On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

Newton Polyhedra of Discriminants of Projections

For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving Euler obstructions of toric varieties) in the unmixed case, suggests certain open questions in general, and generalizes a number of similar known results (Gelfand et al. in Discriminants, resultants, and multidimensional determinants. Birkhäuser, Boston, 1994; Sturmfels in J. Algebraic Comb. 32(2):207–236, 1994; McDonald in Discrete Comput. Geom. 27:501–529, 2002; Gonzalez-Perez in Can. J. Math. 52(2):348-368, 2000; Esterov and Khovanskii in Funct. Anal. Math. 2(1), 2008).     

An inverse-free Newton-Jarratt-type iterative method for solving equations under the gamma condition

A local as well as a semilocal convergence analysis for Newton–Jarratt–type iterative method for solving equations in a Banach space setting is studied here using information only at a point via a gamma-type condition (Argyros in Approximate Solution of Operator Equations with Applications, [2005]; Wang in Chin. Sci. Bull. 42(7):552–555, [1997]). This method has already been examined by us in (Argyros et al. in J. Comput. Appl. Math. 51:103–106, [1994]; Argyros in Comment. Mat. XXIII:97–108, [1994]), where the order of convergence four was established using however information on the domain of the operator. In this study we also establish the same order of convergence under weaker conditions. Moreover we show that all though we use weaker conditions the results obtained here can be used to solve equations in cases where the results in (Argyros et al. in J. Comput. Appl. Math. 51:103–106, [1994]; Argyros in Comment. Mat. XXIII:97–108, [1994]) cannot apply. Our method is inverse free, and therefore cheaper at the second step in contrast with the corresponding two–step Newton methods. Numerical Examples are also provided.     

Linearly Implicit Approximations of Diffusive Relaxation Systems

Diffusive relaxation systems provide a general framework to approximate nonlinear diffusion problems, also in the degenerate case (Aregba-Driollet et al. in Math. Comput. 73(245):63–94, 2004; Boscarino et al. in Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, 2011; Cavalli et al. in SIAM J. Sci. Comput. 34:A137–A160, 2012; SIAM J. Numer. Anal. 45(5):2098–2119, 2007; Naldi and Pareschi in SIAM J. Numer. Anal. 37:1246–1270, 2000; Naldi et al. in Surveys Math. Indust. 10(4):315–343, 2002). Their discretization is usually obtained by explicit schemes in time coupled with a suitable method in space, which inherits the standard stability parabolic constraint. In this paper we combine the effectiveness of the relaxation systems with the computational efficiency and robustness of the implicit approximations, avoiding the need to resolve nonlinear problems and avoiding stability constraints on time step. In particular we consider an implicit scheme for the whole relaxation system except for the nonlinear source term, which is treated though a suitable linearization technique. We give some theoretical stability results in a particular case of linearization and we provide insight on the general case. Several numerical simulations confirm the theoretical results and give evidence of the stability and convergence also in the case of nonlinear degenerate diffusion.     

An improved local convergence analysis for a two-step Steffensen-type method

We study a class of Steffensen-type algorithm for solving nonsmooth variational inclusions in Banach spaces. We provide a local convergence analysis under ω-conditioned divided difference, and the Aubin continuity property. This work on the one hand extends the results on local convergence of Steffensen’s method related to the resolution of nonlinear equations (see Amat and Busquier in Comput. Math. Appl. 49:13–22, 2005; J. Math. Anal. Appl. 324:1084–1092, 2006; Argyros in Southwest J. Pure Appl. Math. 1:23–29, 1997; Nonlinear Anal. 62:179–194, 2005; J. Math. Anal. Appl. 322:146–157, 2006; Rev. Colomb. Math. 40:65–73, 2006; Computational Theory of Iterative Methods, 2007). On the other hand our approach improves the ratio of convergence and enlarges the convergence ball under weaker hypotheses than one given in Hilout (Commun. Appl. Nonlinear Anal. 14:27–34, 2007).     

