Instability of the finite‐difference split‐step method applied to the nonlinear Schrödinger equation. II. moving soliton

We analyze a mechanism and features of a numerical instability (NI) that can be observed in simulations of moving solitons of the nonlinear Schrödinger equation (NLS). This NI is completely different than the one for the standing soliton. We explain how this seeming violation of the Galilean invariance of the NLS is caused by the finite‐difference approximation of the spatial derivative. Our theory extends beyond the von Neumann analysis of numerical methods; in fact, it critically relies on the coefficients in the equation for the numerical error being spatially localized. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1024–1040, 2016     

Ground States to the Generalized Nonlinear Schr(o)dinger Equations with Bernstein Symbols

This paper concerns with existence and qualitative properties of ground states to generalized nonlinear Schr(o)dinger equations (gNLS) with abstract symbols.Under some structural assumptions on the symbol,we prove a ground state exists and it satisfies several fundamental properties that the ground state to the standard NLS enjoys.Furthermore,by imposing additional assumptions,we construct,in small mass case,a nontrivial radially symmetric solution to gNLS with H1-subcritical nonlinearity,even if the natural energy space does not control the H1-subcritical nonlinearity.     

Ground States to the Generalized Nonlinear Schr?dinger Equations with Bernstein Symbols

This paper concerns with existence and qualitative properties of ground states to generalized nonlinear Schr?dinger equations(gNLS) with abstract symbols.Under some structural assumptions on the symbol, we prove a ground state exists and it satisfies several fundamental properties that the ground state to the standard NLS enjoys. Furthermore, by imposing additional assumptions, we construct, in small mass case, a nontrivial radially symmetric solution to gNLS with H~1-subcritical nonlinearity,even if the natural energy space does not control the H~1-subcritical nonlinearity.     

A robust algebraic multilevel preconditioner for non‐symmetric M‐matrices

Stable finite difference approximations of convection‐diffusion equations lead to large sparse linear systems of equations whose coefficient matrix is an M‐matrix, which is highly non‐symmetric when the convection dominates. For an efficient iterative solution of such systems, it is proposed to consider in the non‐symmetric case an algebraic multilevel preconditioning method formerly proposed for pure diffusion problems, and for which theoretical results prove grid independent convergence in this context. These results are supplemented here by a Fourier analysis that applies to constant coefficient problems with periodic boundary conditions whenever using an ‘idealized’ version of the two‐level preconditioner. Within this setting, it is proved that any eigenvalue λ of the preconditioned system satisfies for some real constant c such that . This result holds independently of the grid size and uniformly with respect to the ratio between convection and diffusion. Extensive numerical experiments are conducted to assess the convergence of practical two‐ and multi‐level schemes. These experiments, which include problems with highly variable and rotating convective flow, indicate that the convergence is grid independent. It deteriorates moderately as the convection becomes increasingly dominating, but the convergence factor remains uniformly bounded. This conclusion is supported for both uniform and some non‐uniform (stretched) grids. Copyright © 2000 John Wiley & Sons, Ltd.     

Instability of the finite‐difference split‐step method applied to the nonlinear Schrödinger equation. I. standing soliton

We consider numerical instability that can be observed in simulations of solitons of the nonlinear Schrödinger equation (NLS) by a split‐step method (SSM) where the linear part of the evolution is solved by a finite‐difference discretization. The von Neumann analysis predicts that this method is unconditionally stable on the background of a constant‐amplitude plane wave. However, simulations show that the method can become unstable on the background of a soliton. We present an analysis explaining this instability. Both this analysis and the features and threshold of the instability are substantially different from those of the Fourier SSM, which computes the linear part of the NLS by a spectral discretization. For example, the modes responsible for the numerical instability are not similar to plane waves, as for the Fourier SSM or, more generally, in the von Neumann analysis. Instead, they are localized at the sides of the soliton. This also makes them different from “physical” (as opposed to numerical) unstable modes of nonlinear waves, which (the modes) are localized around the “core” of a solitary wave. Moreover, the instability threshold for thefinite‐difference split‐step method is considerably relaxed compared with that for the Fourier split‐step. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1002–1023, 2016     

A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

The Wright-Fisher model is an It? stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0,1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0,1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0,1].     

A continuous–discontinuous second‐order transition in the satisfiability of random Horn‐SAT formulas

We compute the probability of satisfiability of a class of random Horn‐SAT formulae, motivated by a connection with the nonemptiness problem of finite tree automata. In particular, when the maximum clause length is three, this model displays a curve in its parameter space, along which the probability of satisfiability is discontinuous, ending in a second‐order phase transition where it is continuous but its derivative diverges. This is the first case in which a phase transition of this type has been rigorously established for a random constraint satisfaction problem. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

