Stochastic Element Research Articles

Complementing CO<sub>2</sub> emission reduction by solar radiation management might strongly enhance future welfare

Abstract. Solar radiation management (SRM) has been proposed as a means to reduce global warming in spite of high greenhouse-gas concentrations and to lower the chance of warming-induced tipping points. However, SRM may cause economic damages and its feasibility is still uncertain. To investigate the trade-off between these (economic) gains and damages, we incorporate SRM into a stochastic dynamic integrated assessment model and perform the first rigorous cost–benefit analysis of sulfate-based SRM under uncertainty, treating warming-induced climate tipping and SRM failure as stochastic elements. We find that within our model, SRM has the potential to greatly enhance future welfare and merits being taken seriously as a policy option. However, if only SRM and no CO2 abatement is used, global warming is not stabilised and will exceed 2 K. Therefore, even if successful, SRM can not replace but only complement CO2 abatement. The optimal policy combines CO2 abatement and modest SRM and succeeds in keeping global warming below 2 K.

Erratum: Variability of collective dynamics in random tree networks of strongly coupled stochastic excitable elements [Phys. Rev. E 98 , 052303 (2018)

Received 5 June 2019DOI:https://doi.org/10.1103/PhysRevE.99.069902©2019 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAction potential propagationStochastic processesSynchronizationPhysical SystemsCoupled oscillatorsTree networkTechniquesHodgkin-Huxley modelSpiking neuron modelsStatistical methodsBiological Physics

Stochastic PEEC Method Based on Polynomial Chaos Expansion

The new stochastic partial element equivalent circuit (PEEC) method is proposed for uncertainty quantification in electromagnetic problems when material parameters are considered as random variables. The proposed formulation is derived using polynomial chaos expansion (PCE) and Galerkin projection. For the first time, the well-known advantages of PEEC are combined with those of PCE techniques, which can tackle also large variations in the random parameters. Volume conductive media are first considered in the formulation, which is then extended to dielectric, magnetic, and surface conductive media.

Optimal threshold density in a stochastic resource management model with pulse intervention

AbstractHuman activities and agricultural practices are having huge impacts on the development of fishery and land resources through different ways. To model such systems that involve harvesting, an impulsive model of natural resources with a stochastic noise perturbation element is formulated to study the relationship between (a) the maximal expectation of biomass after harvesting or fishing events and (b) the minimal expectation of pest biomass and the number of times pesticide is applied. Using a detailed analytical treatment, time estimation, and numerical demonstrations, we establish that the proposed mechanism is capable of maximizing fish populations at the end of a fishing season and minimizing pest numbers after a crop harvesting season once the intensity of the noise is relatively small. Investigations of the effects of different parameters reveal that theoretical predictions from the new stochastic model accord with those from the deterministic case.Recommendations for Resource Managers Various measures can be implemented to manage natural resources, such as adjusting fishing quantity and intensity to maximize fish population. In the natural environment, population growth is inevitably affected by the environment noise. So it is important to understand the noise effect to maintain sustainability of resources. Investigated methods are useful to converse resources and can be widely applied to control pests.

Population-based simulation optimization for urban mass rapid transit networks

In this paper, we present a simulation-based headway optimization for urban mass rapid transit networks. The underlying discrete event simulation model contains several stochastic elements, including time-dependent demand and turning maneuver times as well as direction-dependent vehicle travel and passenger transfer times. Passenger creation is a Poisson process that uses hourly origin–destination-matrices based on anonymous mobile phone and infrared count data. The numbers of passengers on platforms and within vehicles are subject to capacity restrictions. As a microscopic element, passenger distribution along platforms and within vehicles is considered. The bi-objective problem, involving cost reduction and service level improvement, is transformed into a single-objective optimization problem by normalization and scalarization. Population-based evolutionary algorithms and different solution encoding variants are applied. Computational experience is gained from test instances based on real-world data (i.e., the Viennese subway network). A covariance matrix adaptation evolution strategy performs best in most cases, and a newly developed encoding helps accelerate the optimization process by producing better short-term results.

