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The authors would like to correct the proof of Theorem 3(a) in the original paper “On the asymptotic behaviour of the 2D Navier-Stokes equations with Navier friction conditions towards Euler equations.” [ZAMM 89(10), 810-822, 2009].

van Lenthe, Broer, and Rashid made comments on our 2009 paper [Song et al., J. Comput. Chem. 2009, 30, 399] by criticizing that we did not properly reference the work by Broer and Nieuwpoort in 1988 [Broer and Nieuwpoort, Theor. Chim. Acta. 1988, 73, 405], and we favorably compared our valence bond self-consistent field (VBSCF) algorithm with theirs. However, both criticisms are unjustified insignificant. The Broer–Nieuwpoort algorithm, properly cited in our paper, is for the evaluations of matrix elements between determinants of nonorthogonal orbitals. Stating that this algorithm “can be used for an orbital optimization” afterwards [van Lenthe et al., submitted] is not a plausible way to require more credits or even criticize others. While we stand by our statement that our algorithms scales at O(m4) and van Lenthe et al.'s approximate Newton Raphson algorithm scales at O(mN5) (here m and N are the numbers of basis functions and electrons), as we discussed in our original paper, it becomes obvious that any strict comparison among different algorithms is difficult, unproductive, and counteractive. © 2012 Wiley Periodicals, Inc.

This work comments on a recent paper by J. C. Strikwerda in SIAM Journal on Scientific and Statistical Computing, in an attempt to clear up the evident confusion regarding the use of a Poisson equation for pressure in incompressible Navier-Stokes solutions.

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The article deals with the analysis of the fractional COVID‐19 epidemic model (FCEM) with a convex incidence rate. Keeping in view the fading memory and crossover behavior found in many biological phenomena, we study the coronavirus disease by using the noninteger Caputo derivative (CD). Under the Caputo operator (CO), existence and uniqueness for the solutions of the FCEM have been analyzed using fixed point theorems. We study all the basic properties and results including local and global stability. We show the global stability of disease‐free equilibrium using the method of Castillo‐Chavez, while for disease endemic, we use the method of geometrical approach. Sensitivity analysis is carried out to highlight the most sensitive parameters corresponding to basic reproduction number. Simulations are performed via first‐order convergent numerical technique to determine how changes in parameters affect the dynamical behavior of the system.

Max Margules contributed a short paper for the Festschrift published in 1904 to mark the sixtieth birthday of his former teacher, the renowned physicist Ludwig Boltzmann. Margules considered the possibility of predicting pressure changes by means of the continuity equation. He showed that, to obtain an accurate estimate of the pressure tendency, the winds would have to be known to a precision quite beyond the practical limit. He concluded that any attempt to forecast synoptic changes by this means was doomed to failure. We re-examine the numerical weather forecast made by Lewis Fry Richardson in the light of Margules’ findings. Richardson employed the method which Margules had shown to be problematical; as a result, his prediction was completely unrealistic. It appears that Richardson was unaware of Margules’ paper, although a copy was received by the Met Office Library in 1905.

In this paper we present a parametric estimation method for certain multi-parameter heavy-tailed Levy-driven moving averages. The theory relies on recent multivariate central limit theorems obtained in [3] via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to the papers [14, 15], which propose to use the marginal empirical characteristic function to estimate the one-dimensional parameter of the kernel function and the stability index of the driving Levy motion. We extend their work to allow for a multi-parametric framework that in particular includes the important examples of the linear fractional stable motion, the stable Ornstein-Uhlenbeck process, certain CARMA(2, 1) models and Ornstein-Uhlenbeck processes with a periodic component among other models. We present both the consistency and the associated central limit theorem of the minimal contrast estimator. Furthermore, we demonstrate numerical analysis to uncover the finite sample performance of our method.

We present a novel method of analysis and prove finite time asymptotically self-similar blowup of the De Gregorio model \cite{DG90,DG96} for some smooth initial data on the real line with compact support. We also prove self-similar blowup results for the generalized De Gregorio model \cite{OSW08} for the entire range of parameter on $\mathbb{R}$ or $S^1$ for H\"older continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self-similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations., Comment: Added discussion in Section 2.3 and made some minor edits. Main paper 57 pages, Supplementary material 29 pages. In previous arXiv versions, the hyperlinks of the equation number in the main paper are linked to the supplementary material, which is fixed in this version

This paper concerns the existence of transonic shock solutions to the 2-D steady compressible Euler system in an almost flat finite nozzle ( in the sense that it is a generic small perturbation of a flat one ), under physical boundary conditions proposed by Courant-Friedrichs in \cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock-front is one of the most desirable information one would like to determine. However, the location of the normal shock-front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock-front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock-front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived which yields the existence of the solutions to the proposed free boundary problem. Once an initial approximation is obtained, a further nonlinear iteration could be constructed and proved to lead to a transonic shock solution., 53 pages

We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension., 37 pages. First part of a two-papers work aimed at establishing a first general partial regularity theory for area minimizing currents modulo p, for any p and in any dimension and codimension. v3 is the final version, to appear on Comm. Pure Appl. Math

