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In this paper we review Stochastic Response Surface Methodology as a tool for modeling uncertainty in the context of Risk Analysis. An application in the survival analysis in the breast cancer context is implemented with R software.
We consider nonlinear dynamic systems forced by colored noise. Using first approximation systems, we study dynamics of deviations of stochastic solutions from stable deterministic equilibria. Equations for the stationary second moments of deviations of random states are derived. An application of the elaborated theory to Van der Pol system driven by colored noise is given. A dependence of the dispersion on the time correlation of the colored noise is studied.
A nonlinear stochastic control system is considered. We discuss a problem of the synthesis of stochastic attractors and suggest a constructive approach based on the design of the stochastic sensitivity and corresponding confidence domains. Details of this approach are demonstrated for the problem of the control of confidence ellipses near the equilibrium. An example of the control for stochastic Van der Pol equation is presented.
Randomly forced glycolytic oscillations in Higgins model are studied both numerically and analytically. Numerical analysis is based on the direct simulation of the solutions of stochastic system. Non-uniformity of the stochastic bundle along the deterministic cycle is shown. For the analytical investigation of the randomly forced Higgins model, the stochastic sensitivity function technique and confidence domains method are applied. Results of the influence of additive noise on the cycle of this model are given.
A control problem for stochastically forced nonlinear continuous-time systems is considered. We propose a method for construction of the regulator that provides a preassigned probabilistic distribution of random states in stochastic equilibrium. Geometric criteria of the controllability are obtained. Constructive technique for the specification of attainability sets is suggested.
The goal of this article is the formulation of the principles of one of the possible strategies in implementing correlation between input random variables so as to be usable for algorithm development and the evaluation of Sobol’s sensitivity analysis. With regard to the types of stochastic computational models, which are commonly found in structural mechanics, an algorithm was designed for effective use in conjunction with Monte Carlo methods. Sensitivity indices are evaluated for all possible permutations of the decorrelation procedures for input parameters. The evaluation of Sobol’s sensitivity coefficients is illustrated on an example in which a computational model was used for the analysis of the resistance of a steel bar in tension with statistically dependent input geometric characteristics.
We consider continuous random dynamical systems with jumps. We estimate the dimension of the invariant measures and apply the results to a model of stochastic gene expression.
A kind of magnetic shape memory alloy (MSMA) microgripper is proposed in this paper, and its nonlinear dynamic characteristics are studied when the stochastic perturbation is considered. Nonlinear differential items are introduced to explain the hysteretic phenomena of MSMA, and the constructive relationships among strain, stress, and magnetic field intensity are obtained by the partial least-square regression method. The nonlinear dynamic model of a MSMA microgripper subjected to in-plane stochastic excitation is developed. The stationary probability density function of the system’s response is obtained, the transition sets of the system are determined, and the conditions of stochastic bifurcation are obtained. The homoclinic and heteroclinic orbits of the system are given, and the boundary of the system’s safe basin is obtained by stochastic Melnikov integral method. The numerical and experimental results show that the system’s motion depends on its parameters, and stochastic Hopf bifurcation appears in the variation of the parameters; the area of the safe basin decreases with the increase of the stochastic excitation, and the boundary of the safe basin becomes fractal. The results of this paper are helpful for the application of MSMA microgripper in engineering fields.
Stochastic networks of memristive devices were fabricated using a sponge as a skeleton material. Cyclic voltage-current characteristics, measured on the network, revealed properties, similar to the organic memristive device with deterministic architecture. Application of the external training resulted in the adaptation of the network electrical properties. The system revealed an improved stability with respect to the networks, composed from polymer fibers.
