An applied mathematics perspective on stochastic modelling for climate

Systematic strategies from applied mathematics for stochastic modelling in climate are reviewed here. One of the topics discussed is the stochastic modelling of mid-latitude low-frequency variability through a few teleconnection patterns, including the central role and physical mechanisms responsible for multiplicative noise. A new low-dimensional stochastic model is developed here, which mimics key features of atmospheric general circulation models, to test the fidelity of stochastic mode reduction procedures. The second topic discussed here is the systematic design of stochastic lattice models to capture irregular and highly intermittent features that are not resolved by a deterministic parametrization. A recent applied mathematics design principle for stochastic column modelling with intermittency is illustrated in an idealized setting for deep tropical convection; the practical effect of this stochastic model in both slowing down convectively coupled waves and increasing their fluctuations is presented here.     

Hybrid Petri-nets for modelling and performance evaluation of supply chains

Modelling and analysis of complex and co-ordinated supply chains is a crucial task due to its inherent complexity and uncertainty. Therefore, the current research direction is to devise an efficient modelling technique that maps the dynamics of a real life supply chain and assists industrial practitioners in evaluating and comparing their network with other competing networks. Here an effective modelling technique, the hybrid Petri-net, is proposed to efficiently handle the dynamic behaviour of the supply chain. This modelling methodology embeds two enticing features, i.e. cost and batch sizes, in deterministic and stochastic Petri-net for the modelling and performance evaluation of supply chain networks. The model is subsequently used for risk management to investigate the issues of supply chain vulnerability and risk that has become a major research subject in recent years. In the test bed, a simple productive supply chain and an industrial supply chain are modelled with fundamental inventory replenishment policy. Subsequently, its performance is evaluated along with the identification and assessment of risk factors using analytical and simulation techniques respectively. Thus, this paper presents a complete package for industrial practitioners to model, evaluate performance and manage risky events in a supply chain.     

Optimisation approach to target costing under uncertainty with application to ICT-service

Target costing is a modern approach applied during product development that defines cost targets for products and its components. These cost targets are driven by customer requirements and achievable revenues. The intention of this paper is the integration of target costing with modern concepts of modelling uncertainty and management of risk based on optimisation. Contrary to the traditional focus of target costing on cost targets, this paper prefers a strategy for achieving a target profit. Moreover, in this paper target costing is understood as a continuous process with incremental changes of cost drivers, product and component design as well as product prices. Therefore, the change in costs and profit with respect to aforementioned control parameters is modelled by linear approximations. Hence, improved decisions concerning design and prices are derived by linear programming models. In practice, information concerning product and component costs, demand or customer preferences are not given with certainty. Therefore, we apply a stochastic programming approach to manage the risk inherent in the target costing process. After a general presentation, we apply our approach to the provision of an information and communication technology service where the level of uncertainty is considerable.     

Quantification of uncertainty modelling in stochastic analysis of FRP composites

The extensive use of FRP composite materials in a wide range of industries, and their inherent variability, has prompted many researchers to assess their performance from a probabilistic perspective. This paper attempts to quantify the uncertainty in FRP composites and to summarise the different stochastic modelling approaches suggested in the literature. Researchers have considered uncertainties starting at a constituent (fibre/matrix) level, at the ply level or at a coupon or component level. The constituent based approach could be further classified as a random variable based stochastic computational mechanics approach (whose usage is comparatively limited due to complex test data requirements and possible uncertainty propagation errors) and the more widely used morphology based random composite modelling which has been recommended for exploring local damage and failure characteristics. The ply level analysis using either stiffness/strength or fracture mechanics based models is suggested when the ply characteristics influence the composite properties significantly, or as a way to check the propagation of uncertainties across length scales. On the other hand, a coupon or component level based uncertainty modelling is suggested when global response characteristics govern the design objectives. Though relatively unexplored, appropriate cross-fertilisation between these approaches in a multi-scale modelling framework seems to be a promising avenue for stochastic analysis of composite structures. It is hoped that this review paper could facilitate and strengthen this process.     

