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Numerical Algebra, Control & Optimization

2016 , Volume 6 , Issue 1

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Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control
Peter Benner , Jens Saak and  M. Monir Uddin
2016, 6(1): 1-20 doi: 10.3934/naco.2016.6.1 +[Abstract](44) +[PDF](757.9KB)
Abstract:
Stabilizing a flow around an unstable equilibrium is a typical problem in flow control. Model-based designed of modern controllers like LQR/LQG or $H_\infty$ compensators is often limited by the large-scale of the discretized flow models. Therefore, model reduction is usually needed before designing such a controller. Here we suggest an approach based on applying balanced truncation for unstable systems to the linearized flow equations usually used for compensator design. For this purpose, we modify the ADI iteration for Lyapunov equations to deal with the index-2 structure of the underlying descriptor system efficiently in an implicit way. The resulting algorithm is tested for model reduction and control design of a linearized Navier-Stokes system describing von Kármán vortex shedding.
Optimal layer reinsurance on the maximization of the adjustment coefficient
Xuepeng Zhang and  Zhibin Liang
2016, 6(1): 21-34 doi: 10.3934/naco.2016.6.21 +[Abstract](45) +[PDF](418.2KB)
Abstract:
In this paper, we study the optimal retentions for an insurance company, which intends to transfer risk by means of a layer reinsurance treaty. Under the criterion of maximizing the adjustment coefficient, the closed form expressions of the optimal results are obtained for the Brownian motion risk model as well as the compound Poisson risk model. Moreover, we conclude that under the expected value principle there exists a special layer reinsurance strategy, i.e., excess of loss reinsurance strategy which is better than any other layer reinsurance strategies. Whereas, under the variance premium principle, the pure excess of loss reinsurance is not the optimal layer reinsurance strategy any longer. Some numerical examples are presented to show the impacts of the parameters as well as the premium principles on the optimal results.
Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps
Guolin Yu
2016, 6(1): 35-44 doi: 10.3934/naco.2016.6.35 +[Abstract](78) +[PDF](298.3KB)
Abstract:
This paper deals with the characteristics of global proper efficient points and the optimality conditions of vector optimization problems involving generalized convex set-valued maps. Several equivalent properties of global proper efficient points are proposed. Utilizing cone-directed contingent derivative, it presents the unified necessary and sufficient optimality conditions for global proper efficient element in vector optimization problem with cone-arcwise connected set-valued mapping.
A new convergence proof of augmented Lagrangian-based method with full Jacobian decomposition for structured variational inequalities
Xi-Hong Yan
2016, 6(1): 45-54 doi: 10.3934/naco.2016.6.45 +[Abstract](40) +[PDF](309.6KB)
Abstract:
In the work, we present a new proof for global convergence of a classical method, augmented Lagrangian-based method with full Jacobian decomposition, for a special class of variational inequality problems with a separable structure. This work can be regarded as an improvement to work [14]. The convergence result of the work is established under more general conditions and proven in a new way.
Deflation by restriction for the inverse-free preconditioned Krylov subspace method
Qiao Liang and  Qiang Ye
2016, 6(1): 55-71 doi: 10.3934/naco.2016.6.55 +[Abstract](85) +[PDF](496.2KB)
Abstract:
A deflation by restriction scheme is developed for the inverse-free preconditioned Krylov subspace method for computing a few extreme eigenvalues of the definite symmetric generalized eigenvalue problem $Ax = \lambda Bx$. The convergence theory for the inverse-free preconditioned Krylov subspace method is generalized to include this deflation scheme and numerical examples are presented to demonstrate the convergence properties of the algorithm with the deflation scheme.
Projection-based model reduction for time-varying descriptor systems: New results
Mohammad-Sahadet Hossain
2016, 6(1): 73-90 doi: 10.3934/naco.2016.6.73 +[Abstract](86) +[PDF](1007.1KB)
Abstract:
We have presented a Krylov-based projection method for model reduction of linear time-varying descriptor systems in [13] which was based on earlier ideas in the work of J. Philips [17] and others. This contribution continues that work by presenting more details of linear time-varying descriptor systems and new results coming from real fields of application. The idea behind the proposed procedure is based on a multipoint rational approximation of the monodromy matrix of the corresponding differential-algebraic equation. This is realized by orthogonal projection onto a rational Krylov subspace. The algorithmic realization of the method employs recycling techniques for shifted Krylov subspaces and their invariance properties. The proposed method works efficiently for macro-models, such as time varying circuit systems and models arising in network interconnection, on limited frequency ranges. Bode plots and step response are used to illustrate both the performance and accuracy of the reduced-order model.

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