@article {MoNo2007, author="K. Moriya and T. Nodera", title="{Usage of the convergence test of the residual norm in the Tsuno--Nodera version of the GMRES algorithm}", journal="ANZIAM J.", fjournal={ANZIAM Journal}, volume="49", year="2007", number="2", pages="293--308", issn="1446-8735", coden="AJNOA2", language="English", date="Received 13 October, 2005; revised 15 January, 2007", classmath="primary 65F10; secondary 65M12", publisher={AMPAI, Australian Mathematical Society}, keywords={linear system of equations, GMRES($m$) algorithm, parallel computer, adaptive restart}, MRID="MR2376???", url="http://www.austms.org.au/Publ/ANZIAM/V49P2/492-2339-MoNo/index.html", abstract={Tsuno and Nodera proposed a new variant of the GMRES($m$) algorithm. Their algorithm is referred to as the GMRES($\leqslant \! m_{\max }$) algorithm and performs the restart process adaptively, considering the distribution of the zeros of the residual polynomial. However, unless the zeros of the residual polynomial are distributed uniformly, $m_{\max }$ is always chosen and their algorithm becomes almost the same as the GMRES($m$) algorithm with $m=m_{\max }$. \par In this paper, we include a convergence test for the residual norm in the GMRES($\leqslant \! m_{\max }$) algorithm and propose a new restarting technique based on two criteria. Even if the distribution of zeros does not become uniform, the restart can be performed by using the convergence test of the residual norm. Numerical examples simulated on a Compaq Beowulf computer demonstrate that the proposed technique accelerates the convergence of the GMRES($\leqslant \! m_{\max }$) algorithm. } } @article {YaOu2007, author="Runsheng Yang and Yunhua Ou", title="{Inverse coefficient problems for nonlinear elliptic equations}", journal="ANZIAM J.", fjournal={ANZIAM Journal}, volume="49", year="2007", number="2", pages="271--279", issn="1446-8735", coden="AJNOA2", language="English", date="Received 12 June, 2006; revised 3 October, 2006", classmath="primary 35R30; secondary 35J85", publisher={AMPAI, Australian Mathematical Society}, keywords={elliptic equations, inverse coefficient problems, existence of quasisolutions}, MRID="MR2376???", url="http://www.austms.org.au/Publ/ANZIAM/V49P2/492-2429-YaOu/index.html", abstract={This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic equations. The unknown coefficient of the elliptic equations depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic equations are uniquely solvable for the given class of coefficients. Proof of the existence of a quasisolution of the inverse problems is obtained. } } @article {ChLiuSr2007, author="Kung-Yu Chen and Shuoh-Jung Liu and H. M. Srivastava", title="{Some new results for the Lagrange polynomials in several variables}", journal="ANZIAM J.", fjournal={ANZIAM Journal}, volume="49", year="2007", number="2", pages="243--258", issn="1446-8735", coden="AJNOA2", language="English", date="Received 1 March, 2007; revised 29 October, 2007", classmath="primary 33C05, 33C45; secondary 11B73", publisher={AMPAI, Australian Mathematical Society}, keywords={Lagrange polynomials, summation identity, Chan--Chyan--Srivastava polynomials, differential operators, generating functions, Jacobi polynomials, Stirling numbers of the second kind, Cauchy integral formula, Laguerre polynomials}, MRID="MR2376???", url="http://www.austms.org.au/Publ/ANZIAM/V49P2/492-2509-ChLiuSr/index.html", abstract={In some recent investigations involving certain differential operators for a general family of Lagrange polynomials, Chan \textit {et al.} encountered and proved a certain summation identity for the Lagrange polynomials in several variables. In the present paper, we derive some generalizations of this summation identity for the Chan--Chyan--Srivastava polynomials in several variables. We also discuss a number of interesting corollaries and consequences of our main results. } } @article {Verma2007, author="Ram U. Verma", title="{General projection systems and relaxed cocoercive nonlinear variational inequalities}", journal="ANZIAM J.", fjournal={ANZIAM Journal}, volume="49", year="2007", number="2", pages="205--212", issn="1446-8735", coden="AJNOA2", language="English", date="Received 8 April, 2007", classmath="primary 49J40, 65B05; secondary 47H20", publisher={AMPAI, Australian Mathematical Society}, keywords={relaxed cocoercive mappings, approximation solvability, projection system, system of nonlinear relaxed cocoercive variational inequalities}, MRID="MR2376???", url="http://www.austms.org.au/Publ/ANZIAM/V49P2/492-2529-Verma/index.html", abstract={We explore the \emph {solvability} of a general system of nonlinear relaxed cocoercive variational inequality (SNVI) problems based on a new projection system for the direct product of two nonempty closed and convex subsets of real Hilbert spaces. } } @article {AmHa2007, author="Keyvan Amini and Arash Haseli", title="{A new proximity function generating the best known iteration bounds for both large-update and small-update