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We address a numerical methodology for the approximation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a macroscopic description does not exist analytically in a closed form. Thus, the underlying hypothesis is that we have a detailed microscopic simulator (Monte Carlo, molecular dynamics, agent-based model etc.) that describes the dynamics of the subunits of a complex system (or a black-box large-scale simulator) but we do not have explicitly available a dynamical model in a closed form that describes the emergent coarse-grained/macroscopic dynamics. Our numerical scheme is based on the equation-free multiscale framework, and it is a three-tier procedure including (a) the convergence on the coarse-grained saddle equilibrium, (b) its coarse-grained stability analysis, and (c) the approximation of the local invariant stable and unstable manifolds; the later task is achieved by the numerical solution of a set of homological/functional equations for the coefficients of a polynomial approximation of the manifolds.

We investigate a behavioral epidemic model including a partially effective vaccination at all ages. The vaccination is information-dependent, in the sense that the vaccination rate of susceptibles depends on the current and the past information available about the disease prevalence in the population. The weight given to the past history is described by an exponential kernel. The proposed model presents both the possibility of backward bifurcation and that of oscillations triggered by behavioral memory. Furthermore, a forward hysteresis scenario may take place where multiple endemic states are possible when the basic reproduction number P 0 is greater than one. Finally, a stable endemic state may destabilize via Hopf bifurcation not only when P 0 > 1 but also when P 0 1 , depending on the interplay between some relevant information-related parameters.

In this paper we present a minimal integrated environment-economic growth model. The accumulations of capital and pollution are connected by reciprocal feedbacks. Pollution is an undesirable but inevitable by-product of production, whose efficiency is hindered by pollution. The nexus between pollution and production is embodied in a damage function. The evolution of capital is enriched by the specific technology adopted here, an S -shaped production function. The model dynamics is represented by a couple of nonlinear differential equations, whose long run behavior gives rise to multiple stationary points. A global analysis underlines the importance of a meditated choice of the relevant policy parameters. Pollution induced poverty traps emerge for low level of saving ratio and share of abated emission. Periodic behaviour of the economic and environmental variables represents an early signal of the imminent risk of being trapped in an unsatisfactory level of economic performance. Numerical examples corroborate the theoretical results of the paper.

We localize the sources of brain activity of children with epilepsy based on electroencephalograph (EEG) recordings acquired during a visual discrimination working memory task. For the numerical solution of the inverse problem, with the aid of age-specific MRI scans processed from a publicly available database, we use and compare three regularization numerical methods, namely the standardized low resolution brain electromagnetic tomography (sLORETA), the weighted minimum norm estimation (wMNE) and the dynamic statistical parametric mapping (dSPM). We show that all three methods provide the same spatio-temporal patterns of differences between the groups of epileptic and control children. In particular, our analysis reveals statistically significant differences between the two groups in regions of the parietal cortex indicating that these may serve as "biomarkers" for diagnostic purposes and ultimately localized treatment.

Purpose The purpose of this paper is to analyze the thermal and fluid dynamic behaviors of mixed convection in air because of the interaction between a buoyancy flow and a moving plate induced flow in a horizontal no parallel-plates channel to investigate the effects of the minimum channel spacing, wall heat flux, moving plate velocity and converging angle. Design/methodology/approach The horizontal channel is made up of an upper inclined plate heated at uniform wall heat flux and a lower adiabatic moving surface (belt). The belt moves from the minimum channel spacing section to the maximum channel spacing section at a constant velocity so that its effect interferes with the buoyancy effect. The numerical analysis is accomplished by means of the finite volume method, using the commercial code Fluent. Findings Results in terms of heated upper plate and moving lower plate temperatures and stream function fields are presented. The paper underlines the thermal and fluid dynamic differences when natural convection or mixed convection takes place, varying minimum channel spacing, wall heat flux, moving plate velocity and converging angle. Research limitations/implications The hypotheses on which the present analysis is based are two-dimensional, laminar and steady state flow and constant thermo physical properties with the Boussinesq approximation. The minimum distance between the upper heated plate of the channel and its lower adiabatic moving plate is 10 and 20 mm. The moving plate velocity varies in the range 0-1 m/s; the belt moves from the right reservoir to the left one. Three values of the uniform wall heat flux are considered, 30, 60 and 120 W/m2, whereas the inclination angle of the upper plate θ is 2° and 10°. Practical implications Mixed convection because of moving surfaces in channels is present in many industrial applications; examples of processes include continuous casting, extrusion of plastics and other polymeric materials, bonding, annealing and tempering, cooling and/or drying of paper and textiles, chemical catalytic reactors, nuclear waste repositories, petroleum reservoirs, composite materials manufacturing and many others. The investigated configuration is used in applications such as re-heating of billets in furnaces for hot rolling process, continuous extrusion of materials and chemical vapor deposition, and it could also be used in thermal control of electronic systems. Originality/value This paper evaluates the thermal and velocity fields to detect the maximum temperature location and the presence of fluid recirculation. The paper is useful to thermal designers.