Simple geometrical aspects of grain growth in the framework of Herring's analysis and a disclination model

The time change of the area of a single grain is calculated in 2D by applying Herring's analysis. The grain is a regular n-sided polygon with its corners representing triple junctions (TJs). The resulting change of grain area is compared to the Von Neumann–Mullins analysis, where grain boundaries (GB) are curved and angles at TJs are equilibrium angles. The rates of area change are similar with the largest deviation for triangles. For both cases of different angles at TJs polygons with n < 6 shrink and those with n > 6 grow. Pieces of evidence are provided for describing the GB-movement by disclination generation and motion due to thermal fluctuations. Thus besides energy considerations a stochastic element comes into play.

Implications of Realistic Fracture Criteria on Crack Morphology

We study the effects realistic fracture criteria have on crack morphology obtained in numerical simulations with a stochastic discrete element method. Results are obtained with two criteria which are consistent with the theory of elasticity and compared with previous results using the original criterion, chosen when the method was first published. The conventional choice has been to consider the combined loading as an interaction between bending and tensile forces only, leaving out shear forces altogether. Moreover the combination of bending and tension used in the old criterion is correct only for plastic deformations. Our results show that the inclusion of shear forces have a profound effect on crack morphology. We consider two types of external loading, torsion applied to a circular cylinder and tension applied to a cube. In the tensile case, the exponent which characterizes scaling of crack roughness with system size is found to be very close to the experimental value zeta=0.5 when realistic fracture criteria are used. In the present calculations we obtain zeta=0.52, a value which remains constant for all disorders. It is proposed that the small-scale exponent zeta=0.8 appears as a consequence of cleavage between crystal planes and consequently requires a different fracture criterion than that which is used on larger scales.

A methodology for sensitivity analysis of spatial features in forecasts: the stochastic kinetic energy backscatter scheme

Stochastic kinetic energy backscatter schemes (SKEBSs) are introduced in numerical weather forecast models to represent uncertainties related to unresolved subgrid‐scale processes. These schemes are formulated using a set of parameters that must be determined using physical knowledge and/or to obtain a desired outcome. Here, a methodology is developed for assessing the effect of four factors on spatial features of forecasts simulated by the SKEBS‐enabled Weather Research and Forecasting model. The four factors include two physically motivated SKEBS parameters (the determining amplitude of perturbations applied to stream function and potential temperature tendencies), a purely stochastic element (a seed used in generating random perturbations) and a factor reflecting daily variability. A simple threshold‐based approach for identifying coherent objects within forecast fields is employed, and the effect of the four factors on object features (e.g. number, size and intensity) is assessed. Four object types are examined: upper‐air jet streaks, low‐level jets, precipitation areas and frontal boundaries. The proposed method consists of a set of standard techniques in experimental design, based on the analysis of variance, tailored to sensitivity analysis. More specifically, a Latin square design is employed to reduce the number of model simulations necessary for performing the sensitivity analysis. Fixed effects and random effects models are employed to assess the main effects and the percentage of the total variability explained by the four factors. It is found that the two SKEBS parameters do not have an appreciable and/or statistically significant effect on any of the examined object features.

Simple Geometrical Aspects of Grain Growth in the Framework of Herring’s Analysis and a Disclination Model

The time change of the area of a single grain is calculated in 2D by applying Herring’s analysis. The grain is a regular n-sided polygon with its corners representing triple junctions (TJs). The resulting change of grain area is compared to the Von Neumann–Mullins analysis, where grain boundaries (GB) are curved and angles at TJs are equilibrium angles. The rates of area change are similar with the largest deviation for triangles. For both cases of different angles at TJs polygons with n 6 grow. Pieces of evidence are provided for describing the GB-movement by disclination generation and motion due to thermal fluctuations. Thus besides energy considerations a stochastic element comes into play.

A stochastic spectral finite element method for solution of faulting-induced wave propagation in materially random continua without explicitly modeled discontinuities

This paper proposes a new efficient stochastically adapted spectral finite element method to simulate fault dislocation and its wave propagation consequences. For this purpose, a dynamic form of the split node technique is formulated and developed to stochastic spectral finite element method in order to model fault dislocation happening within a random media without increasing computational demand caused by discontinuities. As discontinuities are not modeled explicitly herein, no additional degrees of freedom are implemented in the proposed method due to the discontinuities, while effects of these discontinuities are preserved. Therefore, the present method simultaneously includes merits of stochastic finite element method, spectral finite element method and the spilt node technique, thereby providing a new numerical tool for analysis of wave propagation under fault dislocation in random media. Several numerical simulations are solved by the proposed method, which present stochastic analysis of fault slip-induced wave propagation in layered random media. Formulations and numerical results demonstrate capability, application and efficiency of this novel method.