SummaryWe consider the problem of jointly modeling multiple covariance matrices by partial common principal component analysis (PCPCA), which assumes a proportion of eigenvectors to be shared across covariance matrices and the rest to be individual-specific. This paper proposes consistent estimators of the shared eigenvectors in PCPCA as the number of matrices or the number of samples to estimate each matrix goes to infinity. We prove such asymptotic results without making any assumptions on the ranks of eigenvalues that are associated with the shared eigenvectors. When the number of samples goes to infinity, our results do not require the data to be Gaussian distributed. Furthermore, this paper introduces a sequential testing procedure to identify the number of shared eigenvectors in PCPCA. In simulation studies, our method shows higher accuracy in estimating the shared eigenvectors than competing methods. Applied to a motor-task functional magnetic resonance imaging data set, our estimator identifies meaningful brain networks that are consistent with current scientific understandings of motor networks during a motor paradigm.

It is well known that when the geometry and/or coefficients allow stable trapped rays, the solution operator of the Helmholtz equation (a.k.a. the resolvent of the Laplacian) grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest-possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial-growth assumption on the solution operator (e.g. $hp$-finite elements, $hp$-boundary elements, certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest-possible trapping, for most frequencies.

The second part of our study is devoted to an analysis of the exactness of penalty functions for optimal control problems with terminal and pointwise state constraints. We demonstrate that with the use of the exact penalty function method one can reduce fixed-endpoint problems for linear time-varying systems and linear evolution equations with convex constraints on the control inputs to completely equivalent free-endpoint optimal control problems, if the terminal state belongs to the relative interior of the reachable set. In the nonlinear case, we prove that a local reduction of fixed-endpoint and variable-endpoint problems to equivalent free-endpoint ones is possible under the assumption that the linearised system is completely controllable, and point out some general properties of nonlinear systems under which a global reduction to equivalent free-endpoint problems can be achieved. In the case of problems with pointwise state inequality constraints, we prove that such problems for linear time-varying systems and linear evolution equations with convex state constraints can be reduced to equivalent problems without state constraints, provided one uses the $L^{\infty}$ penalty term, and Slater's condition holds true, while for nonlinear systems a local reduction is possible, if a natural constraint qualification is satisfied. Finally, we show that the exact $L^p$-penalisation of state constraints with finite $p$ is possible for convex problems, if Lagrange multipliers corresponding to the state constraints belong to $L^{p'}$, where $p'$ is the conjugate exponent of $p$, and for general nonlinear problems, if the cost functional does not depend on the control inputs explicitly., This is a second part of the paper arXiv: 1903.00236. In the second version of this paper, a new section on variable-endpoint problems was added. In the third version, a much simpler proof of Theorem 14 is given

This paper presents the application of variationally consistent selective homogenization applied to a polycrystal with a subscale model of gradient-enhanced crystal inelasticity. Although the full gradient problem is solved on Statistical Volume Elements (SVEs), the resulting macroscale problem has the formal character of a standard local continuum. A semi-dual format of gradient inelasticity is exploited, whereby the unknown global variables are the displacements and the energetic micro-stresses on each slip-system. The corresponding time-discrete variational formulation of the SVE-problem defines a saddle point of an associated incremental potential. Focus is placed on the computation of statistical bounds on the effective energy, based on virtual testing on SVEs and an argument of ergodicity. As it turns out, suitable combinations of Dirichlet and Neumann conditions pertinent to the standard equilibrium and the micro-force balance, respectively, will have to be imposed. Whereas arguments leading to the upper bound are quite straightforward, those leading to the lower bound are significantly more involved; hence, a viable approximation of the lower bound is computed in this paper. Numerical evaluations of the effective strain energy confirm the theoretical predictions. Furthermore, heuristic arguments for the resulting macroscale stress-strain relations are numerically confirmed.

In this paper, we investigate the exponential stability of highly nonlinear hybrid neutral pantograph stochastic differential equations(NPSDEs). The aim of this paper is to establish exponential stability criteria for a class of hybrid NPSDEs without the linear growth condition. The methods of Lyapunov functions and M-matrix are used to study exponential stability and boundedness of the hybrid NPSDEs.

Parallel-in-time algorithms have been successfully employed for reducing time-to-solution of a variety of partial differential equations, especially for diffusive (parabolic-type) equations. A major failing of parallel-in-time approaches to date, however, is that most methods show instabilities or poor convergence for hyperbolic problems. This paper focuses on the analysis of the convergence behavior of multigrid methods for the parallel-in-time solution of hyperbolic problems. Three analysis tools are considered that differ, in particular, in the treatment of the time dimension: (1) space-time local Fourier analysis, using a Fourier ansatz in space and time, (2) semi-algebraic mode analysis, coupling standard local Fourier analysis approaches in space with algebraic computation in time, and (3) a two-level reduction analysis, considering error propagation only on the coarse time grid. In this paper, we show how insights from reduction analysis can be used to improve feasibility of the semi-algebraic mode analysis, resulting in a tool that offers the best features of both analysis techniques. Following validating numerical results, we investigate what insights the combined analysis framework can offer for two model hyperbolic problems, the linear advection equation in one space dimension and linear elasticity in two space dimensions.