New probabilistic approach to estimate vehicle failure trajectories in curve driving

The vehicle trajectories analysis on dangerous bends is an important task to improve road safety. This paper proposes a new methodology to predict failure trajectories of light vehicles in curve driving. It consists to use a stochastic modelling and reliability analysis in order to estimate the failure probability of vehicle trajectories.Firstly, we build probabilistic models able to describe real trajectories in a given bend. The models are transforms of scalar normalized second order stochastic processes which are stationary, ergodic and non-Gaussian. The process is characterized by its probability density function and its power spectral density estimated starting from the experimental trajectories. The probability density is approximated by using a development on the basis of Hermite polynomials.The second part is devoted to apply a reliability strategy intended to associate a risk level to each class of trajectories. Based on the joint use of probabilistic methods for modelling uncertainties, reliability analysis for assessing risk levels and statistics for classifying the trajectories, this approach provides a realistic answer to the tackled problem. Experiments show the relevance and effectiveness of this method.     

Modelling and estimating the reliability of stochastic dynamical systems with Markovian switching

In this paper, a general framework for the modelling of physical phenomena with stochastic dynamical systems switched by jump Markov processes is given. A methodology of the associated estimation procedures is provided. A particular attention is paid to the estimation of the underlying jump process, which is not observable.As an application, a stochastic model is proposed for the fatigue crack growth problem. The estimation of the model parameters is made on a real crack growth data set. We are thus able to simulate some crack growth paths which are used for reliability analysis through Monte Carlo techniques.     

The use of a grey-based Taguchi method for optimizing multi-response simulation problems

Simulation modelling is a widely accepted tool in system design and analysis, particularly when the system or environment has stochastic and nonlinear behaviour. However, it does not provide a method for optimization. In general, problems contain more than one response, which are often in conflict with each other. This article proposes a grey-based Taguchi method to solve the multi-response simulation problem. The grey-based Taguchi method is based on the optimizing procedure of the Taguchi method, and adopts grey relational analysis (GRA) to transfer multi-response problems into single-response problems. A practical case study from an integrated-circuit packaging company illustrates that differences in performance of the proposed grey-based Taguchi method and other methods found in the literature were not significant. The grey-based Taguchi method thus provides a new option when solving a multi-response simulation-optimization problem.     

Stabilization of linear systems by noise: application to flow induced oscillations

In a recent paper, Popp and Romberg reported on stabilization by grid generated turbulence of a smooth circular cylinder immersed in the wake from an identical cylinder in an array of aluminium tubes. Although these results were obtained experimentally, so far they have not been explained from a stochastic point of view on a rigorous theoretical basis. This paper provides analytical results which may explain this stabilization phenomenon by modelling the immersed cylinder as a two degree of freedom oscillator and the turbulence as a stochastic process. We obtain general asymptotic approximation for the moment Lyapunov exponent, g (p), and the Lyapunov exponent, lambda, for a four-dimensional system with one critical mode and another asymptotically stable mode driven by a small intensity stochastic process. These results, pertaining to p th moment stability and almost-sure stability, explain how the stochastic components that couple the stable and the critical modes play an important role in determining whether a noisy excitation can stabilize or destabilize the oscillatory critical mode. They are then applied to a prototypical flow induced oscillation model to justify the experimental results.     

Morphology-controllable modeling approach for a porous scaffold structure in tissue engineering

Heterogeneous structures represent an important new frontier for twenty-first-century engineering. In this paper, based on the shape function in the finite element method, a morphology-controllable modelling approach for constructing tissue engineering (TE) bone scaffold with various irregular pores is presented. The modelling approach consists of both irregular element modelling and the whole bone scaffold modelling. Accepting the elements’ information after all-hex mesh generation as inputs, the basic pore-making element can be mapped into various irregular elements based on the shape function. In the bone scaffold modelling, the Boolean difference between the contour model of the solid entity and the pore model which can be constructed by the Boolean operation union would generate a porous bone scaffold model. Compared to the stochastic geometry method and the discrete element packing method, the bone scaffold model obtained in this paper has a continuous, smooth contour and various irregular pores. Moreover, a decrease in computational complexity is achieved in this paper.     