Despite the variety of existing methods, efficient generation of turbulent inflow conditions for large-eddy simulation (LES) models remains a challenging and active research area. Herein, we extend our previous research on the cell perturbation method, which uses a novel stochastic approach based upon finite amplitude perturbations of the potential temperature field applied within a region near the inflow boundaries of the LES domain [Muñoz-Esparza et al., “Bridging the transition from mesoscale to microscale turbulence in numerical weather prediction models,” Boundary-Layer Meteorol., 153, 409–440 (2014)]. The objective was twofold: (i) to identify the governing parameters of the method and their optimum values and (ii) to generalize the results over a broad range of atmospheric large-scale forcing conditions, U{sub g} = 5 − 25 m s{sup −1}, where U{sub g} is the geostrophic wind. We identified the perturbation Eckert number, Ec=U{sub g}{sup 2}/ρc{sub p}θ{sup ~}{sub pm}, to be the parameter governing the flow transition to turbulence in neutrally stratified boundary layers. Here, θ{sup ~}{sub pm} is the maximum perturbation amplitude applied, c{sub p} is the specific heat capacity at constant pressure, and ρ is the density. The optimal Eckert number was found for nonlinear perturbations allowed by Ec ≈ 0.16, which instigate formation of hairpin-like vortices that most rapidly transition to a developed turbulent state. Larger Ec numbers (linear small-amplitude perturbations) result in streaky structures requiring larger fetches to reach the quasi-equilibrium solution, while smaller Ec numbers lead to buoyancy dominated perturbations exhibiting difficulties for hairpin-like vortices to emerge. Cell perturbations with wavelengths within the inertial range of three-dimensional turbulence achieved identical quasi-equilibrium values of resolved turbulent kinetic energy, q, and Reynolds-shear stress, . In contrast, large-scale perturbations acting at the production range exhibited reduced levels of , due to the formation of coherent streamwise structures, while q was maintained, requiring larger fetches for the turbulent solution to stabilize. Additionally, the cell perturbation method was compared to a synthetic turbulence generator. The proposed stochastic approach provided at least the same efficiency in developing realistic turbulence, while accelerating the formation of large-scales associated with production of turbulent kinetic energy. Also, it is computationally inexpensive and does not require any turbulent information.
The effect of non-uniformity in bulk particle mass loading on the linear development of a particle-laden shear layer is analyzed by means of a stochastic Eulerian-Eulerian model. From the set of governing equations of the two-fluid model, a modified Rayleigh equation is derived that governs the linear growth of a spatially periodic disturbance. Eigenvalues for this Rayleigh equation are determined numerically using proper conditions at the co-flowing gas and particle interface locations. For the first time, it is shown that non-uniform loading of small-inertia particles (Stokes number (St) <0.2) may destabilize the inviscid mixing layer development as compared to the pure-gas flow. The destabilization is triggered by an energy transfer rate that globally flows from the particle phase to the gas phase. For intermediate St (1 < St < 10), a maximum stabilizing effect is computed, while at larger St, two unstable modes may coexist. The growth rate computations from linear stability analysis are verified numerically through simulations based on an Eulerian-Lagrangian (EL) model based on the inviscid Euler equations and a point particle model. The growth rates found in numerical experiments using the EL method are in very good agreement with growth rates from the linear stability analysis and validate the destabilizing effect induced by the presence of particles with low St.
The probability distributions of large-scale atmospheric and oceanic variables are generally skewed and heavy-tailed. We argue that their distinctive departures from Gaussianity arise fundamentally from the fact that in a quadratically nonlinear system with a quadratic invariant, the coupling coefficients between system components are not constant but depend linearly on the system state in a distinctive way. In particular, the skewness arises from a tendency of the system trajectory to linger near states of weak coupling. We show that the salient features of the observed non-Gaussianity can be captured in the simplest such nonlinear 2-component system. If the system is stochastically forced and linearly damped, with one component damped much more strongly than the other, then the strongly damped fast component becomes effectively decoupled from the weakly damped slow component, and its impact on the slow component can be approximated as a stochastic noise forcing plus an augmented nonlinear damping. In the limit of large time-scale separation, the nonlinear augmentation of the damping becomes small, and the noise forcing can be approximated as an additive noise plus a correlated additive and multiplicative noise (CAM noise) forcing. Much of the diversity of observed large-scale atmospheric and oceanic probability distributions can be interpreted in this minimal framework.
