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Numerical Algebra, Control & Optimization

2016 , Volume 6 , Issue 2

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On bounds of the Pythagoras number of the sum of square magnitudes of Laurent polynomials
Thanh Hieu Le and  Marc Van Barel
2016, 6(2): 91-102 doi: 10.3934/naco.2016001 +[Abstract](66) +[PDF](392.1KB)
This paper presents a lower and upper bound of the Pythagoras number of sum of square magnitudes of Laurent polynomials (sosm-polynomials). To prove these bounds, properties of the corresponding system of quadratic polynomial equations are used. Applying this method, a new proof for the best (known until now) upper bound of the Pythagoras number of real polynomials is also presented.
Index-proper nonnegative splittings of matrices
Chinmay Kumar Giri
2016, 6(2): 103-113 doi: 10.3934/naco.2016002 +[Abstract](51) +[PDF](348.3KB)
The theory of splitting is a useful tool for finding solution of a system of linear equations. Many woks are going on for singular system of linear equations. In this article, we have introduced a new splitting called index-proper nonnegative splitting for singular square matrices. Several convergence and comparison results are also established. We then apply the same theory to double splitting.
Derivatives of eigenvalues and Jordan frames
Manuel V. C. Vieira
2016, 6(2): 115-126 doi: 10.3934/naco.2016003 +[Abstract](33) +[PDF](336.9KB)
Every element in a Euclidean Jordan algebra has a spectral decomposition. This spectral decomposition is generalization of the spectral decompositions of a matrix. In the context of Euclidean Jordan algebras, this is written using eigenvalues and the so-called Jordan frame. In this paper we deduce the derivative of eigenvalues in the context of Euclidean Jordan algebras. We also deduce the derivative of the elements of a Jordan frame associated to the spectral decomposition.
Output feedback overlapping control design of interconnected systems with input saturation
Magdi S. Mahmoud
2016, 6(2): 127-151 doi: 10.3934/naco.2016004 +[Abstract](44) +[PDF](763.8KB)
In this paper, we establish new results to the problem of output feedback control design for a class of nonlinear interconnected continuous-time systems subject to input saturation. New schemes based on overlapping design methodology are developed for both static and dynamic output feedback control structures. The theoretical developments are illustrated by numerical simulations of a linearized nuclear power plant model.
Solving Malfatti's high dimensional problem by global optimization
Rentsen Enkhbat , M. V. Barkova and  A. S. Strekalovsky
2016, 6(2): 153-160 doi: 10.3934/naco.2016005 +[Abstract](36) +[PDF](347.7KB)
We generalize Malfatti's problem which dates back to 200 years ago as a global optimization problem in a high dimensional space. The problem has been formulated as the convex maximization problem over a nonconvex set. Global optimality condition by Strekalovsky [11] has been applied to this problem. For solving numerically Malfatti's problem, we propose the algorithm in [3] which converges globally. Some computational results are provided.
A new smoothing approach to exact penalty functions for inequality constrained optimization problems
Ahmet Sahiner , Gulden Kapusuz and  Nurullah Yilmaz
2016, 6(2): 161-173 doi: 10.3934/naco.2016006 +[Abstract](35) +[PDF](356.7KB)
In this study, we introduce a new smoothing approximation to the non-differentiable exact penalty functions for inequality constrained optimization problems. Error estimations are investigated between non-smooth penalty function and smoothed penalty function. In order to demonstrate the effectiveness of proposed smoothing approach the numerical examples are given.
On general form of the Tanh method and its application to nonlinear partial differential equations
Ali Hamidoǧlu
2016, 6(2): 175-181 doi: 10.3934/naco.2016007 +[Abstract](139) +[PDF](280.0KB)
The tanh method is used to compute travelling waves solutions of one-dimensional nonlinear wave and evolution equations. The technique is based on seeking travelling wave solutions in the form of a finite series in tanh. In this article, we introduce a new general form of tanh transformation and solve well-known nonlinear partial differential equations in which tanh method becomes weaker in the sense of obtaining general form of solutions.
Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -- efficient solution strategies based on homogenization theory
Erik Kropat , Silja Meyer-Nieberg and  Gerhard-Wilhelm Weber
2016, 6(2): 183-219 doi: 10.3934/naco.2016008 +[Abstract](161) +[PDF](2229.8KB)
Boundary value problems on large periodic networks arise in many applications such as soil mechanics in geophysics or the analysis of photonic crystals in nanotechnology. As a model example, singularly perturbed elliptic differential equations of second order are addressed. Typically, the length of periodicity is very small compared to the size of the covered region. The overall complexity of the networks raises serious problems on the computational side. The high density of the graph, the huge number of edges and vertices and highly oscillating coefficients necessitate solution schemes, where even a numerical approximation is no longer feasible. Realizing that such a system depends on two spatial scales - global scale (full domain) and local scale (microstructure) - a two-scale asymptotic analysis for network differential equations is applied. The limit process leads to a homogenized model on the full domain. The homogenized coefficients cover the micro-oscillations and the topology of the periodic network and characterize the effective behaviour. The approximate model's quality is guaranteed by error estimates. Furthermore, singularly perturbed microscopic models with a decreasing diffusion part and transport-dominant problems are discussed. The effectiveness of the two-scale limit analysis is demonstrated by numerical examples of diffusion-advection-reaction problems on large periodic grids.




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