Stochastic interval analysis for structural natural frequencies based on stochastic hybrid perturbation edge-based smoothing finite element method

Realistically calculating of natural frequencies is crucial for studying dynamical characteristics of structures in engineering. In this paper, the gradient smoothing technique is first applied into the conventional finite element method (named ES-FEM), to improve the accuracy of the deterministic calculation of natural frequencies. Then, the hybrid random and interval uncertain parameters are introduced into the proposed ES-FEM based on hybrid stochastic and interval perturbation method. Expressions for the mean value and variance of natural frequencies are derived by combining the random interval perturbation method and the random interval moment method. The upper and lower bounds of mean value and variance of natural frequencies are calculated based on hybrid perturbation vertex method. The proposed method (denoted ESHPVM) is compared with the different methods. The high accuracy and efficiency of the proposed methods are verified by two numerical examples.

Prey availability, environmental constraints, and aggregation dictate population distribution of an imperiled fish

AbstractAccelerated global loss of fish species underscores an urgent need to elucidate constraints on spatial and temporal population patterns and to mitigate human‐mediated impacts accordingly. We analyzed 20 yr of trawl data using a spatiotemporally explicit, hierarchically Bayesian model to define density distributions of an imperiled anadromous fish—the delta smelt, Hypomesus transpacificus—in the heavily modified and highly used San Francisco Estuary. To reduce management reliance on an estuary‐wide annual population index that minimizes data resolution and ignores uncertainty, our model took a spatiotemporal approach and explicitly accounted for sampling and observation variation through both stochastic and deterministic elements. Our model demonstrated that juvenile smelt density has decreased over annual time and displays a distinct seasonal pattern (the combination of growth into the sampled size class, mortality, and movement into and out of the sampling frame) of increases in March–June followed by decreases in June–August. Smelt density was highly spatiotemporally autocorrelated and strongly tracked prey availability yet was also constrained by local hydrological factors (salinity, turbidity, velocity). Specifically, juvenile smelt preferred slightly saline, turbid, generally slow‐moving water with ample copepod prey. However, poor swimming capabilities reduced the capacity of juvenile smelt to mix throughout the estuary and find optimal habitat, emphasizing the importance of accounting for spatiotemporal autocorrelation in species distribution models. Further, whole‐estuary outflow appeared to influence the spatial distribution of covariates and juvenile smelt, such that smelt densities tended to peak closer to the marine zone at a lower maximal value when outflow was high. The predictions of our model accurately matched observed patterns of juvenile delta smelt catch while embracing uncertainty (although the predictions tended to exaggerate the total smelt catch variation via over‐predicted tail values). These results highlight that data‐driven improvements can be made to the analytical methods currently used to guide delta smelt management and lay the groundwork for mitigating the impacts of water use, flow regimes, habitat alteration, and climate change on an endangered species in a highly impacted and economically consequential habitat.

Model order reduction accelerated Monte Carlo stochastic isogeometric method for the analysis of structures with high-dimensional and independent material uncertainties

Structural stochastic analysis is vital to engineering. However, current material related uncertainty methods are mostly limited to low dimension, and they mostly remain unable to account for spatially uncorrelated material uncertainties. They are not representative of realistic and practical engineering situations. In particular, it is more serious for composite structures comprised of dissimilar materials. Therefore, we propose a novel model order reduction via proper orthogonal decomposition accelerated Monte Carlo stochastic isogeometric method (IGA-POD-MCS) for stochastic analysis of exactly represented (composite) structures. This approach particularly enables high-dimensional material uncertainties wherein the characteristics of each element are independent. And the novelties include: (1) the structural geometry is exactly modeled thanks to isogeometric analysis (IGA), as well as providing more accurate deterministic and stochastic solutions, (2) we innovatively consider high-dimensional and independent material uncertainties by separating the stochastic mesh from the IGA mesh, and modeling different stochastic elements to have different (independent) uncertainty behaviors, (3) the classical Monte Carlo simulation (MCS) is employed to universally solve the high-dimensional uncertainty problem. However, to circumvent its computational expense, we employ model order reduction via proper orthogonal decomposition (POD) into the IGA coupled MCS stochastic analysis. In particular, we observe that this work decouples all IGA elements and hence permits independent uncertainty models easily, thereby the engineering problem is modeled to be more realistic and authentic. Several illustrative numerical examples verify the proposed IGA-POD-MCS approach is effective and efficient; and the larger the scale of the problem is, the more advantageous the method will become.

