Optimization of the Runge-Kutta iteration with residual smoothing

Iterative solvers in combination with multi-grid have been used extensively to solve large algebraic systems. One of the best known is the Runge-Kutta iteration. Previously (Haelterman et al. (2009) [3]) we reformulated the Runge-Kutta scheme and established a model of a complete V-cycle which was used to optimize the coefficients of the multi-stage scheme and resulted in a better overall performance. We now look into aspects of central and upwind residual smoothing within the same optimization framework. We consider explicit and implicit residual smoothing and either apply it within the Runge-Kutta time-steps, as a filter for restriction or as a preconditioner for the discretized equations. We also shed a different light on the very high CFL numbers obtained by upwind residual smoothing and point out that damping the high frequencies by residual smoothing is not necessarily a good idea.     

Convergence and optimization of the parallel method of simultaneous directions for the solution of elliptic problems

For the solution of elliptic problems, fractional step methods and in particular alternating directions (ADI) methods are iterative methods where fractional steps are sequential. Therefore, they only accept parallelization at low level. In [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708], Lu et al. proposed a method where the fractional steps can be performed in parallel. We can thus speak of parallel fractional step (PFS) methods and, in particular, simultaneous directions (SDI) methods. In this paper, we perform a detailed analysis of the convergence and optimization of PFS and SDI methods, complementing what was done in [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708]. We describe the behavior of the method and we specify the good choice of the parameters. We also study the efficiency of the parallelization. Some 2D, 3D and high-dimensional tests confirm our results.     

A blow-up criterion for 3D compressible viscoelastic flow with large initial data

In this paper, we consider a Cauchy problem for the three-dimensional compressible viscoelastic flow with large initial data. We establish a blow-up criterion for the strong solutions in terms of the gradient of velocity only, which is similar to the Beale-Kato-Majda criterion for ideal incompressible flow (cf. Beale et al. (1984) [20]) and the blow-up criterion for the compressible Navier-Stokes equations (cf. Huang et al. (2011) [21]).     

Multilevel Monte Carlo simulation for Lévy processes based on the Wiener–Hopf factorisation

In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was introduced for a large family of Lévy processes that is based on the Wiener–Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. Moreover, we provide here for the first time a theoretical analysis of the new Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its multilevel variant for computing expectations of functions depending on the historical trajectory of a Lévy process. We derive rates of convergence for both methods and show that they are uniform with respect to the “jump activity” (e.g. characterised by the Blumenthal–Getoor index). We also present a modified version of the algorithm in Kuznetsov et al. (2011) which combined with the multilevel methodology obtains the optimal rate of convergence for general Lévy processes and Lipschitz functionals. This final result is only a theoretical one at present, since it requires independent sampling from a triple of distributions which is currently only possible for a limited number of processes.     

On recent developments in the theory of abstract differential equations with fractional derivatives

This note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives. We show that the existence results in [Agarwal et al. (2009) [1], Belmekki and Benchohra (2010) [2], Darwish et al. (2009) [3], Hu et al. (2009) [4], Mophou and N’Guérékata (2009) [6] and [7], Mophou (2010) [8] and [9], Muslim (2009) [10], Pandey et al. (2009) [11], Rashid and El-Qaderi (2009) [12] and Tai and Wang (2009) [13]] are incorrect since the considered variation of constant formulas is not appropriate. In this note, we also consider a different approach to treat a general class of abstract fractional differential equations.     

On pricing arithmetic average reset options with multiple reset dates in a lattice framework

We develop a straightforward algorithm to price arithmetic average reset options with multiple reset dates in a Cox et al. (CRR) (1979) [10] framework. The use of a lattice approach is due to its adaptability and flexibility in managing arithmetic average reset options, as already evidenced by Kim et al. (2003) [9]. Their model is based on the Hull and White (1993) [5] bucketing algorithm and uses an exogenous exponential function to manage the averaging feature, but their choice of fictitious values does not guarantee the algorithm’s convergence (cfr., Forsyth et al. (2002) [11]). We propose to overcome this drawback by selecting a limited number of trajectories among the ones reaching each node of the lattice, where we compute effective averages. In this way, the computational cost of the pricing problem is reduced, and the convergence of the discrete time model to the corresponding continuous time one is guaranteed.     

On the global regularity of generalized Leray-alpha type models

We generalize Leray-alpha type models studied in Cheskidov et al. (2005) [1] and Linshiz and Titi (2007) [4] via fractional Laplacians and employ Besov space techniques to obtain global regularity results with the logarithmically supercritical dissipation.     