Summary In this paper, we present a detailed report on a revised form of simplified and highly stable lattice Boltzmann method (SHSLBM) and its boundary treatment as well as stability analysis. The SHSLBM is a recently developed scheme within lattice Boltzmann framework, which utilizes lattice properties and relationships given by Chapman-Enskog expansion analysis to reconstruct solutions of macroscopic governing equations recovered from lattice Boltzmann equation and resolved in a predictor-corrector scheme. Formulations of original SHSLBM are slightly adjusted in the present work to facilitate implementation on body-fitted mesh. The boundary treatment proposed in this paper offers an analytical approach to interpret no-slip boundary condition, and the stability analysis in this paper fixes flaws in previous works and reveals a very nice stability characteristic in high Reynolds number scenarios. Several benchmark tests are conducted for comprehensive evaluation of the boundary treatment and numerical validation of stability analysis. It turns out that by adopting the modifications suggested in this work, lower numerical error can be expected.

In this paper, the author derives an O ( h 4 ) -superconvergence for the piecewise linear Ritz-Galerkin finite element approximations for the second-order elliptic equation − ∇ · ( A ∇ u ) = f equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor A and the usual shape functions on each element, called A-equilateral assumption in this paper. Several examples are presented for the coefficient tensor A and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence.

Summary There is increasing interest in the Material Point Method (MPM) as a means of modelling solid mechanics problems in which very large deformations occur, e.g. in the study of landslides and metal forming, however some aspects vital to wider use of the method have to date been ignored, in particular methods for imposing essential boundary conditions in the case where the problem domain boundary does not coincide with the background grid element edges. In this paper we develop a simple procedure originally devised for standard finite elements for the imposition of essential boundary conditions, to the MPM, expanding its capabilities to boundaries of any inclination. To the authors' knowledge this is the first time that a method has been proposed that allows arbitrary Dirichlet boundary conditions (zero and non-zero values at any inclination) to be imposed in the MPM. The method presented in this paper is different from other MPM boundary approximation approaches, in that: (i) the boundaries are independent of the background mesh, (ii) artificially stiff regions of material points are avoided and (iii) the method does not rely on spurious mirroring of the problem domain to imposed symmetry. The main contribution of this work is equally applicable to standard finite elements and the MPM. This article is protected by copyright. All rights reserved.

In this paper, an iteration process is considered to solve linear ill-posed problems. Based on the randomness of the involved variables, this kind of problems is regarded as simulation problems of the posterior distribution of the unknown variable given the noise data. We construct a new ensemble Kalman filter-based method to seek the posterior target distribution. Despite the ensemble Kalman filter method having widespread applications, there has been little analysis of its theoretical properties, especially in the field of inverse problems. This paper analyzes the propagation of the error with the iteration step for the proposed algorithm. The theoretical analysis shows that the proposed algorithm is convergence. We compare the numerical effect with the Bayesian inversion approach by two numerical examples: backward heat conduction problem and the first kind of integral equation. The numerical tests show that the proposed algorithm is effective and competitive with the Bayesian method. Copyright © 2017 John Wiley & Sons, Ltd.

This paper is about vector autoregressive-moving average (VARMA) models with time-dependent coefficients to represent non-stationary time series. Contrary to other papers in the univariate case, the coefficients depend on time but not on the series’ length n. Under appropriate assumptions, it is shown that a Gaussian quasi-maximum likelihood estimator is almost surely consistent and asymptotically normal. The theoretical results are illustrated by means of two examples of bivariate processes. It is shown that the assumptions underly- ing the theoretical results apply. In the second example the innovations are marginally heteroscedastic with a correlation ranging from −0.8 to 0.8. In the two examples, the asymptotic information matrix is obtained in the Gaussian case. Finally, the finite-sample behavior is checked via a Monte Carlo simulation study for n from 25 to 400. The results confirm the validity of the asymptotic properties even for short series and the asymptotic information matrix deduced from the theory.

Summary This paper proposes a most probable point (MPP)-based dimension reduction method (DRM) using the Hessian matrix called HeDRM to improve accuracy of reliability analysis in existing MPP-based DRM methods. Conventional MPP-based DRMs contain two types of errors: (1) error due to eliminating cross-terms of a performance function by using the univariate DRM; (2) error because of dependency of an axis direction after a rotational transformation. The proposed method minimizes the aforementioned errors by utilizing the Hessian matrix of a performance function. By performing an orthogonal transformation using the eigenvectors of the Hessian matrix, the cross-term effect of the performance function is minimized and the axis direction that results in the most accurate calculation is obtained because the Gaussian quadrature points for numerical integration are arranged along the eigenvector directions. In this way, the error incurred by exiting MPP-based DRMs can be reduced that leads to more accurate probability of failure estimation. In addition, this paper proposes to allocate the Gaussian quadrature points using the magnitude of the eigenvalues of the Hessian matrix. This allocation makes it possible to predetermine the number of function evaluations required to estimate the probability of failure accurately and efficiently. Copyright © 2016 John Wiley & Sons, Ltd.