A simple Gause‐type predator–prey model considering social predation

The consumer–resource relationships are among the most fundamental of all ecological relationships and have been the focus of ecology since its beginnings. Usually are described by nonlinear differential equation systems, putting the emphasis in the effect of antipredator behavior (APB) by the prey; nevertheless, a minor quantity of articles has considered the social behavior of predators. In this work, two predator–prey models derived from the Volterra model are analyzed, in which the equation of predators is modified considering cooperation or collaboration among predators. It is well known that competition among predators produces a stabilizing effect on system describing the model, since there exists a wide set in the parameter space where the system has a unique equilibrium point in the phase plane, which is globally asymptotically stable. Meanwhile, the cooperation can originate more complex and unusual dynamics. As we will show, it is possible to prove that for certain subset of parameter values the predator population sizes tend to infinite when the prey population goes to extinct. This apparently contradicts the idea of a realistic model, when it is implicitly assumed that the predators are specialist, ie, the prey is its unique source of food. However, this could be a desirable effect when the prey constitutes a plague. To reinforce the analytical result, numerical simulations are presented.     

An enhanced‐physics‐based scheme for the NS‐α turbulence model

We study a new enhanced‐physics‐based numerical scheme for the NS‐alpha turbulence model that conserves both energy and helicity. Although most turbulence models (in the continuous case) conserve only energy, NS‐alpha is one of only a very few that also conserve helicity. This is one reason why it is becoming accepted as the most physically accurate turbulence model. However, no numerical scheme for NS‐alpha, until now, conserved both energy and helicity, and thus the advantage gained in physical accuracy by modeling with NS‐alpha could be lost in a computation. This report presents a finite element numerical scheme, and gives a rigorous analysis of its conservation properties, stability, solution existence, and convergence. A key feature of the analysis is the identification of the discrete energy and energy dissipation norms, and proofs that these norms are equivalent (provided a careful choice of filtering radius) in the discrete space to the usual energy and energy dissipation norms. Numerical experiments are given to demonstrate the effectiveness of the scheme over usual (helicity‐ignoring) schemes. A generalization of this scheme to a family of high‐order NS‐alpha‐deconvolution models, which combine the attractive physical properties of NS‐alpha with the high accuracy gained by combining α‐filtering with van Cittert approximate deconvolution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On the Moore–Penrose inverse in solving saddle‐point systems with singular diagonal blocks

This paper deals with the role of the generalized inverses in solving saddle‐point systems arising naturally in the solution of many scientific and engineering problems when finite‐element tearing and interconnecting based domain decomposition methods are used to the numerical solution. It was shown that the Moore–Penrose inverse may be obtained in this case at negligible cost by projecting an arbitrary generalized inverse using orthogonal projectors. Applying an eigenvalue analysis based on the Moore–Penrose inverse, we proved that for simple model problems, the number of conjugate gradient iterations required for the solution of associate dual systems does not depend on discretization norms. The theoretical results were confirmed by numerical experiments with linear elasticity problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Two‐strata rotatability in split‐plot central composite designs

A rotatable design (Ann. Math. Stat. 1957; 28 :195–241) for k factors is one such that the prediction variance is purely a function of distance from the design center. Of special interest in this paper is the rotatable central composite design (CCD), which most software packages use as the typical default choice for a second‐order design. In many cases some factors are hard to change while others are easy to change, which creates a split‐plot experiment. This paper establishes that the split‐plot structure precludes the possibility of any second‐order design being rotatable in the traditional sense. As an alternative this paper proposes the two‐strata rotatable split‐plot CCD, where the resulting prediction variance is a function of the whole plot (WP) distance and the subplot (SP) distance separately instead of the sum of them. The resulting design is rotatable in the WP space when the SP factors are held fixed, and vice versa. In the special case where the WP variance component is zero, the two‐strata rotatable split‐plot CCD becomes the standard rotatable CCD. Copyright © 2009 John Wiley & Sons, Ltd.     

关于零对合类算子

本文定义了ＮＩ算子和ＡＮＩ算子，给出共轭算子下有界时，一个算子为ＮＩ类算子的等价条件，并在一定条件下证明了ＮＩ类算子具有闭的数值值域．最后，本文研究了ＮＩ类算子对，尤其是Ｈ２空间上的Ｔｏｅｐｌｉｔｚ算子对，刻划了Ｔ＝（Ｔφ，Ｔ＊φ）的联合谱     

SPH method applied to compression of solid materials for a variety of loading conditions

Smoothed Particle Hydrodynamics (SPH) is a numerical method that does not use a mesh or grid when solving a set of partial differential equations. This makes it particularly useful in application to solid mechanics problems where the sample undergoes large deformation. Whereas mesh-based methods have difficulty when the sample becomes severely distorted, SPH naturally deals with this important engineering scenario. We implement the SPH method for compressional deformation of solid samples and focus on uniaxial, biaxial and triaxial loading. We develop numerical procedures that naturally deal with these three different sets of boundary conditions and apply it to both small and larger strains in elastic and more complex materials. The method is shown to be robust up to large strains of 30%. Under uniaxial loading, a cylindrical sample tends to deform by bulging while under triaxial loading the cylindrical sample will remain cylindrical, but the diameter of the sample increases accordingly.     