Improved sparse fourier approximation results: faster implementations and stronger guarantees

We study the problem of quickly estimating the best k-term Fourier representation for a given periodic function f: [0, 2π] → ?. Solving this problem requires the identification of k of the largest magnitude Fourier series coefficients of f in worst case k ² · log ^O(1) N time. Randomized sublinear-time Monte Carlo algorithms, which have a small probability of failing to output accurate answers for each input signal, have been developed for solving this problem (Gilbert et al. 2002, 2005). These methods were implemented as the Ann Arbor Fast Fourier Transform (AAFFT) and empirically evaluated in Iwen et al. (Commun Math Sci 5(4):981–998, 2007). In this paper we present a new implementation, called the Gopher Fast Fourier Transform (GFFT), of more recently developed sparse Fourier transform techniques (Iwen, Found Comput Math 10(3):303–338, 2010, Appl Comput Harmon Anal, 2012). Experiments indicate that GFFT is faster than AAFFT. In addition to our empirical evaluation, we also consider the existence of sublinear-time Fourier approximation methods with deterministic approximation guarantees for functions whose sequences of Fourier series coefficents are compressible. In particular, we prove the existence of sublinear-time Las Vegas Fourier Transforms which improve on the recent deterministic Fourier approximation results of Iwen (Found Comput Math 10(3):303–338, 2010, Appl Comput Harmon Anal, 2012) for Fourier compressible functions by guaranteeing accurate answers while using an asymptotically near-optimal number of function evaluations.     

General convergence conditions of Newton’s method for m-Fréchet differentiable operators

We present a local as well as a semilocal convergence analysis for Newton’s method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. Our hypotheses involve m-Fréchet-differentiable operators and general Lipschitz-type hypotheses, where m≥2 is a positive integer. The new convergence analysis unifies earlier results; it is more flexible and provides a finer convergence analysis than in earlier studies such as Argyros in J. Comput. Appl. Math. 131:149–159, 2001, Argyros and Hilout in J. Appl. Math. Comput. 29:391–400, 2009, Argyros and Hilout in J. Complex. 28:364–387, 2012, Argyros et al. Numerical Methods for Equations and Its Applications, CRC Press/Taylor & Francis, New York, 2012, Gutiérrez in J. Comput. Appl. Math. 79:131–145, 1997, Ren and Argyros in Appl. Math. Comput. 217:612–621, 2010, Traub and Wozniakowski in J. Assoc. Comput. Mech. 26:250–258, 1979. Numerical examples are presented further validating the theoretical results.     

Defect and Area in Beltrami–Klein Model of Hyperbolic Geometry

Ungar (Beyond the Einstein addition law and its gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrouector Spaces, 2001; Comput Math Appl 49:187–221, 2005; Comput Math Appl 53, 2007) introduced into hyperbolic geometry the concept of defect based on relativity addition of A. Einstein. Another approach is from Karzel (Resultate Math. 47:305–326, 2005) for the relation between the K-loop and the defect of an absolute plane in the sense (Karzel in Einführung in die Geometrie, 1973). Our main concern is to introduce a systematical exact definition for defect and area in the Beltrami–Klein model of hyperbolic geometry. Combining the ideas and methods of Karzel and Ungar give an elegant concept for defect and area in this model. In particular we give a rigorous and elementary proof for the defect formula stated (Ungar in Comput Math Appl 53, 2007). Furthermore, we give a formulary for area of circle in the Beltrami–Klein model of hyperbolic geometry.     

Blow-up of Compressible Navier–Stokes–Korteweg Equations

Based on the results of Xin (Commun. Pure Appl. Math. 51(3):229–240, 1998), Zhang and Tan (Acta Math. Sin. Engl. Ser. 28(3):645–652, 2012), we show the blow-up phenomena of smooth solutions to the non-isothermal compressible Navier–Stokes–Korteweg equations in arbitrary dimensions, under the assumption that the initial density has compact support. Here the coefficients are generalized to a more general case which depends on density and temperature. Our work extends the previous corresponding results.     

