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Numerical Algebra, Control & Optimization

March 2019 , Volume 9 , Issue 1

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A brief survey of methods for solving nonlinear least-squares problems
Hassan Mohammad, Mohammed Yusuf Waziri and Sandra Augusta Santos
2019, 9(1): 1-13 doi: 10.3934/naco.2019001 +[Abstract](328) +[HTML](93) +[PDF](397.82KB)

In this paper, we present a brief survey of methods for solving nonlinear least-squares problems. We pay specific attention to methods that take into account the special structure of the problems. Most of the methods discussed belong to the quasi-Newton family (i.e. the structured quasi-Newton methods (SQN)). Our survey comprises some of the traditional and modern developed methods for nonlinear least-squares problems. At the end, we suggest a few topics for further research.

Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA*=B
Sihem Guerarra
2019, 9(1): 15-22 doi: 10.3934/naco.2019002 +[Abstract](141) +[HTML](74) +[PDF](327.08KB)

This work is devoted to establish the extremal inertias ofthe two submatrices $X_{1}$ and $X_{4}$ in a Hermitian least rank solution $X$of the matrix equation $AXA^{*}=B$. From these formulas, necessary andsufficient conditions for these submatrices to be positive (nonpositive,negative, nonnegative) definite are achieved.

Stability preservation in Galerkin-type projection-based model order reduction
Roland Pulch
2019, 9(1): 23-44 doi: 10.3934/naco.2019003 +[Abstract](114) +[HTML](74) +[PDF](838.1KB)

We consider linear dynamical systems consisting of ordinary differential equations with high dimensionality. The aim of model order reduction is to construct an approximating system of a much lower dimension. Therein, the reduced system may be unstable, even though the original system is asymptotically stable. We focus on projection-based model order reduction of Galerkin-type. A transformation of the original system guarantees an asymptotically stable reduced system. This transformation requires the numerical solution of a high-dimensional Lyapunov equation. We specify an approximation of the solution, which allows for an efficient iterative treatment of the Lyapunov equation under a certain assumption. Furthermore, we generalize this strategy to preserve the asymptotic stability of stationary solutions in model order reduction of nonlinear dynamical systems. Numerical results for high-dimensional examples confirm the computational feasibility of the stability-preserving approach.

Local smooth representation of solution sets in parametric linear fractional programming problems
Rui Qian, Rong Hu and Ya-Ping Fang
2019, 9(1): 45-52 doi: 10.3934/naco.2019004 +[Abstract](122) +[HTML](62) +[PDF](284.38KB)

The purpose of this paper is to investigate the structure of the solution sets in parametric linear fractional programming problems. It is shown that the solution set of a parametric linear fractional programming problem with smooth data has a local smooth representation. As a consequence, the corresponding marginal function is differentiable and the solution map admits a differentiable selection. We also give an example to illustrate the result.

On construction of upper and lower bounds for the HOMO-LUMO spectral gap
Soña Pavlíková and Daniel Ševčovič
2019, 9(1): 53-69 doi: 10.3934/naco.2019005 +[Abstract](172) +[HTML](65) +[PDF](678.7KB)

In this paper we study spectral properties of graphs which are constructed from two given invertible graphs by bridging them over a bipartite graph. We analyze the so-called HOMO-LUMO spectral gap which is the difference between the smallest positive and largest negative eigenvalue of the adjacency matrix of a graph. We investigate its dependence on the bridging bipartite graph and we construct a mixed integer semidefinite program for maximization of the HOMO-LUMO gap with respect to the bridging bipartite graph. We also derive upper and lower bounds for the optimal HOMO-LUMO spectral graph by means of semidefinite relaxation techniques. Several computational examples are also presented in this paper.

A fourth order implicit symmetric and symplectic exponentially fitted Runge-Kutta-Nyström method for solving oscillatory problems
Wenjuan Zhai and Bingzhen Chen
2019, 9(1): 71-84 doi: 10.3934/naco.2019006 +[Abstract](132) +[HTML](68) +[PDF](3582.26KB)

In this paper, we derive an implicit symmetric, symplectic and exponentially fitted Runge-Kutta-Nyström (ISSEFRKN) method. The new integrator ISSEFRKN2 is of fourth order and integrates exactly differential systems whose solutions can be expressed as linear combinations of functions from the set $\{\exp(λ t), \exp(-λ t)|λ∈ \mathbb{C}\}$, or equivalently $\{\sin(ω t), \cos(ω t)|λ = iω, ~ω∈ \mathbb{R}\}$. We analysis the periodicity stability of the derived method ISSEFRKN2. Some the existing implicit RKN methods in the literature are used to compare with ISSEFRKN2 for several oscillatory problems. Numerical results show that the method ISSEFRKN2 possess a more accuracy among them.

Semi-local convergence of the Newton-HSS method under the center Lipschitz condition
Hongxiu Zhong, Guoliang Chen and Xueping Guo
2019, 9(1): 85-99 doi: 10.3934/naco.2019007 +[Abstract](132) +[HTML](68) +[PDF](405.18KB)

Newton-type methods have gained much attention in the past decades, especially for the semilocal convergence based on no information around the solution $x_*$ of the target nonlinear equation. For large sparse non-Hermitian positive definite systems of nonlinear equation, assuming that the nonlinear operator satisfies the center Lipschitz condition, which is wider than usual Lipschtiz condition and H$ö$lder continuous condition, we establish a new Newton-Kantorovich convergence theorem for the Newton-HSS method. Once the convergence criteria is satisfied, the iteration sequence $\{x_k\}_{k = 0}^∞$ generated by the Newton-HSS method is well defined, and converges to the solution $x_*$. Numerical results illustrate the effect.

Solving optimal control problem using Hermite wavelet
Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri and Sohrab Effati
2019, 9(1): 101-112 doi: 10.3934/naco.2019008 +[Abstract](163) +[HTML](75) +[PDF](359.78KB)

In this paper, we derive the operational matrices of integration, derivative and production of Hermite wavelets and use a direct numerical method based on Hermite wavelet, for solving optimal control problems. The properties of Hermite polynomials are used for finding these matrices. First, we approximate the state and control variables by Hermite wavelets basis; then, the operational matrices is used to transfer the given problem into a linear system of algebraic equations. In fact, operational matrices of Hermite wavelet are employed to achieve a linear algebraic equation, in place of the dynamical system in terms of the unknown coefficients. The solution of this system gives us the solution of the original problem. Numerical examples with time varying and time invariant coefficient are given to demonstrate the applicability of these matrices.




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