A new active error compensatory method of machinery accuracy†

A new active error compensatory method for on-line cutting control has been developed for reduction of the form errors in machining. This approach is a combination of in-process gauging and model active compensatory control. In this approach, there are two features of substantial importance: stochastic modelling and optimum forecasting. Through stochastic modelling, the cutting tool error motions can be represented by a simple model without the necessity of obtaining the complex cause-and-effect relationships between various errors and error sources, and more importantly, it is possible to account for both repeatable and non-repeatable parts of errors. Optimum forecasting is an important prerequisite for a rational control strategy, considering the inevitable time delay associate with sensing, computation and actuation. The proposed control method was implemented for the control of cylindricity in boring operations. Through the controller simulation based on experimental measurements, the improvement in cylindricity accuracy confirms the effectiveness of this proposed strategy     

Static condensation based reduced order modelling of stochastically parametered large ordered systems

A surrogate stochastic reduced order model is developed for the analysis of randomly parametered structural systems with complex geometries. It is assumed that the mathematical model is available in terms of large ordered finite element (FE) matrices. The structure material properties are assumed to have spatial random inhomogeneities and are modelled as non-Gaussian random fields. A polynomial chaos expansion (PCE) based framework is developed for modelling the random fields directly from measurements and for uncertainty quantification of the response. Difficulties in implementing PCE due to geometrical complexities are circumvented by adopting PCE on a geometrically regular domain that bounds the physical domain and are shown to lead to mathematically equivalent representation. The static condensation technique is subsequently extended for stochastic cases based on PCE formalism to obtain reduced order stochastic FE models. The efficacy of the method is illustrated through two numerical examples.     

Stochastic hybrid systems for studying biochemical processes

Many protein and mRNA species occur at low molecular counts within cells, and hence are subject to large stochastic fluctuations in copy numbers over time. Development of computationally tractable frameworks for modelling stochastic fluctuations in population counts is essential to understand how noise at the cellular level affects biological function and phenotype. We show that stochastic hybrid systems (SHSs) provide a convenient framework for modelling the time evolution of population counts of different chemical species involved in a set of biochemical reactions. We illustrate recently developed techniques that allow fast computations of the statistical moments of the population count, without having to run computationally expensive Monte Carlo simulations of the biochemical reactions. Finally, we review different examples from the literature that illustrate the benefits of using SHSs for modelling biochemical processes.     

Introducing two hyperparameters in Bayesian estimation of wave spectra

An estimate of the on-site wave spectrum can be obtained from measured ship responses by use of Bayesian modelling, which means that the wave spectrum is found as the optimum solution from a probabilistic viewpoint. The paper describes the introduction of two hyperparameters into Bayesian modelling so that the prior information included in the modelling is based on two constraints: the wave spectrum must be smooth directional-wise as well as frequency-wise. Traditionally, only one hyperparameter has been used to control the amount of smoothing applied in both the frequency and directional ranges. From numerical simulations of stochastic response measurements, it is shown that the optimal hyperparameters, determined by use of ABIC (a Bayesian Information Criterion), correspond to the best estimate of the wave spectrum, which is not always the case when only one hyperparameter is included in the Bayesian modelling. The paper includes also an analysis of full-scale motion measurements where wave spectra estimated by the Bayesian modelling are compared with results from ocean surface measurements by satellite and from a wave radar. The agreement is found to be reasonable.     

Risk control for staff planning in e-commerce warehouses

Internet sale supply chains often need to fulfil quickly small orders for many customers. The resulting high demand and planning uncertainties pose new challenges for e-commerce warehouse operations. Here, we develop a decision support tool to assist managers in selecting appropriate risk policies and making staff planning decisions in uncertain conditions. Multistage stochastic modelling has been used to analyse risk optimisation approaches and expected value-based optimisation. Exhaustive numerical and practical validations have been performed to test the tool’s applicability. We demonstrate, using a Dutch e-commerce warehouse, that the multi-period conditional value at risk appears to be most applicable.     

Structure-Preserving Dynamics of Stochastic Epidemic Model with the Saturated Incidence Rate

The structure-preserving features of the nonlinear stochastic models are positivity, dynamical consistency and boundedness. These features have a significant role in different fields of computational biology and many more. Unfortunately, the existing stochastic approaches in literature do not restore aforesaid structure-preserving features, particularly for the stochastic models. Therefore, these gaps should be occupied up in literature, by constructing the structure-preserving features preserving numerical approach. This writing aims to describe the structure-preserving dynamics of the stochastic model. We have analysed the effect of reproduction number in stochastic modelling the same as described in the literature for deterministic modelling. The usual explicit stochastic numerical approaches are time-dependent. We have developed the implicitly driven explicit approach for the stochastic epidemic model. We have proved that the newly developed approach is preserving the structural, dynamical properties as positivity, boundedness and dynamical consistency. Finally, convergence analysis of a newly developed approach and graphically illustration is also presented.     