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq Equations

We establish a connection between optimal transport theory (see Villani in Topics in optimal transportation. Graduate studies in mathematics, vol. 58, AMS, Providence, 2003, for instance) and classical convection theory for geophysical flows (Pedlosky, in Geophysical fluid dynamics, Springer, New York, 1979). Our starting point is the model designed few years ago by Angenent, Haker, and Tannenbaum (SIAM J. Math. Anal. 35:61–97, 2003) to solve some optimal transport problems. This model can be seen as a generalization of the Darcy–Boussinesq equations, which is a degenerate version of the Navier–Stokes–Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized hydrostatic-Boussinesq equations) to various models involving optimal transport (and the related Monge–Ampère equation, Brenier in Commun. Pure Appl. Math. 64:375–417, 1991; Caffarelli in Commun. Pure Appl. Math. 45:1141–1151, 1992). This includes the 2D semi-geostrophic equations (Hoskins in Annual review of fluid mechanics, vol. 14, pp. 131–151, Palo Alto, 1982; Cullen et al. in SIAM J. Appl. Math. 51:20–31, 1991, Arch. Ration. Mech. Anal. 185:341–363, 2007; Benamou and Brenier in SIAM J. Appl. Math. 58:1450–1461, 1998; Loeper in SIAM J. Math. Anal. 38:795–823, 2006) and some fully nonlinear versions of the so-called high-field limit of the Vlasov–Poisson system (Nieto et al. in Arch. Ration. Mech. Anal. 158:29–59, 2001) and of the Keller–Segel for Chemotaxis (Keller and Segel in J. Theor. Biol. 30:225–234, 1971; Jäger and Luckhaus in Trans. Am. Math. Soc. 329:819–824, 1992; Chalub et al. in Mon. Math. 142:123–141, 2004). Mathematically speaking, we establish some existence theorems for local smooth, global smooth or global weak solutions of the different models. We also justify that the inertia terms can be rigorously neglected under appropriate scaling assumptions in the generalized Navier–Stokes–Boussinesq equations. Finally, we show how a “stringy” generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology (see Arnold and Khesin in Topological methods in hydrodynamics. Applied mathematical sciences, vol. 125, Springer, Berlin, 1998; Moffatt in J. Fluid Mech. 159:359–378, 1985, Topological aspects of the dynamics of fluids and plasmas. NATO adv. sci. inst. ser. E, appl. sci., vol. 218, Kluwer, Dordrecht, 1992; Schonbek in Theory of the Navier–Stokes equations, Ser. adv. math. appl. sci., vol. 47, pp. 179–184, World Sci., Singapore, 1998; Vladimirov et al. in J. Fluid Mech. 390:127–150, 1999; Nishiyama in Bull. Inst. Math. Acad. Sin. (N.S.) 2:139–154, 2007).     

On average values of convex functions

The aim of this paper is to give an extension of an inequality proved by Wulbert (Math Comput Model 37:1383–1391, 2003, Lemma 2.5) and to define Stolarsky type means as an application of this inequality. Further, we discuss some properties of averages of a continuous convex function, some consequences of a double inequality given by Wulbert (Math Comput Model 37:1383–1391, 2003, Theorem 3.3) and obtain improvement results of Wulbert (Math Comput Model 37:1383–1391, 2003, Corollary 4.3).     

A Kantorovich-type analysis of Broyden’s method using recurrent functions

We provide a semilocal convergence analysis for Broyden’s method for approximating locally unique solutions of nonlinear operator equations. Using the majorant principle we show that under the same or weaker hypotheses, in combination with our new idea of recurrent functions, we can find weaker sufficient conditions for the convergence of Broyden’s method as well as finer error bounds on the distances involved, and a more precise information on the location of the solution than before (Broyden, Math. Comput. 19:577–593, 1965; Chen, Ann. Inst. Stat. Math. 42:387–401, 1990; Dennis, Nonlinear Functional Analysis and Applications, pp. 425–472, Academic Press, San Diego, 1971; Li and Fukushima, Ann. Oper. Res. 103:71–97, 2001). Numerical examples are also provided involving polynomial, integral, and differential equations.     

The strong convergence of a three-step algorithm for the split feasibility problem

Based on the very recent work by Dang and Gao (Invers Probl 27:1–9, 2011) and Wang and Xu (J Inequal Appl, doi:10.1155/2010/102085, 2010), and inspired by Yao (Appl Math Comput 186:1551–1558, 2007), Noor (J Math Anal Appl 251:217–229, 2000), and Xu (Invers Probl 22:2021–2034, 2006), we suggest a three-step KM-CQ-like method for solving the split common fixed-point problems in Hilbert spaces. Our results improve and develop previously discussed feasibility problem and related algorithms.     

The best papers from BRIMS 2011: models of users and teams interacting

This special issue is similar to our previous special issues (Kennedy et al. in Comput. Math. Organ. Theory 16(3):217–219, 2010; 17(3):225–228, 2011) in that it includes articles based on the award winning conference papers of the, here, 2011 BRiMS Annual Conference. These articles were reviewed by the editors, extended to journal article length, and then peer-reviewed and revised before being accepted. The articles include a new way to evaluate designs of interfaces for safety critical systems (Bolton in Comput. Math. Organ. Theory, 2012), an article that extends our understanding of how to model situation awareness (SA) in a cognitive architecture (Rodgers et al. in Comput. Math. Organ. Theory, 2012), an article that presents electroencephalography (EEG) data used to derive dynamic neurophysiologic models of engagement in teamwork (Stevens et al. in Comput. Math. Organ. Theory, 2012), and an article that demonstrates using machine learning to generate models and an example application of that tool (Best in Comput. Math. Organ. Theory, 2012). After presenting a brief summary of each paper we will see some recurrent themes of task analysis, team and individual models, spatial reasoning, usability issues, and particularly that they are models that interact with each other or systems.     