A nonlocal nonlinear Schr\"odinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced "potential" is $PT$ symmetric thus the nonlocal NLS equation is also $PT$ symmetric. In this paper, new {\it reverse space-time} and {\it reverse time} nonlocal nonlinear integrable equations are introduced. They arise from remarkably simple symmetry reductions of general AKNS scattering problems where the nonlocality appears in both space and time or time alone. They are integrable infinite dimensional Hamiltonian dynamical systems. These include the reverse space-time, and in some cases reverse time, nonlocal nonlinear Schr\"odinger, modified Korteweg-deVries (mKdV), sine-Gordon, $(1+1)$ and $(2+1)$ dimensional three-wave interaction, derivative NLS, "loop soliton", Davey-Stewartson (DS), partially $PT$ symmetric DS and partially reverse space-time DS equations. Linear Lax pairs, an infinite number of conservation laws, inverse scattering transforms are discussed and one soliton solutions are found. Integrable reverse space-time and reverse time nonlocal discrete nonlinear Schr\"odinger type equations are also introduced along with few conserved quantities. Finally, nonlocal Painlev\'e type equations are derived from the reverse space-time and reverse time nonlocal NLS equations., Comment: This paper will be published in Studies in Applied Mathematics in a special volume dedicated to Professor David J. Benny

Summary In this paper we present a new family of direct Arbitrary-Lagrangian-Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high order accurate both in space and time and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space-time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy [26, 25] that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. According to [19, 20] we rely on an element-local space-time Galerkin finite element predictor on moving curved meshes to obtain a high order accurate one-step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tn is connected to the new one at tn + 1 by straight edges, hence providing unstructured space-time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt the quadrature-free integration proposed in [21], in which the space-time boundaries of the control volumes are split into simplex sub-elements that yield constant space-time normal vectors and Jacobian matrices. In this way the integrals over the simplex sub-elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics (HD) equations) as well as to the magnetohydrodynamics (MHD) equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm [19, 20] that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the three-dimensional case. This article is protected by copyright. All rights reserved.

Summary The paper deals with the problem of delay-dependent output feedback L1 control for positive Markovian jump systems with mode-dependent time-varying delays and partly known transition rates. First, by constructing an appropriate co-positive type Lyapunov function, sufficient conditions for stochastic stability and L1-gain performance of the open-loop system are developed. Then, an effective method is proposed to construct the output feedback controller. These sufficient criteria are derived in the form of linear programming. A key point of this paper is to extend the special requirement of completely known transition rates to more general form that covers completely known and completely unknown transition rates as two special cases. Finally, a numerical example is given to illustrate the validity of the main results. Copyright © 2016 John Wiley & Sons, Ltd.

This paper builds on recent work developed by the authors for the numerical analysis of large strain solid dynamics, by introducing an upwind cell centred hexahedral Finite Volume framework implemented within the open source code OpenFOAM [http://www.openfoam.com/http://www.openfoam.com/]. In Lee, Gil and Bonet [1], a first order hyperbolic system of conservation laws was introduced in terms of the linear momentum and the deformation gradient tensor of the system, leading to excellent behaviour in two dimensional bending dominated nearly incompressible scenarios. The main aim of this paper is the extension of this algorithm into three dimensions, its tailor-made implementation into OpenFOAM and the enhancement of the formulation with three key novelties. First, the introduction of two different strategies in order to ensure the satisfaction of the underlying involutions of the system, that is, that the deformation gradient tensor must be curl-free throughout the deformation process. Second, the use of a discrete angular momentum projection algorithm and a monolithic Total Variation Diminishing Runge-Kutta time integrator combined in order to guarantee the conservation of angular momentum. Third, and for comparison purposes, an adapted Total Lagrangian version of the Hyperelastic-GLACE nodal scheme of Kluth and Despr´es [2] is presented. A series of challenging numerical examples are examined in order to assess the robustness and accuracy of the proposed algorithm, benchmarking it against an ample spectrum of alternative numerical strategies developed by the authors in recent publications.

Staircase tableaux are combinatorial objects that were first introduced due to a connection with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since their introduction, staircase tableaux have been the object of study in many recent papers. Relevant to this paper, Hitczenko and Janson proved that distribution of parameters on the first diagonal is asymptotically normal. In addition, they conjectured that other diagonals would be asymptotically Poisson. Since then, only the second and the third diagonal were proven to follow the conjecture. This paper builds upon those results to prove the conjecture for the kth diagonal where k is fixed. In particular, we prove that the distribution of the number of α's (β's) on the kth diagonal, k > 1, is asymptotically Poisson with parameter 1/2. In addition, we prove that symbols on the kth diagonal are asymptotically independent and thus, collectively follow the Poisson distribution with parameter 1. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 795–818, 2016