We present a systematic approach to the construction of starting procedures for general linear methods (GLMs) of order p and stage order q = p . Order conditions for starting procedures based on the generalized Runge–Kutta (RK) are derived using the theory of rooted trees, elementary differentials, and elementary weights, and examples of generalized RK formulas are given up to the order p = 4 .

This paper introduces a study on twisted viscoelastic beams, having considered fractional calculus to capture the viscoelastic behaviour. Further another novelty of this paper is extending a recent numerical approach, labelled line elementless method (LEM), to viscoelastic beams. The latter does not require any discretization neither in the domain nor in the boundary. Some numerical applications have been reported to demonstrate the efficiency and accuracy of the method.

International audience; We introduce a class of preconditioners for general sparse matrices based on the Birkhoff-von Neumann decomposition of doubly stochastic matrices. These precondi- tioners are aimed primarily at solving challenging linear systems with highly unstructured and indefinite coefficient matrices. We present some theoretical results and numerical experiments on linear systems from a variety of applications.; Nous introduisons une classe de préconditionnement pour matrices creuses générales fondées sur la décomposition de Birkhoff–von Neumann des matrices bistochastiques. Ces préconditionnements visent principalement à résoudre des systèmes linéaires difficiles dont les matrices sont non-structurées et non-définies. Nous présentons quelques résultats théoriques et des expériences numériques pour des systèmes linéaires issus de tout un ensemble d’applications.

The absorption inverse, studied in Jacobsen and Tien (2018) [13] , is a generalized inverse specifically introduced for the analysis of graphs with absorption. In this paper we consider numerical methods for the efficient computation of the absorption inverse and related quantities. Both direct and iterative methods are developed. We also consider different centrality measures for graphs with absorption, as well as fast updating/downdating techniques. Numerical experiments show that computations on graphs with up to 36 million edges can be performed quickly on a standard laptop.

In this paper, motivated by applications to multigrid, we present families of anisotropic, bivariate, interpolating and approximating subdivision schemes. We study the minimality and polynomial generation/reproduction properties of both families and Holder regularity of their prominent representatives. From the symbols of the proposed subdivision schemes, we define bivariate grid transfer operators for anisotropic multigrid methods. We link the generation/reproduction properties of subdivision to the convergence and optimality of the corresponding multigrid methods. We illustrate the performance of our subdivision based grid transfer operators on examples of anisotropic Laplacian and biharmonic problems.

Within the framework of the displacement-based virtual element method (VEM), namely, for plane elasticity, an important topic is the development of optimal techniques for the evaluation of the stress field. In fact, in the classical VEM formulation, the same projection operator used to approximate the strain field (and then evaluate the stiffness matrix) is employed to recover, via constitutive law, the stress field. Considering a first-order formulation, strains are locally mapped onto constant functions, and stresses are piecewise constant. However, the virtual displacements might engender more complex strain fields for polygons, which are not triangles. This leads to an undesirable loss of information with respect to the underlying virtual stress field. The recovery by compatibility in patches, originally proposed for finite element schemes, is here extended to VEM, aiming at mitigating such an effect. Stresses are recovered by minimizing the complementary energy of patches of elements over an assumed set of equilibrated stress modes. The procedure is simple, efficient, and can be readily implemented in existing codes. Numerical tests confirm the good performance of the proposed technique in terms of accuracy and indicate an increase of convergence rate with respect to the classical approach in many cases.