Superprocesses with spatial interactions in a random medium

We construct a class of interactive measure-valued diffusions driven by a historical super-Brownian motion and an independent white noise by solving a certain stochastic equation. In doing so, we show that the approach of Perkins (2002) [3] can be used to study the problem examined by Dawson et al. (2001) [1]. This unifies and extends both Dawson et al. (2001) [1] and Perkins (2002) [3] and establishes a new class of measure-valued diffusions. The existence and pathwise uniqueness of the solutions are proved, and the solutions are shown to satisfy the natural martingale problem.     

Error estimate for the upwind finite volume method for the nonlinear scalar conservation law

In this paper we estimate the error of upwind first order finite volume schemes applied to scalar conservation laws. As a first step, we consider standard upwind and flux finite volume scheme discretization of a linear equation with space variable coefficients in conservation form. We prove that, in spite of their lack of consistency, both schemes lead to a first order error estimate. As a final step, we prove a similar estimate for the nonlinear case. Our proofs rely on the notion of geometric corrector, introduced in our previous paper by Bouche et al. (2005) [24] in the context of constant coefficient linear advection equations.     

A mathematical modeling for the lookback option with jump-diffusion using binomial tree method

The binomial tree method (BTM), first proposed by Cox et al. (1979) [4] in diffusion models and extended by Amin (1993) [9] to jump-diffusion models, is one of the most popular approaches to pricing options. In this paper, we present a binomial tree method for lookback options in jump-diffusion models and show its equivalence to certain explicit difference scheme. We also prove the existence and convergence of the optimal exercise boundary in the binomial tree approximation to American lookback options and give the terminal value of the genuine exercise boundary. Further, numerical simulations are performed to illustrate the theoretical results.     

Optimal admission control for two station tandem queues with loss

We consider a two-station tandem queue with a buffer size of one at the first station and a finite buffer size at the second station. Silva et al. (2013) gave a criterion determining the optimal admission control policy for this model. In this paper, we improve the results of Silva et al. (2013) and also solve the problem conjectured by Silva et al. (2013).     

A note on the integrality gap of an ILP formulation for the periodic maintenance problem

We address the integrality gap of the integer linear program introduced by Grigoriev et al. (2006) [3] for the periodic maintenance problem. We prove that the integrality gap of this program is bounded by a constant.     

Shortest paths in linear time on minor-closed graph classes, with an application to Steiner tree approximation

We generalize the linear-time shortest-paths algorithm for planar graphs with nonnegative edge-weights of Henzinger et al. (1994) to work for any proper minor-closed class of graphs. We argue that their algorithm can not be adapted by standard methods to all proper minor-closed classes. By using recent deep results in graph minor theory, we show how to construct an appropriate recursive division in linear time for any graph excluding a fixed minor and how to transform the graph and its division afterwards, so that it has maximum degree three. Based on such a division, the original framework of Henzinger et al. can be applied. Afterwards, we show that using this algorithm, one can implement Mehlhorn’s (1988) 2-approximation algorithm for the Steiner tree problem in linear time on these graph classes.     

New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces

Recently, Ceng, Guu and Yao introduced an iterative scheme by viscosity-like approximation method to approximate the fixed point of nonexpansive mappings and solve some variational inequalities in Hilbert space (see Ceng et al. (2009) [9]). Takahashi and Takahashi proposed an iteration scheme to solve an equilibrium problem and approximate the fixed point of nonexpansive mapping by viscosity approximation method in Hilbert space (see Takahashi and Takahashi (2007) [12]). In this paper, we introduce an iterative scheme by viscosity approximation method for finding a common element of the set of a countable family of nonexpansive mappings and the set of an equilibrium problem in a Hilbert space. We prove the strong convergence of the proposed iteration to the unique solution of a variational inequality.     