Summary This paper proposes an efficient metamodeling approach for uncertainty quantification of complex system based on Gaussian process model (GPM). The proposed GPM-based method is able to efficiently and accurately calculate the mean and variance of model outputs with uncertain parameters specified by arbitrary probability distributions. Because of the use of GPM, the closed form expressions of mean and variance can be derived by decomposing high-dimensional integrals into one-dimensional integrals. This paper details on how to efficiently compute the one-dimensional integrals. When the parameters are either uniformly or normally distributed, the one-dimensional integrals can be analytically evaluated, while when parameters do not follow normal or uniform distributions, this paper adopts the effective Gaussian quadrature technique for the fast computation of the one-dimensional integrals. As a result, the developed GPM method is able to calculate mean and variance of model outputs in an efficient manner independent of parameter distributions. The proposed GPM method is applied to a collection of examples. And its accuracy and efficiency is compared with Monte Carlo simulation, which is used as benchmark solution. Results show that the proposed GPM method is feasible and reliable for efficient uncertainty quantification of complex systems in terms of the computational accuracy and efficiency. Copyright © 2016 John Wiley & Sons, Ltd.

Reactivity ratio estimation was carried out in various nonlinear models using Markov Chain Monte Carlo (MCMC) technique and an error-in-variables (EVM) regression model. The implementation steps for three different polymerization case studies are discussed in detail and the results from this work are compared to previously used approximation methods. Approximation techniques that rely on linear regression theory are shown to produce inaccurate joint confidence regions (JCRs). Therefore, in this paper, we explore MCMC techniques that can be used to produce JCRs with correct shape and probability content. In addition, the paper illustrates how an EVM model can be used in tackling any type of regression problem, including multi-response problems.

Summary A stochastic Lotka–Volterra predator–prey system driven by both Brownian motion and Poisson counting measure is modeled and studied in this paper. A new ergodic method is proposed to solve the classical optimal harvesting problem. Equivalency between time averaged yield function and sustained yield function is proved by this new approach. The optimal harvesting strategy and the corresponding maximum yield with respect to stationary probability density are obtained. Several examples are taken to show that results in this paper are new even in the deterministic case. The method proposed in this paper can avoid trouble of solving the corresponding partial differential equations, and it can be extended to a more general high-dimensional case or some other stochastic system. Copyright © 2015 John Wiley & Sons, Ltd.

This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non-linear differential-difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential-difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich-type theorem and Malmquist-type theorem about differential equations to the case of differential-difference equations. Copyright © 2015 John Wiley & Sons, Ltd.

This paper discusses the state estimation and optimal control problem of a class of partially-observable stochastic hybrid systems (POSHS). The POSHS has interacting continuous and discrete dynamics with uncertainties. The continuous dynamics are given by a Markov-jump linear system and the discrete dynamics are defined by a Markov chain whose transition probabilities are dependent on the continuous state via guard conditions. The only information available to the controller are noisy measurements of the continuous state. To solve the optimal control problem, a separable control scheme is applied: the controller estimates the continuous and discrete states of the POSHS using noisy measurements and computes the optimal control input from the state estimates. Since computing both optimal state estimates and optimal control inputs are intractable, this paper proposes computationally efficient algorithms to solve this problem numerically. The proposed hybrid estimation algorithm is able to handle state-dependent Markov transitions and compute Gaussian- mixture distributions as the state estimates. With the computed state estimates, a reinforcement learning algorithm defined on a function space is proposed. This approach is based on Monte Carlo sampling and integration on a function space containing all the probability distributions of the hybrid state estimates. Finally, the proposed algorithm is tested via numerical simulations.

Summary The computational efficiency of a typical, projection-based, nonlinear model reduction method hinges on the efficient approximation, for explicit computations, of the scalar projections onto a subspace of a residual vector. For implicit computations, it also hinges on the additional efficient approximation of similar projections of the Jacobian of this residual with respect to the solution. The computation of both approximations is often referred to in the literature as hyper reduction. To this effect, this paper focuses on the analysis and comparative performance study for nonlinear finite element reduced-order models of solids and structures of the recently developed energy-conserving mesh sampling and weighting (ECSW) hyper reduction method. Unlike most alternative approaches, this method approximates the scalar projections of residuals and/or Jacobians directly, instead of approximating first these vectors and matrices then projecting the resulting approximations onto the subspaces of interest. In this paper, it is shown that ECSW distinguishes itself furthermore from other hyper reduction methods through its preservation of the Lagrangian structure associated with Hamilton's principle. For second-order dynamical systems, this enables it to also preserve the numerical stability properties of the discrete system to which it is applied. It is also shown that for a fixed set of parameter values, the approximation error committed online by ECSW is bounded by its counterpart error committed off-line during the training of this method. Therefore, this error can be estimated in this case a priori and controlled. The performance of ECSW is demonstrated first for two academic but nevertheless interesting nonlinear dynamic response problems. For both of them, ECSW is shown to preserve numerical stability and deliver the desired level of accuracy, whereas the discrete empirical interpolation method and its recently introduced unassembled variant are shown to be susceptible to failure because of numerical instability. The potential of ECSW for enabling the effective reduction of nonlinear finite element dynamic models of solids and structures is also highlighted with the realistic simulation of the nonlinear transient dynamic response of a complete car engine to thermal and combustion pressure loads using an implicit scheme. For this simulation, ECSW is reported to enable the reduction of the CPU time required by the high-dimensional nonlinear finite element dynamic analysis by more than four orders of magnitude, while achieving a very good level of accuracy. Copyright © 2015 John Wiley & Sons, Ltd.