Estimating upper bounds on the limit points of majorizing sequences for Newton’s method

We present new sufficient conditions for the semilocal convergence of Newton’s method to a locally unique solution of an equation in a Banach space setting. Upper bounds on the limit points of majorizing sequences are also given. Numerical examples are provided, where our new results compare favorably to earlier ones such as Argyros (J Math Anal Appl 298:374–397, 2004), Argyros and Hilout (J Comput Appl Math 234:2993-3006, 2010, 2011), Ortega and Rheinboldt (1970) and Potra and Pták (1984).     

A discontinuous Galerkin scheme for front propagation with obstacles

We are interested in front propagation problems in the presence of obstacles. We extend a previous work (Bokanowski et al. SIAM J Sci Comput 33(2):923–938, 2011), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of Bokanowski et al. (SIAM J Control Optim 48(7):4292–4316, (2010)), leading to a level set formulation driven by $\min (u_t + H(x,\nabla u), u-g(x))=0$ , where $g(x)$ is an obstacle function. The DG scheme is motivated by the variational formulation when the Hamiltonian $H$ is a linear function of $\nabla u$ , corresponding to linear convection problems in the presence of obstacles. The scheme is then generalized to nonlinear equations, written in an explicit form. Stability analysis is performed for the linear case with Euler forward, a Heun scheme and a Runge-Kutta third order time discretization using the technique proposed in Zhang and Shu (SIAM J Numer Anal 48:1038–1063, 2010). Several numerical examples are provided to demonstrate the robustness of the method. Finally, a narrow band approach is considered in order to reduce the computational cost.     

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\) ). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\) – \(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\) -bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\) – \(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin carathéodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Hölder spaces, but our approach is different from them.     

Local well-posedness for the periodic higher order KdV type equations

Higher order KdV type equations are the equation replaced by a higher order derivative ${\partial_{x}^{2k+1}}$ for the KdV equation. Recently, the local well-posedness result for these equations on torus have been given by Gorsky and Himonas (Math. Comput. Simul. 80:173–183, 2009). We extend this result by improving a bilinear estimate used in the Fourier restriction norm method.     

Complementary Lidstone interpolation on scattered data sets

Recently we have introduced a new technique for combining classical bivariate Shepard operators with three point polynomial interpolation operators (Dell’Accio and Di Tommaso, On the extension of the Shepard-Bernoulli operators to higher dimensions, unpublished). This technique is based on the association, to each sample point, of a triangle with a vertex in it and other ones in its neighborhood to minimize the error of the three point interpolation polynomial. The combination inherits both degree of exactness and interpolation conditions of the interpolation polynomial at each sample point, so that in Caira et al. (J Comput Appl Math 236:1691–1707, 2012) we generalized the notion of Lidstone Interpolation (LI) to scattered data sets by combining Shepard operators with the three point Lidstone interpolation polynomial (Costabile and Dell’Accio, Appl Numer Math 52:339–361, 2005). Complementary Lidstone Interpolation (CLI), which naturally complements Lidstone interpolation, was recently introduced by Costabile et al. (J Comput Appl Math 176:77–90, 2005) and drawn on by Agarwal et al. (2009) and Agarwal and Wong (J Comput Appl Math 234:2543–2561, 2010). In this paper we generalize the notion of CLI to bivariate scattered data sets. Numerical results are provided.     

Multivariate Generalized Marshall–Olkin Distributions and Copulas

The multivariate generalized Marshall–Olkin distributions, which include the multivariate Marshall–Olkin exponential distribution due to Marshall and Olkin (J Am Stat Assoc 62:30–41, 1967) and multivariate Marshall–Olkin type distribution due to Muliere and Scarsini (Ann Inst Stat Math 39:429–441, 1987) as special cases, are studied in this paper. We derive the survival copula and the upper/lower orthant dependence coefficient, build the order of these survival copulas, and investigate the evolution of dependence of the residual life with respect to age. The main conclusions developed here are both nice extensions of the main results in Li (Commun Stat Theory Methods 37:1721–1733, 2008a, Methodol Comput Appl Probab 10:39–54, 2008b) and high dimensional generalizations of some results on the bivariate generalized Marshall–Olkin distributions in Li and Pellerey (J Multivar Anal 102:1399–1409, 2011).