interior-point methods}", journal="ANZIAM J.", fjournal={ANZIAM Journal}, volume="49", year="2007", number="2", pages="259--270", issn="1446-8735", coden="AJNOA2", language="English", date="Received 16 December, 2006; revised 6 August, 2007", classmath="primary 90C05; secondary 90C51", publisher={AMPAI, Australian Mathematical Society}, keywords={linear optimization, interior-point method, primal-dual method, kernel function, large-update, small-update, polynomial complexity}, MRID="MR2376???", url="http://www.austms.org.au/Publ/ANZIAM/V49P2/492-2533-AmHa/index.html", abstract={Interior-Point Methods (IPMs) are not only very effective in practice for solving linear optimization problems but also have polynomial-time complexity. Despite the practical efficiency of large-update algorithms, from a theoretical point of view, these algorithms have a weaker iteration bound with respect to small-update algorithms. In fact, there is a significant gap between theory and practice for large-update algorithms. By introducing self-regular barrier functions, Peng, Roos and Terlaky improved this gap up to a factor of $\log n$. However, checking these self-regular functions is not simple and proofs of theorems involving these functions are very complicated. Roos \textit {et al.} by presenting a new class of barrier functions which are not necessarily self-regular, achieved very good results through some much simpler theorems. In this paper we introduce a new kernel function in this class which yields the best known complexity bound, both for large-update and small-update methods. } } @article {WuHu2007, author="Shi-Liang Wu and Ting-Zhu Huang", title="{A modified AOR-type iterative method for $L$-matrix linear systems}", journal="ANZIAM J.", fjournal={ANZIAM Journal}, volume="49", year="2007", number="2", pages="281--292", issn="1446-8735", coden="AJNOA2", language="English", date="Received 14 April, 2007; revised 22 October, 2007", classmath="primary 65F10; secondary 15A06", publisher={AMPAI, Australian Mathematical Society}, keywords={preconditioner; $L$-matrix; AOR method; spectral radius}, MRID="MR2376???", url="http://www.austms.org.au/Publ/ANZIAM/V49P2/492-2544-WuHu/index.html", abstract={Both Evans \textit {et al.} and Li \textit {et al.} have presented preconditioned methods for linear systems to improve the convergence rates of AOR-type iterative schemes. In this paper, we present a new preconditioner. Some comparison theorems on preconditioned iterative methods for solving $L$-matrix linear systems are presented. Comparison results and a numerical example show that convergence of the preconditioned Gauss--Seidel method is faster than that of the preconditioned AOR iterative method. } } @article {ChEGORAg2007, author="A. Cherid and M. A. El-Gebeily and Donal O'Regan and Ravi Agarwal", title="{Deblurring and denoising of images with minimization of variation and negative norms}", journal="ANZIAM J.", fjournal={ANZIAM Journal}, volume="49", year="2007", number="2", pages="171--185", issn="1446-8735", coden="AJNOA2", language="English", date="Received March 28, 2007; revised October 16, 2007", classmath="primary 68U10; secondary 94A08", publisher={AMPAI, Australian Mathematical Society}, keywords={image deblurring, image restoration, variation minimization, negative norms}, MRID="MR2376???", url="http://www.austms.org.au/Publ/ANZIAM/V49P2/492-2549-ChEGORAg/index.html", abstract={A method based on the minimization of variation is presented for the identification of a completely unknown blur operator. We assume the knowledge of a blurred image and its original version. The class of blurring operators is identified in the class of compact operators. A variational method with negative norms is then used for the restoration of a blurred and noised image. The restoration method works for a wide class of blurring operators and we do not assume that the blur operator commutes with the Laplacian. } } @article {LiFeTeoDu2007, author="R. Li and Z. G. Feng and K. L. Teo and G. R. Duan", title="{Tracking control of linear switched systems}", journal="ANZIAM J.", fjournal={ANZIAM Journal}, volume="49", year="2007", number="2", pages="187--203", issn="1446-8735", coden="AJNOA2", language="English", date="Received 9 May, 2007", classmath="primary 49N90; secondary 90C11, 93C95", publisher={AMPAI, Australian Mathematical Society}, keywords={optimal tracking control, switched systems, a time scaling transformation, linear quadratic control theory, discrete filled function}, MRID="MR2376???", url="http://www.austms.org.au/Publ/ANZIAM/V49P2/492-2578-LiFeTeoDu/index.html", abstract={This paper deals with the optimal tracking problem for switched systems, where the control input, the switching times and the switching index are all design variables. We propose a three-stage method for solving this problem. First, we fix the switching times and switching index sequence, which leads to a linear tracking problem, except different subsystems are defined in their respective time intervals. The optimal