Numerical techniques are suggested in this paper, in order to improve the computational efficiency of the spectral boundary integral method, initiated by Clamond & Grue [D. Clamond and J. Grue. A fast method for fully nonlinear water-wave computations. J. Fluid Mech. 2001; 447: 337-355] for simulating nonlinear water waves. This method involves dealing with the high order convolutions by using Fourier transform or inverse Fourier transform and evaluating the integrals with weakly singular integrands. A de-singularity technique is proposed here to help in efficiently evaluating the integrals with weak singularity. An anti-aliasing technique is developed in this paper to overcome the aliasing problem associated with Fourier transform or inverse Fourier transform with a limited resolution. This paper also presents a technique for determining a critical value of the free surface, under which the integrals can be neglected. Numerical tests are carried out on the numerical techniques and on the improved method equipped with the techniques. The tests will demonstrate that the improved method can significantly accelerate the computation, in particular when waves are strongly nonlinear.

An efficient solution strategy for the simulation of incompressible fluids needs adequate and accurate space and time discretisation schemes. In this paper for the space discretisation we use an inf--sup stable finite element method and for the time discretisation Radau-IIA methods of higher order, which have the advantage that the pressure component has convergence order $s$ in time, where $s$ is the number of internal stages. The disadvantage of this approach is that we have a high computational amount of work, since large nonlinear systems of equations have to solved. In this paper we use a transformation of the coefficient matrix and the simplified Newton method. This approach has the effect that our large nonlinear systems split into smaller ones, which can now also be solved in parallel. For the parallelisation of the code we use the software component technology and the Component emplate Library (CTL). Numerical examples show that high order in the pressure component can be achieved and that the proposed solution technique is very effective.

Summary The current paper is focused on investigating a Jacobian-free Newton–Krylov (JFNK) method to obtain a fully implicit solution for two-phase flows. In the JFNK formulation, the Jacobian matrix is not directly evaluated, potentially leading to major computational savings compared with a simple Newton's solver. The objectives of the present paper are as follows: (i) application of the JFNK method to two-fluid models; (ii) investigation of the advantages and disadvantages of the fully implicit JFNK method compared with commonly used explicit formulations and implicit Newton–Krylov calculations using the determination of the Jacobian matrix; and (iii) comparison of the numerical predictions with those obtained by the Canadian Algorithm for Thermaulhydraulics Network Analysis 4. Two well-known benchmarks are considered, the water faucet and the oscillating manometer. An isentropic two-fluid model is selected. Time discretization is performed using a backward Euler scheme. A Crank–Nicolson scheme is also implemented to check the effect of temporal discretization on the predictions. Advection Upstream Splitting Method+ is applied to the convective fluxes. The source terms are discretized using a central differencing scheme. One explicit and two implicit formulations, one with Newton's solver with the Jacobian matrix and one with JFNK, are implemented. A detailed grid and model parameter sensitivity analysis is performed. For both cases, the JFNK predictions are in good agreement with the analytical solutions and explicit profiles. Further, stable results can be achieved using high CFL numbers up to 200 with a suitable choice of JFNK parameters. The computational time is significantly reduced by JFNK compared with the calculations requiring the determination of the Jacobian matrix. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4+ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12: 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4+ T cells. Copyright © 2014 John Wiley & Sons, Ltd.

Summary In this paper, we take a different perspective on the derivation of artificial viscosity. Heretofore, the development of artificial viscosity has been based on the paper published in Journal of Applied Physics in 1950 authored by John von Neumann and Robert Richtmyer [1]. Earlier, in 1948, Richtmyer published a report at Los Alamos Scientific Laboratory documenting the original concept [2]. This report was the true origin of shock capturing methods and contains several key ideas that are conceptually different than the 1950 journal article. Unfortunately, this report (LA-671) was classified until 1993. This has resulted in two issues: the misattribution of the invention of artificial viscosity as primarily being the work of von Neumann and the loss of the structurally different ideas in the original report. We seek to right the record of history here and use the ideas contained in Richtmyer's report to good effect in deriving a new shock viscosity. The focus of previous development has been the Hugoniot curve describing the locus of states connected by a single shock wave. Here we follow a path more focused upon the Rayleigh line, which is strongly guided by Richtmyer's line of development of the original artificial viscosity formulation. We provide an implementation of the method resulting from this perspective and computational results for simple shock problems. Copyright © 2014 John Wiley & Sons, Ltd.