We perform one and two-parameter numerical bifurcation analysis of a mechanotransduction model approximating the dynamics of mesenchymal stem cell differentiation into neurons, adipocytes, myocytes and osteoblasts. For our analysis, we use as bifurcation parameters the stiffness of the extracellular matrix and parameters linked with the positive feedback mechanisms that up-regulate the production of the YAP/TAZ transcriptional regulators (TRs) and the cell-substrate adhesion area. Our analysis reveals regimes of hysteresis and multistability, stable oscillations of the effective adhesion area, the YAP/TAZ TRs and the PPAR γ receptors associated with the adipogenic fate, as well as homoclinic bifurcations that interrupt high-amplitude oscillations abruptly. The two-parameter bifurcation analysis of the Andronov–Hopf points that give birth to the oscillating patterns predicts their existence for soft extracellular substrates ( 1 k P a ), a regime that favours the neurogenic and the adipogenic cell fate. Furthermore, in these regimes, the analysis reveals the presence of homoclinic bifurcations that result in the sudden loss of stable oscillations of the cell-substrate adhesion towards weaker adhesions and high expression levels of the gene encoding Tubulin beta-3 chain, thus favouring the phase transition from the adipogenic to the neurogenic fate.

This paper presents a mathematical approach to model the effects of phenomena with random nature such as turbulence and fire-spotting into the existing wildfire simulators. The formulation proposes that the propagation of the fire-front is the sum of a drifting component (obtained from an existing wildfire simulator without turbulence and fire-spotting) and a random fluctuating component. The modelling of the random effects is embodied in a probability density function accounting for the fluctuations around the fire perimeter which is given by the drifting component. In past, this formulation has been applied to include these random effects into a wildfire simulator based on an Eulerian moving interface method, namely the Level Set Method (LSM), but in this paper the same formulation is adapted for a wildfire simulator based on a Lagrangian front tracking technique, namely the Discrete Event System Specification (DEVS). The main highlight of the present study is the comparison of the performance of a Lagrangian and an Eulerian moving interface method when applied to wild-land fire propagation. Simple idealised numerical experiments are used to investigate the potential applicability of the proposed formulation to DEVS and to compare its behaviour with respect to the LSM. The results show that DEVS based wildfire propagation model qualitatively improves its performance (e.g., reproducing flank and back fire, increase in fire spread due to pre-heating of the fuel by hot air and firebrands, fire propagation across no fuel zones, secondary fire generation, \dots). Though the results presented here are devoid of any validation exercise and provide only a proof of concept, they show a strong inclination towards an intended operational use. The existing LSM or DEVS based operational simulators like WRF-SFIRE and ForeFire respectively can serve as an ideal basis for the same., Accepted for publication in Commun. Nonlinear Sci. Numer. Simulat

We present a discrepancy-like stopping criterium for iterative regularization methods for the solution of linear discrete ill-posed problems. The presented criterium terminates the iterations of the iterative method when the residual norm of the computed solution becomes less or equal to the residual norm of a regularized Truncated Singular Value Decomposition (TSVD) solution. We present two algorithms for the automatic computation of the TSVD residual norm using the Discrete Picard Condition. The first algorithm uses the SVD coefficients while the second one uses the Fourier coefficients. In this work, we mainly focus on the Conjugate Gradient Least Squares method, but the proposed criterium can be used for terminating the iterations of any iterative regularization method. Many numerical tests on some selected one dimensional and image deblurring problems are presented and the results are compared with those obtained by state-of-the-art parameter selection rules. The numerical results show the efficiency and robustness of the proposed criterium.

We present the essential tools to deal with virtual element method (VEM) for the approximation of solutions of partial differential equations in mixed form. Functional spaces, degrees of freedom, projectors and differential operators are described emphasizing how to build them in a virtual element framework and for a general approximation order. To achieve this goal, it was necessary to make a deep analysis on polynomial spaces and decompositions. We exploit such “bricks” to construct virtual element approximations of Stokes, Darcy and Navier–Stokes problems and we provide a series of examples to numerically verify the theoretical behaviour of high-order VEM.

We revisit the theorem stating that it is possible to approximate with any accuracy any real continuous function with a class of relational maps. In other words, relational maps are universal approximators. We review the key works that have proved this property, highlighting their limitations and providing yet another proof that it is not restricted by certain assumptions considered in early proofs. We also show how one can go inversely to approximate these systems with a series of polynomials. This provides us with analytical expressions of these maps which can facilitate a series of important analysis tasks such as modeling and numerical analysis of ill-defined-uncertain complex systems.