Duality and optimality conditions for generalized equilibrium problems involving DC functions

We consider a generalized equilibrium problem involving DC functions which is called (GEP). For this problem we establish two new dual formulations based on Toland-Fenchel-Lagrange duality for DC programming problems. The first one allows us to obtain a unified dual analysis for many interesting problems. So, this dual coincides with the dual problem proposed by Martinez-Legaz and Sosa (J Glob Optim 25:311–319, 2006) for equilibrium problems in the sense of Blum and Oettli. Furthermore it is equivalent to Mosco’s dual problem (Mosco in J Math Anal Appl 40:202–206, 1972) when applied to a variational inequality problem. The second dual problem generalizes to our problem another dual scheme that has been recently introduced by Jacinto and Scheimberg (Optimization 57:795–805, 2008) for convex equilibrium problems. Through these schemes, as by products, we obtain new optimality conditions for (GEP) and also, gap functions for (GEP), which cover the ones in Antangerel et al. (J Oper Res 24:353–371, 2007, Pac J Optim 2:667–678, 2006) for variational inequalities and standard convex equilibrium problems. These results, in turn, when applied to DC and convex optimization problems with convex constraints (considered as special cases of (GEP)) lead to Toland-Fenchel-Lagrange duality for DC problems in Dinh et al. (Optimization 1–20, 2008, J Convex Anal 15:235–262, 2008), Fenchel-Lagrange and Lagrange dualities for convex problems as in Antangerel et al. (Pac J Optim 2:667–678, 2006), Bot and Wanka (Nonlinear Anal to appear), Jeyakumar et al. (Applied Mathematics research report AMR04/8, 2004). Besides, as consequences of the main results, we obtain some new optimality conditions for DC and convex problems.     

On the Secant method

We present a new semilocal convergence analysis for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been ubiquitously employed in other studies such as Amat et al. (2004)  [2], Bosarge and Falb (1969)  [9], Dennis (1971)  [10], Ezquerro et al. (2010)  [11], Hernández et al. (2005, 2000)   and , Kantorovich and Akilov (1982)  [14], Laasonen (1969)  [15], Ortega and Rheinboldt (1970)  [16], Parida and Gupta (2007)  [17], Potra (1982, 1984–1985, 1985)  ,  and , Proinov (2009, 2010)   and , Schmidt (1978) [23], Wolfe (1978)  [24] and Yamamoto (1987)  [25] for computing the inverses of the linear operators. We also provide lower and upper bounds on the limit point of the majorizing sequences for the Secant method. Under the same computational cost, our error analysis is tighter than that proposed in earlier studies. Numerical examples illustrating the theoretical results are also given in this study.     

On sparse estimation for semiparametric linear transformation models

Semiparametric linear transformation models have received much attention due to their high flexibility in modeling survival data. A useful estimating equation procedure was recently proposed by Chen et al. (2002) [21] for linear transformation models to jointly estimate parametric and nonparametric terms. They showed that this procedure can yield a consistent and robust estimator. However, the problem of variable selection for linear transformation models has been less studied, partially because a convenient loss function is not readily available under this context. In this paper, we propose a simple yet powerful approach to achieve both sparse and consistent estimation for linear transformation models. The main idea is to derive a profiled score from the estimating equation of Chen et al. [21], construct a loss function based on the profile scored and its variance, and then minimize the loss subject to some shrinkage penalty. Under regularity conditions, we have shown that the resulting estimator is consistent for both model estimation and variable selection. Furthermore, the estimated parametric terms are asymptotically normal and can achieve a higher efficiency than that yielded from the estimation equations. For computation, we suggest a one-step approximation algorithm which can take advantage of the LARS and build the entire solution path efficiently. Performance of the new procedure is illustrated through numerous simulations and real examples including one microarray data.     

The nonlinear boundary layer to the Boltzmann equation for cutoff soft potential with physical boundary condition

We consider the nonlinear boundary layer to the Boltzmann equation for cutoff soft potential with physical boundary condition, i.e., the Dirichlet boundary condition with weak diffuse effect. Under the assumption that the distribution function of gas particles tends to a global Maxwellian in the far field, we will show the boundary layer exist if the boundary data satisfy the solvability condition. Moreover, the codimensions of the boundary data which satisfies the solvability condition change with the Mach number of the far field Maxwellian like Chen et al. (2004) [5], Ukai et al. (2003) [6] and Wang et al. (2007) [7].     

On a fluid model of neutron star

We consider an initial boundary value problem for the equations of a fluid spherical model of neutron star considered by Lattimer et al. We estimate the asymptotic decay of the solution, which may serve as a crude estimate of a “thermalization time” for the system.      

A note on the efficient implementation of Hamiltonian BVMs

We discuss the efficient implementation of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems (see Brugnano et al. [8] and references therein), also sketching their blended formulation. We also discuss the case of separable problems, for which the structure of the problem can be exploited to gain efficiency.