In the paper at hand, co-simulation approaches are analyzed for coupling two solvers. The solvers are assumed to be coupled by algebraic constraint equations. We discuss 2 different coupling methods. Both methods are semi-implicit, i.e. they are based on a predictor/corrector approach. Method 1 makes use of the well-known Baumgarte-stabilization technique. Method 2 is based on a weighted multiplier approach. For both methods, we investigate formulations on index-3, index-2 and index-1 level and analyze the convergence, the numerical stability and the numerical error. The presented approaches require Jacobian matrices. Since only partial derivatives with respect to the coupling variables are needed, calculation of the Jacobian matrices may very easily be calculated numerically and in parallel with the predictor step. For that reason, the presented methods can in a straightforward manner be applied to couple commercial simulation tools without full solver access. The only requirement on the subsystem solvers is that the macro-time step can be repeated once in order to accomplish the corrector step. Within the paper, we introduce methods for coupling mechanical systems. The presented approaches can, however, also be applied to couple arbitrary non-mechanical dynamical systems.

16 pages; International audience; This paper concerns the second order marginals of biphased random media. We give discriminating necessary conditions for a bivariate function to be such a valid marginal, and illustrate our study with two practical applications: (1) the spherical variograms are valid indicator variograms if and only if they are multiplied by a sufficiently small constant, which upper bound is estimated, and (2) not every covariance/indicator variogram can be obtained with a Gaussian level set. The theoretical results backing this study are contained in a companion paper.

SUMMARY Convergence theories and a deluxe dual and primal finite element tearing and interconnecting algorithm are developed for a hybrid staggered DG finite element approximation of H(curl) elliptic problems in two dimensions. In addition to the advantages of staggered DG methods, the basis functions of the new hybrid staggered DG method are all locally supported in the triangular elements, and a Lagrange multiplier approach is applied to enforce the global connections of these basis functions. The interface problem on the Lagrange multipliers is further reduced to the resulting problem on the subdomain interfaces, and a dual and primal finite element tearing and interconnecting algorithm with an enriched weight factor is then applied to the resulting problem. Our algorithm is shown to give a condition number bound ofC.1C log.H=h// 2 , independent of the two parameters, whereH=his the number of triangles across each subdomain. Numerical results are included to confirm our theoretical bounds. Copyright © 2013 John Wiley & Sons, Ltd. Received 12 September 2013; Accepted 11 November 2013 In this paper, a dual and primal finite element tearing and interconnecting (FETI-DP) algorithm is developed for a fast and stable solution of a new hybrid staggered DG (HSDG) method applied to H(curl) problems. Staggered DG (SDG) methods were first introduced in [1–3] for wave propagation problems. The idea was subsequently applied to other problems, such as convection-diffusion equations [4], electromagnetic problems [5–9], Stokes equations [10], and multiscale wave simulations [11, 12]. Similar to [5], the advantages of using SDG for H(curl) problems are the preservation of the structures of the differential operators, the local conservation property, and the optimal convergence. In particular, the discretizations of the two curl operators by the SDG method are adjoint to each other, and the null space of the discrete curl operator is exactly the gradients. Moreover, the divergence condition is automatically satisfied in an appropriate weak sense. Despite the aforementioned advantages, the implementation of SDG methods requires careful numbering of the supports of the basis functions. To allow an easier implementation of the SDG method, the HSDG method is introduced in this paper. One distinctive advantage of the HSDG method is that the basis functions are all locally supported in the triangular elements, hence, the numbering can be easily performed. The fact that the basis functions are totally discontinuous requires some global couplings. In particular, we will use the Lagrange multiplier approach to enforce the global connection of the basis functions. This idea is also related to the hybridized DG methods [13–16]. In fact, it is shown that the SDG method is the limit of a hybridized DG method

At the University of Liverpool, the Helicopter Multi-Block (HMB2) CFD code is used for studies of various subsonic and transonic flows. This paper presents the implicit implementation in HMB2 of the AUSM + and AUSM + up, with a fully analytical Jacobian, so that a wider range of Mach numbers can be modelled, including high-speed flows. A description of the derivation of the analytical Jacobian is given in this paper along with an evaluation of the performance of the implicit schemes for different test cases, including turbulent flows. As examples of high-speed flows of aerospace interest, a blunt body, a single cone, a shock wave/turbulent boundary-layer interaction and the Orion spacecraft have been considered. For the transonic regime, the RAE2822 aerofoil and the ONERA M6 wing have been used as test cases. Finally, for low Mach flows, the S809 wind turbine aerofoil and the MEXICO project wind turbine blade have been chosen. The shear stress transport and κ– ω turbulence models have been employed for the turbulent cases. The proposed implicit implementation of the AUSM + and AUSM + up schemes proved to have good efficiency and robustness without affecting the reliability of the original schemes

The focus of this paper is two-dimensional computational modelling of water flow in unsaturated soils consisting of weakly conductive disconnected inclusions embedded in a highly conductive connected matrix. When the inclusions are small, a two-scale Richards’ equation-based model has been proposed in the literature taking the form of an equation with effective parameters governing the macroscopic flow coupled with a microscopic equation, defined at each point in the macroscopic domain, governing the flow in the inclusions. This paper is devoted to a number of advances in the numerical implementation of this model. Namely, by treating the micro-scale as a two-dimensional problem, our solution approach based on a control volume finite element method can be applied to irregular inclusion geometries, and, if necessary, modified to account for additional phenomena (e.g. imposing the macroscopic gradient on the micro-scale via a linear approximation of the macroscopic variable along the microscopic boundary). This is achieved with the help of an exponential integrator for advancing the solution in time. This time integration method completely avoids generation of the Jacobian matrix of the system and hence eases the computation when solving the two-scale model in a completely coupled manner. Numerical simulations are presented for a two-dimensional infiltration problem.

