 2017) states that approximately 415 million people in the world are living with the disease and that this number will rise to 629 million by the year 2045. It is a very serious problem of the world. A major part of the world population is affected by this disease and its resulting complications. In this paper, we propose to investigate a fractional‐order model of diabetes and its resulting complications. The mathematical model's parameters define the population of diabetic patients and those who are diabetic with complications at a given time t. We have also discussed the existence, uniqueness, and stability of the fractional‐order model, which we consider here. We make use of the homotopy decomposition method (HDM) in order to solve the problem.