We study the dynamics of Fermi-Pasta-Ulam chains with both harmonic and anharmonic power-law long-range interactions. We show that the dynamics is described in the continuum limit by a generalized fractional Boussinesq differential equation, whose derivation is performed in full detail. We also discuss a version of the model where couplings are alternating in sign., Comment: Corrected coefficients of the nonlinear terms of the generalized fractional Boussinesq equations, minor typos amended

We analyze the pattern formation due to dislocations under cyclic loading resulting from the Walgraef–Aifantis model. The model consists of a set of partial differential equations of the reaction–diffusion type in the one dimensional finite space with two different diffusion-like coefficients, for the mobile (free to move when the applied resolved shear stress in the slip plane exceeds a certain threshold) and for the immobile (of slow movement or trapped) dislocations. We derive analytically the Turing spatial and Andronov–Hopf temporal instabilities emanating from the homogeneous solutions and construct the complete bifurcation diagram of the far-from-equilibrium spatio-temporal patterns, with respect to the applied stress and the size of the domain. Finally, we analyze the symmetric properties of all branches of both steady and oscillating far-from-equilibrium regimes.

A family of virtual element methods for the two-dimensional Navier-Stokes equations is proposed and analyzed. The schemes provide a discrete velocity field which is pointwise divergence-free. A rigorous error analysis is developed, showing that the methods are stable and optimally convergent. Several numerical tests are presented, confirming the theoretical predictions. A comparison with some mixed finite elements is also performed.

In the rank of behavioral rules, imitation-based heuristics has received special attention in economics (see [14] and [12]). In particular, imitative behavior is considered in order to understand the evidences arising in experimental oligopolies which reveal that the Cournot-Nash equilibrium does not emerge as unique outcome and show that an important component of the production at the competitive level is observed (see e.g.[1,3,9] or [7,10]). By considering the pioneering groundbreaking approach of [2], we build a dynamical model of linear oligopolies where heterogeneous decision mechanisms of players are made explicit. In particular, we consider two different types of quantity setting players characterized by different decision mechanisms that coexist and operate simultaneously: agents that adaptively adjust their choices towards the direction that increases their profit are embedded with imitator agents. The latter ones use a particular form of proportional imitation rule that considers the awareness about the presence of strategic interactions. It is noteworthy that the Cournot-Nash outcome is a stationary state of our models. Our thesis is that the chaotic dynamics arousing from a dynamical model, where heterogeneous players are considered, are capable to qualitatively reproduce the outcomes of experimental oligopolies.

We show some integral formulas involving the characteristic curvature for closed real hypersurfaces in complex spaces.

We consider, as a simple model problem, the application of virtual element methods (VEMs) to the linear magnetostatic three-dimensional problem in the formulation of Kikuchi. In doing so, we also introduce new serendipity VEM spaces, where the serendipity reduction is made only on the faces of a general polyhedral decomposition (assuming that internal degrees of freedom could be more easily eliminated by static condensation). These new spaces are meant, more generally, for the combined approximation of $H^1$-conforming (0-forms), $H(curl)$-conforming (1-forms), and $H(div)$-conforming (2-forms) functional spaces in three dimensions, and they could surely be useful for other problems and in more general contexts.

The Modified Finite Particle Method (MFPM) is a numerical method belonging to the class of meshless methods, nowadays widely investigated due to their characteristic of being capable to easily model large deformation and fluid-dynamic problems. Here we use the MFPM to approximate the Stokes problem. Since the classical formulation of the Stokes problem may lead to pressure spurious oscillations, we investigate alternative formulations and focus on how MFPM discretization behaves in those situations. Some of the investigated formulations, in fact, do not enforce strongly the incompressibility constraint, and therefore an important issue of the present work is to verify if the MFPM is able to correctly reproduce the incompressibility in those cases. The numerical results show that for the formulations in which the incompressibility constraint is properly satisfied from a numerical point of view, the expected second-order is achieved, both in static and in dynamic problems.

In this paper, we develop a mixed isogeometric analysis approach based on subdivision stabilization to study strongly coupled diffusion in solids in both small and large deformation ranges. Coupling the fluid pressure and the solid deformation, the mixed formulation suffers from numerical instabilities in the incompressible and the nearly incompressible limit due to the violation of the inf-sup condition. We investigate this issue using subdivision-stabilized nonuniform rational B-spline (NURBS) elements, as well as different families of mixed isogeometric analysis techniques, and assess their stability through a numerical inf-sup test. Furthermore, the validity of the inf-sup stability test in poromechanics is supported by a mathematical proof concerning the corresponding stability estimate. Finally, two numerical examples involving a rigid strip foundation on saturated soil and a swelling hydrogel structure are presented to validate the stability and to demonstrate the robustness of the proposed approach.