A new approach in modelling groundwater pollution under uncertainty

This article presents a new methodology to model the time and space evolution of a contaminant in a system of aquifers when certain components of the model, such as the geohydrologic information, the boundary conditions, the magnitude and variability of the sources or physical parameters are uncertain and defined in stochastic terms. The method is based on applications of modern mathematics to the solution of the resulting stochastic transport equations. This procedure exhibits considerable advantages over the existing stochastic modelling techniques. In particular, the semigroup solutions are not restricted to small variances in the stochastic elements (perturbation techniques), unsteady dynamic conditions are specifically considered, time and space randomness may be considered in the sources, the boundary conditions or the parameters, and the methodology reflects a well-posed functional-analytic theory. Several basic example problems are presented in order to illustrate the application of the methodology to the modelling of complex spatially and temporally distributed sources of interest in engineering hydrology today. Further potential applications of the method are very promising.     

The use of Monte Carlo techniques in statistical finite element methods for the determination of the structural behaviour of composite materials structural components

The main purpose of this paper is to develop an alternative approach to the classical deterministic design to account for uncertainties encountered during design, construction and lifetime of structures. This approach is based on the use of statistical tools in material characterisation and structural design by means of the finite element method combined with Monte Carlo techniques. In the first instance, the mechanical behaviour of different materials, including composite materials, is characterised by means of stochastic tools. A procedure based on the combination of various methods for estimating distribution parameters has been set up to ensure correct estimation. The second part of the paper focuses on the finite element modelling of structures combined with Monte Carlo simulation to deal with the stochastic aspects of the input parameters (material properties, structure geometry and loading conditions) and determine the probability distribution characterising the structural response.     

Accuracy of the Michaelis–Menten approximation when analysing effects of molecular noise

Quantitative biology relies on the construction of accurate mathematical models, yet the effectiveness of these models is often predicated on making simplifying approximations that allow for direct comparisons with available experimental data. The Michaelis–Menten (MM) approximation is widely used in both deterministic and discrete stochastic models of intracellular reaction networks, owing to the ubiquity of enzymatic activity in cellular processes and the clear biochemical interpretation of its parameters. However, it is not well understood how the approximation applies to the discrete stochastic case or how it extends to spatially inhomogeneous systems. We study the behaviour of the discrete stochastic MM approximation as a function of system size and show that significant errors can occur for small volumes, in comparison with a corresponding mass-action system. We then explore some consequences of these results for quantitative modelling. One consequence is that fluctuation-induced sensitivity, or stochastic focusing, can become highly exaggerated in models that make use of MM kinetics even if the approximations are excellent in a deterministic model. Another consequence is that spatial stochastic simulations based on the reaction–diffusion master equation can become highly inaccurate if the model contains MM terms.     

A stochastic optimization formulation for the transition from open pit to underground mining

As open pit mining of a mineral deposit deepens, the cost of extraction may increase up to a threshold where transitioning to mining through underground methods is more profitable. This paper provides an approach to determine an optimal depth at which a mine should transition from open pit to underground mining, based on managing technical risk. The value of a set of candidate transition depths is calculated by optimizing the production schedules for each depth’s unique open pit and underground operations which provide yearly discounted cash flow projections. By considering the sum of the open pit and underground mining portion’s value, the most profitable candidate transition depth is identified. The optimization model presented is based on a stochastic integer program that integrates geological uncertainty and manages technical risk. The proposed approach is tested on a gold deposit. Results show the benefits of managing geological uncertainty in long-term strategic decision-making frameworks. Additionally, the stochastic result produces a 9% net present value increase over a similar deterministic formulation. The risk-managing stochastic framework also produces operational schedules that reduce a mining project`s susceptibility to geological risk. This work aims to approve on previous attempts to solve this problem by jointly considering geological uncertainty and describing the optimal transition depth effectively in 3-dimensions.     

Stochastic Numerical Analysis for Impact of Heavy Alcohol Consumption on Transmission Dynamics of Gonorrhoea Epidemic

This paper aims to perform a comparison of deterministic and stochastic models. The stochastic modelling is a more realistic way to study the dynamics of gonorrhoea infection as compared to its corresponding deterministic model. Also, the deterministic solution is itself mean of the stochastic solution of the model. For numerical analysis, first, we developed some explicit stochastic methods, but unfortunately, they do not remain consistent in certain situations. Then we proposed an implicitly driven explicit method for stochastic heavy alcohol epidemic model. The proposed method is independent of the choice of parameters and behaves well in all scenarios. So, some theorems and simulations are presented in support of the article.