control and the corresponding cost function obtained depend on the switching signal. This gives rise to an optimal parameter selection problem for which the switching instants and the switching index are to be chosen optimally. In the second stage, the switching index is fixed. A reverse time transformation followed by a time scaling transform are introduced to convert this subproblem into an equivalent standard optimal parameter selection problem. The gradient formula of the cost function is derived. Then the discrete filled function is used in the third stage to search for the optimal switching index. On this basis, a computational method, which combines a gradient-based method, a local search algorithm and a filled function method, is developed for solving this problem. A numerical example is solved, showing the effectiveness of the proposed approach. } } @article {BoOrRo2007, author="J. Fern\'andez Bonder and R. Orive and J. D. Rossi", title="{The best Sobolev trace constant in domains with holes for critical or subcritical exponents}", journal="ANZIAM J.", fjournal={ANZIAM Journal}, volume="49", year="2007", number="2", pages="213--230", issn="1446-8735", coden="AJNOA2", language="English", date="Received 8 November, 2006", classmath="primary 35B27, 35J65; secondary 46E35", publisher={AMPAI, Australian Mathematical Society}, keywords={homogenization, nonlinear boundary conditions, Sobolev trace embedding}, MRID="MR2376???", url="http://www.austms.org.au/Publ/ANZIAM/V49P2/492-2580-BoOrRo/index.html", abstract={In this paper we study the best constant in the Sobolev trace embedding $H^1(\Omega ) \hookrightarrow L^q (\partial \Omega )$ in a bounded smooth domain for $1< q\le 2_*= 2(N-1)/(N-2)$, that is, critical or subcritical $q$. First, we consider a domain with periodically distributed holes inside which we impose that the involved functions vanish. There exists a critical size of the holes for which the limit problem has an extra term. For sizes larger than critical the best trace constant diverges to infinity and for sizes smaller than critical it converges to the best constant in the domain without holes. Also, we study the problem with the holes located on the boundary of the domain. In this case another critical exists and its extra term appears on the boundary. } } @article {HouDoShi2007, author="Zhenting Hou and Hailing Dong and Peng Shi", title="{Asymptotic stability in the distribution of nonlinear stochastic systems with semi-Markovian switching}", journal="ANZIAM J.", fjournal={ANZIAM Journal}, volume="49", year="2007", number="2", pages="231--241", issn="1446-8735", coden="AJNOA2", language="English", date="Received March 17, 2007; revised August 6, 2007", classmath="primary 34A34; secondary 60K15", publisher={AMPAI, Australian Mathematical Society}, keywords={phase distribution, semi-Markovian switching, associated Markov chain, asymptotic stability in distribution}, MRID="MR2376???", url="http://www.austms.org.au/Publ/ANZIAM/V49P2/492-2581-HouDoShi/index.html", abstract={In this paper, finite phase semi-Markov processes are introduced. By introducing variables and a simple transformation, every finite phase semi-Markov process can be transformed to a finite Markov chain which is called its associated Markov chain. A consequence of this is that every phase semi-Markovian switching system may be equivalently expressed as its associated Markovian switching system. Existing results for Markovian switching systems may then be applied to analyze phase semi-Markovian switching systems. In the following, we obtain asymptotic stability for the distribution of nonlinear stochastic systems with semi-Markovian switching. The results can also be extended to general semi-Markovian switching systems. Finally, an example is given to illustrate the feasibility and effectiveness of the theoretical results obtained. } } @article {ChPlJaBa2007, author="S. J. Chapman and M. J. Plank and A. James and B. Basse", title="{A nonlinear model of age and size-structured populations with applications to cell cycles}", journal="ANZIAM J.", fjournal={ANZIAM Journal}, volume="49", year="2007", number="2", pages="151--169", issn="1446-8735", coden="AJNOA2", language="English", date="Received 15 July, 2007; revised 20 October, 2007", classmath="primary 37N25; secondary 92D25", publisher={AMPAI, Australian Mathematical Society}, keywords={age-structure, size-structure, populations, cell cycle, mathematical model}, MRID="MR2376???", url="http://www.austms.org.au/Publ/ANZIAM/V49P2/492-2584-ChPlJaBa/index.html", abstract={The Sharpe--Lotka--McKendrick (or von Foerster) equations for an age-structured population, with a nonlinear term to represent overcrowding or competition for resources, are considered. The model is extended to include a growth term, allowing the population to be structured by size or weight rather than age, and a general solution is presented. Various examples are then considered, including the case of cell growth where cells divide at a given size. } }