The conventional strong form collocation approach known as Differential Quadrature (DQ) method has been applied in the past to a vast type of engineering problems. It is well-known that its application is strictly limited to regular regions where derivatives are approximated along mesh lines. Generally, its accuracy increases when the number of collocation points is large and the method tends to be stable. However, for some numerical problems several points are needed in order to obtain an accurate solution. Changing the basis functions another numerical technique was developed called Radial Basis Functions (RBFs) method, which has the advantage of approximating derivatives using irregular point distributions and the basis functions depend on the mutual radial distance of the grid points. In order to extend the idea of DQ method to a general case a Radial Basis Function based on Differential Quadrature (RBF-DQ) method has been recently developed. This method merges the advantages of both techniques. Furthermore, this work proposes the application of RBF-DQ when a domain decomposition technique is considered. In this way it will be shown that, using some kind of basis functions the number of grid points per element can be reduced compared to other classical approaches. Furthermore, once the shape parameter is fixed for one case, it is not needed to calculate it again for other applications.

The anomalous transport of a viscous fluid across a porous media with power-law scaling of the geometrical features of the pores is dealt with in the paper. It has been shown that, assuming a linear force–flux relation for the motion in a porous solid, then a generalized version of the Hagen–Poiseuille equation has been obtained with the aid of Riemann–Liouville fractional derivative. The order of the derivative is related to the scaling property of the considered media yielding an appropriate mechanical picture for the use of generalized fractional-order relations, as recently used in scientific literature.

In this paper we consider some nonlinear discrete-time dynamic models proposed in the literature to represent marketing competition, and we use these models to critically discuss the statement, often made in economic literature, that identical agents behave identically and quasi-identical ones behave in a similar way. We show, through examples and some general mathematical statements, that the one-dimensional model of a representative agent, whose dynamics summarize the common behavior of identical interacting agents, may be misleading. In order to discuss these topics some simple methods for the study of local stability and bifurcations are employed, as well as numerical examples where some results taken from the literature on chaos synchronization are applied to two-dimensional marketing models that exhibit riddling, blowout and other global phenomena related to the existence of measure-theoretic attractors. © 2014 Elsevier B.V.

We study the minimizers of an energy functional with a self-consistent magnetic field, which describes a quantum gas of almost-bosonic anyons in the average-field approximation. For the homogeneous gas we prove the existence of the thermodynamic limit of the energy at fixed effective statistics parameter, and the independence of such a limit from the shape of the domain. This result is then used in a local density approximation to derive an effective Thomas--Fermi-like model for the trapped anyon gas in the limit of a large effective statistics parameter (i.e., "less-bosonic" anyons)., Minor typo corrected, version accepted in Analysis and PDEs

We deal with non negative functions satisfying \[ \left\{ \begin{array}{ll} (-��)^s u_s=0 & \mathrm{in}\quad C, u_s=0 & \mathrm{in}\quad \mathbb{R}^n\setminus C, \end{array}\right. \] where $s\in(0,1)$ and $C$ is a given cone on $\mathbb R^n$ with vertex at zero. We consider the case when $s$ approaches $1$, wondering whether solutions of the problem do converge to harmonic functions in the same cone or not. Surprisingly, the answer will depend on the opening of the cone through an auxiliary eigenvalue problem on the upper half sphere. These conic functions are involved in the study of the nodal regions in the case of optimal partitions and other free boundary problems and play a crucial role in the extension of the Alt-Caffarelli-Friedman monotonicity formula to the case of fractional diffusions., 37 pages, 3 figures

The aim of this research is to analyse a Keynesian goods market closed economy by considering a continuous-time setup with fixed delays. The work compares dynamic results based on linear and nonlinear adjustment mechanisms through which the aggregate supply (production) reacts to a disequilibrium in the goods market and consumption depends on income at a preceding date. Both analytical and geometrical (stability switching curves) techniques are used to characterise the stability properties of the stationary equilibrium.

