Renato Spigler

Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators

A mean-field model of nonlinearly coupled oscillators with randomly distributed frequencies and subject to independent external white noises is analyzed in the thermodynamic limit. When the frequency distribution isbimodal, new results include subcritical spontaneous stationary synchronization of the oscillators, supercritical time-periodic synchronization, bistability, and hysteretic phenomena. Bifurcating synchronized states are asymptotically constructed near bifurcation values of the coupling strength, and theirnonlinear stability properties ascertained.

Time-periodic phases in populations of nonlinearly coupled oscillators with bimodal frequency distributions

Abstract The mean field Kuramoto model describing the synchronization of a population of phase oscillators with a bimodal frequency distribution is analyzed (by the method of multiple scales) near regions in its phase diagram corresponding to synchronization to phases with a time-periodic order parameter. The richest behavior is found near the tricritical point where the incoherent, stationarily synchronized, “traveling wave” and “standing wave” phases coexist. The behavior near the tricritical point can be extrapolated to the rest of the phase diagram. Direct Brownian simulation of the model confirms our findings.

Approximation results for neural network operators activated by sigmoidal functions

In this paper, we study pointwise and uniform convergence, as well as the order of approximation, for a family of linear positive neural network operators activated by certain sigmoidal functions. Only the case of functions of one variable is considered, but it can be expected that our results can be generalized to handle multivariate functions as well. Our approach allows us to extend previously existing results. The order of approximation is studied for functions belonging to suitable Lipschitz classes and using a moment-type approach. The special cases of neural network operators activated by logistic, hyperbolic tangent, and ramp sigmoidal functions are considered. In particular, we show that for C(1)-functions, the order of approximation for our operators with logistic and hyperbolic tangent functions here obtained is higher with respect to that established in some previous papers. The case of quasi-interpolation operators constructed with sigmoidal functions is also considered.

Convergence and stability of implicit runge-kutta methods for systems with multiplicative noise

A class ofimplicit Runge-Kutta schemes for stochastic differential equations affected bymultiplicative Gaussian white noise is shown to be optimal with respect to global order of convergence in quadratic mean. A test equation is proposed in order to investigate the stability of discretization methods for systems of this kind. Herestability is intended in a truly probabilistic sense, as opposed to the recently introduced extension of A-stability to the stochastic context, given for systems with additive noise. Stability regions for the optimal class are also given.

Multivariate neural network operators with sigmoidal activation functions

In this paper, we study pointwise and uniform convergence, as well as order of approximation, of a family of linear positive multivariate neural network (NN) operators with sigmoidal activation functions. The order of approximation is studied for functions belonging to suitable Lipschitz classes and using a moment-type approach. The special cases of NN operators, activated by logistic, hyperbolic tangent, and ramp sigmoidal functions are considered. Multivariate NNs approximation finds applications, typically, in neurocomputing processes. Our approach to NN operators allows us to extend previous convergence results and, in some cases, to improve the order of approximation. The case of multivariate quasi-interpolation operators constructed with sigmoidal functions is also considered.

Domain Decomposition Solution of Elliptic Boundary-Value Problems via Monte Carlo and Quasi-Monte Carlo Methods

Domain decomposition of two-dimensional domains on which boundary-value elliptic problems are formulated is accomplished by probabilistic (Monte Carlo) as well as by quasi-Monte Carlo methods, generating only a few interfacial values and interpolating on them. Continuous approximations for the trace of solution are thus obtained, to be used as boundary data for the subproblems. The numerical treatment can then proceed by standard deterministic algorithms, separately in each of the so obtained subdomains. Monte Carlo and quasi-Monte Carlo simulations may naturally exploit multiprocessor architectures, leading to parallel computing, as well as the ensuing domain decomposition does. The advantages such as scalability obtained by increasing the number of processors are shown, both theoretically and experimentally, in a number of test examples, and the possibility of using clusters of computers (grid computing) is emphasized.

A collocation method for solving nonlinear Volterra integro-differential equations of neutral type by sigmoidal functions

A numerical collocation method is developed for solving nonlinear Volterra integro-differential equations (VIDEs) of the neutral type, as well as other non-standard and classical VIDEs. A sigmoidal functions approximation is used to suitably represent the solutions. Special computational advantages are obtained using unit step functions, and important applications can also be obtained by using other sigmoidal functions, such as logistic and Gompertz functions. The method allows one to obtain a simultaneous approximation of the solution to a given VIDE and its first derivative, by means of an explicit formula. A priori as well as a posteriori estimates are derived for the numerical errors, and numerical examples are given for the purpose of illustration. A comparison is made with the classical piecewise polynomial collocation method as for accuracy and CPU time.

A -stability of Runge-Kutta methods for systems with additive noise

Numerical stability of both explicit and implicit Runge-Kutta methods for solving ordinary differential equations with an additive noise term is studied. The concept of numerical stability of deterministic schemes is extended to the stochastic case, and a stochastic analogue of DahlquistsA-stability is proposed. It is shown that the discretization of the drift term alone controls theA-stability of the whole scheme. The quantitative effect of implicitness uponA-stability is also investigated, and stability regions are given for a family of implicit Runge-Kutta methods with optimal order of convergence.

Inequalities and numerical bounds for zeros of ultraspherical polynomials

We improve previous results concerning the monotonicity in

Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees

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