Composite‐grid multigrid for diffusion on the sphere

Recently, there has been much interest in the solution of differential equations on surfaces and manifolds, driven by many applications whose dynamics take place on such domains. Although increasingly powerful algorithms have been developed in this field, many straightforward questions remain, particularly in the area of coupling advanced discretizations with efficient linear solvers. In this paper, we develop a structured refinement algorithm for octahedral triangulations of the surface of the sphere. We explain the composite‐grid finite‐element discretization of the Laplace–Beltrami operator on such triangulations and extend the fast adaptive composite‐grid scheme to provide an efficient solution of the resulting linear system. Supporting numerical examples are presented, including the recovery of second‐order accuracy in the case of a nonsmooth solution.     

Goal achieving probabilities of cone‐constrained mean‐variance portfolios

In this paper, we establish closed‐form formulas for key probabilistic properties of the cone‐constrained optimal mean‐variance strategy, in a continuous market model driven by a multidimensional Brownian motion and deterministic coefficients. In particular, we compute the probability to obtain to a point, during the investment horizon, where the accumulated wealth is large enough to be fully reinvested in the money market, and safely grow there to meet the investor's financial goal at terminal time. We conclude that the result of Li and Zhou [Ann. Appl. Prob., v.16, pp.1751–1763, (2006)] in the unconstrained case carries over when conic constraints are present: the former probability is lower bounded by 80% no matter the market coefficients, trading constraints, and investment goal. We also compute the expected terminal wealth given that the investor's goal is underachieved, for both the mean‐variance strategy and the aforementioned hybrid strategy where transfer to the money market occurs if it allows to safely achieve the goal. The former probabilities and expectations are also provided in the case where all risky assets held are liquidated if financial distress is encountered. These results provide investors with novel practical tools to support portfolio decision‐making and analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

On the average‐case complexity of Shellsort

We prove a lower bound expressed in the increment sequence on the average‐case complexity of the number of inversions of Shellsort. This lower bound is sharp in every case where it could be checked. A special case of this lower bound yields the general Jiang‐Li‐Vitányi lower bound. We obtain new results, for example, determining the average‐case complexity precisely in the Yao‐Janson‐Knuth 3‐pass case.     

The Stability of Large‐Amplitude Shallow Interfacial Non‐Boussinesq Flows

The system of equations describing the shallow‐water limit dynamics of the interface between two layers of immiscible fluids of different densities is formulated. The flow is bounded by horizontal top and bottom walls. The resulting equations are of mixed type: hyperbolic when the shear is weak and the behavior of the system is internal‐wave like, and elliptic for strong shear. This ellipticity, or ill‐posedness is shown to be a manifestation of large‐scale shear instability. This paper gives sharp nonlinear stability conditions for this nonlinear system of equations. For initial data that are initially hyperbolic, two different types of evolution may occur: the system may remain hyperbolic up to internal wave breaking, or it may become elliptic prior to wave breaking. Using simple waves that give a priori bounds on the solutions, we are able to characterize the condition preventing the second behavior, thus providing a long‐time well‐posedness, or nonlinear stability result. Our formulation also provides a systematic way to pass to the Boussinesq limit, whereby the density differences affect buoyancy but not momentum, and to recover the result that shear instability cannot occur from hyperbolic initial data in that case.     

ℋ︁‐matrices for the convection‐diffusion equation

Hierarchical matrices (ℋ︁‐matrices) provide a technique for the sparse approximation of large, fully populated matrices. This technique has been shown to be applicable to stiffness matrices arising in boundary element method applications where the kernel function displays certain smoothness properties. The error estimates for an approximation of the kernel function by a separable function can be carried over directly to error estimates for an approximation of the stiffness matrix by an ℋ︁‐matrix, using a certain standard partitioning and admissibility condition for matrix blocks. Similarly, ℋ︁‐matrix techniques can be applied in the finite element context where it is the inverse of the stiffness matrix that is fully populated. Here one needs a separable approximation of Green's function of the underlying boundary value problem in order to prove approximability by matrix blocks of low rank. Unfortunately, Green's function for the convection‐diffusion equation does not satisfy the required smoothness properties, hence prohibiting a straightforward generalization of the separable approximation through Taylor polynomials. We will use Green's function to motivate a modification in the (hierarchical) partitioning of the index set and as a consequence the resulting hierarchy of block partitionings as well as the admissibility condition. We will illustrate the effect of the proposed modifications by numerical results.     

Stabilization of infinite‐dimensional second‐order system by adaptive PI‐controllers

In this paper adaptive stabilization of infinite‐dimensional undamped second‐order systems is considered in the case where the input and output operators are collocated. The systems have an infinite number of poles and zeros on the imaginary axis. The adaptive stabilizer is constructed by an adaptive PI‐controller (proportional plus integral controller). An energy‐like function and a multiplier function are introduced and adaptive stabilization of the linear second‐order systems is analyzed. Copyright © 2001 John Wiley & Sons, Ltd.