We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. The Treves conjecture defines a stratification and states that an operator of this type is analytic hypoelliptic if and only if all the strata in the stratification are symplectic manifolds. ¶ Albano, Bove, and Mughetti (2016) produced an example where the operator has a single symplectic stratum, according to the conjecture, but is not analytic hypoelliptic. ¶ If the characteristic manifold has codimension 2 and if it consists of a single symplectic stratum, defined again according to the conjecture, it has been shown that the operator is analytic hypoelliptic. ¶ We show here that the above assertion is true only if the stratum is single, by producing an example with two symplectic strata which is not analytic hypoelliptic.

We prove some existence results for a class of nonlinear fractional equations driven by a nonlocal operator.

This article considers a Cournot duopoly model in a continuous-time framework and analyze its dynamic behavior when the competitors are heterogeneous in determining their output decision. Specifically the model is expressed in the form of differential equations with discrete delays. The stability conditions of the unique Nash equilibrium of the system are determined and the emergence of Hopf bifurcations is shown. Applying some recent mathematical techniques (stability switching curves) and performing numerical simulations, the paper confirms how different time delays affect the stability of the economy.

In the present paper, strong form finite elements are employed for the free vibration study of laminated arbitrarily shaped plates. In particular, the stability and accuracy of three different Fourier expansion‐based differential quadrature techniques are shown. These techniques are used to solve the partial differential system of equations inside each computational element. The three approaches are called harmonic differential quadrature, Fourier differential quadrature and improved Fourier expansion‐based differential quadrature methods. The improved Fourier expansion‐based differential quadrature method implements auxiliary functions in order to approximate functional derivatives up to the fourth order, with respect to the Fourier differential quadrature method that has a basis made of sines and cosines. All the present applications are related to literature comparisons and the presentation of new results for further investigation within the same topic. A study of such kind has never been proposed in the literature, and it could be useful as a reference for future investigation in this matter.

Given a large square matrix A and a sufficiently regular function f so that f ( A ) is well defined, we are interested in the approximation of the leading singular values and corresponding left and right singular vectors of f ( A ) , and in particular in the approximation of ‖ f ( A ) ‖ , where ‖ ⋅ ‖ is the matrix norm induced by the Euclidean vector norm. Since neither f ( A ) nor f ( A ) v can be computed exactly, we introduce a new inexact Golub–Kahan–Lanczos bidiagonalization procedure, where the inexactness is related to the inaccuracy of the operations f ( A ) v , f ( A ) ⁎ v . Particular outer and inner stopping criteria are devised so as to cope with the lack of a true residual. Numerical experiments with the new algorithm on typical application problems are reported.

In this work, we propose a fast iterative algorithm for the reconstruction of digital breast tomosynthesis images. The algorithm solves a regularization problem, expressed as the minimization of the sum of a least-squares term and a weighted smoothed version of the Total Variation regularization function. We use a Fixed Point method for the solution of the minimization problem, requiring the solution of a linear system at each iteration, whose coefficient matrix is a positive definite approximation of the Hessian of the objective function. We propose an efficient implementation of the algorithm, where the linear system is solved by a truncated Conjugate Gradient method. We compare the Fixed Point implementation with a fast first order method such as the Scaled Gradient Projection method, that does not require any linear system solution. Numerical experiments on a breast phantom widely used in tomographic simulations show that both the methods recover microcalcifications very fast while the Fixed Point is more efficient in detecting masses, when more time is available for the algorithm execution.

Engineering applications in conventional numerical modeling are investigated by domain decomposition techniques because, generally, structural components do not have a regular shape. The most common and versatile numerical approach, in this regard, is the Finite Element (FE) method, which uses low order polynomials and the variational formulation for the solution of the differential problem. In contrast with the classic approach, the present paper shows a strong form collocation method for solving laminated composite plates wherein discontinuities arise. Both approaches need mapping technique for converting arbitrarily shaped elements into the computational space. Since the present strong form technique can deal with variable order approximating polynomials within each element, advanced blending function mapping must be employed for treating any discontinuity and distortion effect using the smallest number of elements. The Differential Quadrature (DQ) method is utilized for solving the mathematical problem and isogeometric mapping is implemented for the nonlinear mapping of complex shapes. The present methodology belongs to the framework of Isogeometric Analysis (IGA), therefore this approach is termed Strong Formulation Isogeometric Analysis (SFIGA). Numerical applications show the accuracy, stability and reliability of the present methodology.