We investigate the effect of a single magnetic bead (MB) on the domain wall (DW) pinning/depinning fields of a DW trapped at the corner of an L-shaped magnetic nanodevice. DW propagation across the device is investigated using magnetoresistance measurements. DW pinning/depinning fields are characterized in as-prepared devices and after placement of a 1 μm-sized MB (Dynabeads{sup ®} MyOne{sup ™}) at the corner. The effect of the MB on the DW dynamics is seen as an increase in the depinning field for specific orientations of the device with respect to the external magnetic field. The shift of the depinning field, ΔB{sub dep} = 4.5–27.0 mT, is highly stable and reproducible, being significantly above the stochastic deviation which is about 0.5 mT. The shift in the deppinning field is inversely proportional to the device width and larger for small negative angles between the device and the external magnetic field. Thus, we demonstrate that DW-based devices can be successfully used for detection of single micron size MB.
Extraction of stochastic and deterministic components from empirical data—necessary for the reconstruction of the dynamics of the system—is discussed. We determine both components using the Kramers-Moyal expansion. In our earlier papers, we obtained large fluctuations in the magnitude of both terms for rare or extreme valued events in the data. Calculations for such events are burdened by an unsatisfactory quality of the statistics. In general, the method is sensitive to the binning procedure applied for the construction of histograms. Instead of the commonly used constant width of bins, we use here a constant number of counts for each bin. This approach—the fixed mass method—allows to include in the calculation events, which do not yield satisfactory statistics in the fixed bin width method. The method developed is general. To demonstrate its properties, here, we present the modified Kramers-Moyal expansion method and discuss its properties by the application of the fixed mass method to four representative heart rate variability recordings with different numbers of ectopic beats. These beats may be rare events as well as outlying, i.e., very small or very large heart cycle lengths. The properties of ectopic beats are important not only for medical diagnostic purposes but the occurrence of ectopic beats is a general example of the kind of variability that occurs in a signal with outliers. To show that the method is general, we also present results for two examples of data from very different areas of science: daily temperatures at a large European city and recordings of traffics on a highway. Using the fixed mass method, to assess the dynamics leading to the outlying events we studied the occurrence of higher order terms of the Kramers-Moyal expansion in the recordings. We found that the higher order terms of the Kramers-Moyal expansion are negligible for heart rate variability. This finding opens the possibility of the application of the Langevin equation to the whole range of empirical signals containing rare or outlying events. Note, however, that the higher order terms are non-negligible for the other data studied here and for it the Langevin equation is not applicable as a model.
No abstract prepared.
Capacitated arc routing problem (CARP) is the youngest generation of graph theory that focuses on solving the edge/arc routing for optimality. Since many years, operational research devoted to CARP counterpart, known as vehicle routing problem (VRP), which does not fit to several real cases such like waste collection problem and road maintenance. In this paper, we highlighted several extensions of capacitated arc routing problem (CARP) that represents the real-life problem of vehicle operation in waste collection. By purpose, CARP is designed to find a set of routes for vehicles that satisfies all pre-setting constraints in such that all vehicles must start and end at a depot, service a set of demands on edges (or arcs) exactly once without exceeding the capacity, thus the total fleet cost is minimized. We also addressed the differentiation between CARP and VRP in waste collection. Several issues have been discussed including stochastic demands and time window problems in order to show the complexity and importance of CARP in the related industry. A mathematical model of CARP and its new version is presented by considering several factors such like delivery cost, lateness penalty and delivery time.
This paper proposes a newly developed one-step derivative-free method, that is 2-stage stochastic Runge-Kutta (SRK2) to approximate the solution of stochastic delay differential equations (SDDEs) with a constant time lag, r > 0. General formulation of stochastic Runge-Kutta for SDDEs is introduced and Stratonovich Taylor series expansion for numerical solution of SRK2 is presented. Local truncation error of SRK2 is measured by comparing the Stratonovich Taylor expansion of the exact solution with the computed solution. Numerical experiment is performed to assure the validity of the method in simulating the strong solution of SDDEs.
In this paper, a Gompertzian model with random effects is introduced to describe the cervical cancer growth. The parameters values of the mathematical model are estimated via maximum likehood estimation. We apply 4-stage Runge-Kutta (SRK4) for solving the stochastic model numerically. The efficiency of mathematical model is measured by comparing the simulated result and the clinical data of the cervical cancer growth. Low values of root mean-square error (RMSE) of Gompertzian model with random effect indicate good fits.