Large deviation theory is a branch of probability theory that is devoted to a study of the “rate” at which empirical estimates of various quantities converge to their true values. The object of study in this paper is the rate at which estimates of the doublet frequencies of a Markov chain over a finite alphabet converge to their true values. In the case where the Markov process is actually an independent and identically distributed (i.i.d.) process, the rate function turns out to be the relative entropy (or Kullback-Leibler divergence) between the true and the estimated probability vectors. This result is a special case of a very general result known as Sanov's theorem and dates back to 1957. Moreover, since the introduction of the “method of types” by Csiszar and his co-workers during the 1980s, the Proof of this version of Sanov's theorem has been “elementary,” using some combinatorial arguments. However, when the i.i.d. process is replaced by a Markov process, the available Proofs are far more complex. The main objective of this paper is therefore to present a first-principles derivation of the LDP for finite state Markov chains, using only simple combinatorial arguments (e.g. the method of types), thus gathering in one place various arguments and estimates that are scattered over the literature. The approach presented here extends naturally to multi-step Markov chains.

This paper is the first in a series papers devoted to the study of the rigorous derivation of the nonlinear Schrodinger (NLS) equation as well as other related systems starting from a model coming from the gravity-capillary water wave system in the long-wave limit. Our main goal is to understand resonances and their effects on having the nonlinear Schrodinger approximation or modification of it or having other models to describe the limit equation. In this first paper, our goal is not to derive NLS but to allow the presence of an arbitrary sequence of frequencies around which we have a modulation and prove local existence on a uniform time. This yields a new class of large data for which we have a large time of existence. © 2012 Wiley Periodicals, Inc.

This paper deals with the estimation of the intensity of a planar point process on the basis of a single point pattern, observed in a rectangular window. If the model assumptions of stationarity and isotropy hold, the method of block bootstrapping can be used to estimate the intensity of the process with confidence bounds. The results of two variants of block bootstrapping are compared with a parametric approximation based on the assumption of a Gaussian distribution of the numbers of points in deterministic subwindows of the original pattern. The studies were performed on patterns obtained by simulation of well-known point process models (Poisson process, two Matern cluster processes, Matern hardcore process, Strauss hardcore process). They were also performed on real histopathological data (point patterns of capillary profiles of 12 cases of prostatic cancer). The methods are presented as worked examples on two cases, where we illustrate their use as a check on stationarity (homogeneity) of a point process with respect to different fields of vision. The paper concludes with various methodological discussions and suggests possible extensions of the block bootstrap approach to other fields of spatial statistics.

This paper discusses the conjecture that Archimedean copulas—mainly Gumbel copulas—provide better stochastic models than Gaussian copulas for multivariate analysis of small wind energy generation clusters (focused on the analysis of microgrid viability). The paper provides guidance on how to model the multivariate Gumbel copula, thus allowing to follow up some recently published results that show that the correlation structure in bivariate models (generator pairing) is best defined by bivariate Gumbel copulas rather than by their Gaussian counterpart. However, it is shown in this paper that the higher the dimension (the larger the number of microgrid generators) the more probably the Gaussian copulas outperform the Gumbel copulas. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. The class of Shifted Laplacian preconditioners are known to significantly speed-up Krylov convergence. However, these preconditioners have a parameter beta, a measure of the complex shift. Due to contradictory requirements for the multigrid and Krylov convergence, the choice of this shift parameter can be a bottleneck in applying the method. In this paper, we propose a wavenumber-dependent minimal complex shift parameter which is predicted by a rigorous k-grid Local Fourier Analysis (LFA) of the multigrid scheme. We claim that, given any (regionally constant) wavenumber, this minimal complex shift parameter provides the reader with a parameter choice that leads to efficient Krylov convergence. Numerical experiments in one and two spatial dimensions validate the theoretical results. It appears that the proposed complex shift is both the minimal requirement for a multigrid V-cycle to converge, as well as being near-optimal in terms of Krylov iteration count.

SUMMARY In this paper, we study the problem of control of discrete-time linear time varying systems over uncertain channels. The uncertainty in the channels is modeled as a stochastic random variable. We use exponential mean square stability of the closed-loop system as a stability criterion. We show that fundamental limitations arise for the mean square exponential stabilization for the closed-loop system expressed in terms of statistics of channel uncertainty and the positive Lyapunov exponent of the open-loop uncontrolled system. Our results generalize the existing results known in the case of linear time invariant systems, where Lyapunov exponents are shown to emerge as the generalization of eigenvalues from linear time invariant systems to linear time varying systems. Simulation results are presented to verify the main results of this paper. Copyright © 2012 John Wiley & Sons, Ltd.