In the present paper the Airy operator on star graphs is defined and studied. The Airy operator is a third order differential operator arising in different contexts, but our main concern is related to its role as the linear part of the Korteweg-de Vries equation, usually studied on a line or a half-line. The first problem treated and solved is its correct definition, with different characterizations, as a skew-adjoint operator on a star graph, a set of lines connecting at a common vertex representing, for example, a network of branching channels. A necessary condition turns out to be that the graph is balanced, i.e. there is the same number of ingoing and outgoing edges at the vertex. The simplest example is that of the line with a point interaction at the vertex. In these cases the Airy dynamics is given by a unitary or isometric (in the real case) group. In particular the analysis provides the complete classification of boundary conditions giving momentum (i.e., $L^2$-norm of the solution) preserving evolution on the graph. A second more general problem here solved is the characterization of conditions under which the Airy operator generates a contraction semigroup. In this case unbalanced star graphs are allowed. In both unitary and contraction dynamics, restrictions on admissible boundary conditions occur if conservation of mass (i.e., integral of the solution) is further imposed. The above well posedness results can be considered preliminary to the analysis of nonlinear wave propagation on branching structures.

In the context of marine application of computational fluid dynamic (CFD), it is well known that the numerical simulations of planing craft are significantly less reliable than that of displacement ships. For this reason, it is important to perform a comprehensive approach to the verification and validation (V&V) methodology and procedures for simulating CFD planing craft. In the first part of this paper, an assessment of the accuracy and effectiveness of different simulation setups and techniques for planing craft is performed. In the second part, the results of the V&V study are reported for three different hull models at four Froude numbers (Fr). The Unsteady Reynolds-Averaged Navier-Stokes code results were validated using benchmark experimental data obtained for three hull models characterized by systematic variation of the length to beam ratio. Grid independence, iteration, and time-step convergence analysis for response variables (resistance coefficients, wetted surfaces, and dynamic trim angles) were conducted using the main error and uncertainty estimation methods available in the literature. The same procedures were followed for the profiles of the wave patterns. The results showed that there was improved reliability of the numerical simulation of the planing craft in terms of the errors and uncertainties, related to the predictions of resistance, running attitude, and wave pattern. The results of the V&V study highlighted the fact that modeling of the planing craft is a critical point to improve the reliability of the numerical simulation. © 2016, Society of Naval Architects and Marine Engineers. All rights reserved.

We prove the convergence of an explicit numerical scheme for the discretization of a coupled hyperbolic-parabolic system in two space dimensions. The hyperbolic part is solved by a Lax−Friedrichs method with dimensional splitting, while the parabolic part is approximated by an explicit finite-difference method. For both equations, the source terms are treated by operator splitting. To prove convergence of the scheme, we show strong convergence of the hyperbolic variable, while convergence of the parabolic part is obtained only weakly* in L ∞ . The proof relies on the fact that the hyperbolic flux depends on the parabolic variable through a convolution function. The paper also includes numerical examples that document the theoretically proved convergence and display the characteristic behaviour of the Lotka−Volterra equations.

The statistical characterization of the oscillator response with non-integer order damping under Gaussian noise represents an important challenge in the modern stochastic mechanics. In fact, this kind of problem appears in several issues of different type (wave propagation in viscoelastic media, Brownian motion, fluid dynamics, RLC circuit, etc.). The aim of this paper is to provide a stochastic characterization of the stationary response of linear fractional oscillator forced by normal white noise. In particular, this paper shows a new method to obtain the correlation function by exact complex spectral moments. These complex quantities contain all the information to describe the random processes but in the considered case their analytical evaluation needs some mathematical manipulations. For this reason such complex spectral moment characterization is used in conjunction with a fractional-order state variable analysis. This kind of analysis permits to find the exact expression of complex spectral moments, and the correlation function by using the Mellin transform. Moreover, the proposed approach provides an analytical expression of the response variance of the fractional oscillator. Capability and efficiency of the present method are shown in the numerical examples in which correlation and variance of fractional oscillator response are found and compared with those obtained by Monte Carlo simulations.

In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier-Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixed-point iteration. The basic iteration is accelerated by a Krylov subspace method like restarted GMRES. The corresponding preconditioner requires at each iteration the solution of a set of discrete scalar elliptic equations, one for each component of the velocity field. Numerical experiments illustrating the convergence behavior for different finite element discretizations of Stokes and Oseen problems are included. © 2010 IMACS. Published by Elsevier B.V. All rights reserved.

In this paper, we introduce and analyze a modification of the Hermitian and skew-Hermitian splitting iteration method for solving a broad class of complex symmetric linear systems. We show that the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method is unconditionally convergent. Each iteration of this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. These two systems can be solved inexactly. We consider acceleration of the MHSS iteration by Krylov subspace methods. Numerical experiments on a few model problems are used to illustrate the performance of the new method. © 2010 Springer-Verlag.

We address the problem of defining early warning indicators of financial crises. To this purpose, we fit the relevant time series through a class of linear models, known as auto-regressive moving-average (ARMA(p, q)) models. By running such a fit on intervals of the time series that can be considered stationary, we first determine the typical ARMA( p ¯ , q ¯ ). Such a model exists over windows of about 60 days and turns out to be an AR(1). For each of them, we estimate the relative parameters, i.e. ϕi and θi on the same running windows. Then, we define a distance ϒ from such typical model in the space of the likelihood functions and compute it on short intervals of stocks indexes. Such a distance is expected to increase when the stock market deviates from its normal state for the modifications of the volatility which happen commonly before a crisis. We observe that ϒ computed for the Dow Jones, Standard and Poor’s and EURO STOXX 50 indexes provides an effective early warning indicator which allows for detection of the crisis events that showed precursors.

In this paper, we study the extent to which any risk measure can lead to superadditive risk assessments, implying the potential for penalizing portfolio diversification. For this purpose we introduce the notion of extreme-aggregation risk measures. The extreme-aggregation measure characterizes the most superadditive behavior of a risk measure, by yielding the worst-possible diversification ratio across dependence structures. One of the main contributions is demonstrating that, for a wide range of risk measures, the extreme-aggregation measure corresponds to the smallest dominating coherent risk measure. In our main result, it is shown that the extreme- aggregation measure induced by a distortion risk measure is a coherent distortion risk measure. In the case of convex risk measures, a general robust representation of coherent extreme-aggregation measures is provided. In particular, the extreme-aggregation measure induced by a convex short- fall risk measure is a coherent expectile. These results show that, in the presence of dependence uncertainty, quantification of a coherent risk measure is often necessary, an observation that lends further support to the use of coherent risk measures in portfolio risk management.

We provide and analyze analytical approximations of backward SDEs in the limit of small nonlinearity and short time, in the case of nonsmooth drivers. We identify the first and second order approximations within these asymptotics and consider two topical financial applications: the two interest rates pricing problem and the funding value adjustment. In the high dimensional diffusion setting, we show how to compute explicitly the first order formula by taking advantage of recent proxy techniques. Numerical tests up to dimension 20 illustrate the efficiency of the numerical schemes. We additionally investigate higher order expansions, which may hold under additional assumptions; we also provide a counterexample where the third order expansion fails to exist.

We study a heterogeneous duopolistic Cournotian game, in which the firms, producing a homogeneous good, have reduced rationality and respectively adopt a “Local Monopolistic Approximation” (LMA) and a gradient-based approach with endogenous reactivity, in an economy characterized by isoelastic demand function and linear total costs. We give conditions on reactivity and marginal costs under which the solution converges to the Cournot–Nash equilibrium. Moreover, we compare the stability regions of the proposed oligopoly to a similar one, in which the LMA firm is replaced by a best response firm, which is more rational than the LMA firm. We show that, depending on costs ratio, the equilibrium can lose its stability in two different ways, through both a flip and a Neimark–Sacker bifurcation. We show that the nonlinear, noninvertible map describing the model can give rise to several coexisting stable attractors (multistability). We analytically investigate the shape of the basins of attractions, in particular proving the existence of regions known in the literature as lobes.

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems. © Cambridge University Press, 2005.

We provide explicit conditions on the distribution of risk-neutral log-returns which yield sharp asymptotic estimates on the implied volatility smile. We allow for a variety of asymptotic regimes, including both small maturity (with arbitrary strike) and extreme strike (with arbitrary bounded maturity), extending previous work of Benaim and Friz [Math. Finance 19 (2009), 1-12]. We present applications to popular models, including Carr-Wu finite moment logstable model, Merton's jump diffusion model and Heston's model., 35 pages, 2 figures. To appear on SIAM Journal on Financial Mathematics

We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods., Comment: 22 pages, 20 figures