Stochastic Constitutive Model of Isotropic Thin Fiber Networks Based on Stochastic Volume Elements.

Thin fiber networks are widely represented in nature and can be found in man-made materials such as paper and packaging. The strength of such materials is an intricate subject due to inherited randomness and size-dependencies. Direct fiber-level numerical simulations can provide insights into the role of the constitutive components of such networks, their morphology, and arrangements on the strength of the products made of them. However, direct mechanical simulation of randomly generated large and thin fiber networks is characterized by overwhelming computational costs. Herein, a stochastic constitutive model for predicting the random mechanical response of isotropic thin fiber networks of arbitrary size is presented. The model is based on stochastic volume elements (SVEs) with SVE size-specific deterministic and stochastic constitutive law parameters. The randomness in the network is described by the spatial fields of the uniaxial strain and strength to failure, formulated using multivariate kernel functions and approximate univariate probability density functions. The proposed stochastic continuum approach shows good agreement when compared to direct numerical simulation with respect to mechanical response. Furthermore, strain localization patterns matched the one observed in direct simulations, which suggests an accurate prediction of the failure location. This work demonstrates that the proposed stochastic constitutive model can be used to predict the response of random isotropic fiber networks of arbitrary size.

An inverse micro-mechanical analysis toward the stochastic homogenization of nonlinear random composites

An inverse Mean-Field Homogenization (MFH) process is developed to improve the computational efficiency of non-linear stochastic multiscale analyzes by relying on a micro-mechanics model. First full-field simulations of composite Stochastic Volume Element (SVE) realizations are performed to characterize the homogenized stochastic behavior. The uncertainties observed in the non-linear homogenized response, which result from the uncertainties of their micro-structures, are then translated to an incremental-secant MFH formulation by defining the MFH input parameters as random effective properties. These effective input parameters, which correspond to the micro-structure geometrical information and to the material phases model parameters, are identified by conducting an inverse analysis from the full-field homogenized responses. Compared to the direct finite element analyzes on SVEs, the resulting stochastic MFH process reduces not only the computational cost, but also the order of uncertain parameters in the composite micro-structures, leading to a stochastic Mean-Field Reduced Order Model (MF-ROM). A data-driven stochastic model is then built in order to generate the random effective properties under the form of a random field used as entry for the stochastic MF-ROM embedded in a Stochastic Finite Element Method (SFEM). The two cases of elastic Unidirectional (UD) fibers embedded in an elasto-plastic matrix and of elastic UD fibers embedded in a damage-enhanced elasto-plastic matrix are successively considered. In order to illustrate the capabilities of the method, the stochastic response of a ply is studied under transverse loading condition.

Surrogate accelerated Bayesian inversion for the determination of the thermal diffusivity of a material

Determination of the thermal properties of a material is an important task in many scientific and engineering applications. How a material behaves when subjected to high or fluctuating temperatures can be critical to the safety and longevity of a system’s essential components. The laser flash experiment is a well-established technique for indirectly measuring the thermal diffusivity, and hence the thermal conductivity, of a material. In previous works, optimization schemes have been used to find estimates of the thermal conductivity and other quantities of interest which best fit a given model to experimental data. Adopting a Bayesian approach allows for prior beliefs about uncertain model inputs to be conditioned on experimental data to determine a posterior distribution, but probing this distribution using sampling techniques such as Markov chain Monte Carlo methods can be incredibly computationally intensive. This difficulty is especially true for forward models consisting of time-dependent partial differential equations. We pose the problem of determining the thermal conductivity of a material via the laser flash experiment as a Bayesian inverse problem in which the laser intensity is also treated as uncertain. We introduce a parametric surrogate model that takes the form of a stochastic Galerkin finite element approximation, also known as a generalized polynomial chaos expansion, and show how it can be used to sample efficiently from the approximate posterior distribution. This approach gives access not only to the sought-after estimate of the thermal conductivity but also important information about its relationship to the laser intensity, and information for uncertainty quantification. Moreover, this approach leads to significant speed up over traditional methods by orders of magnitude. We also investigate the effects of the spatial profile of the laser on the estimated posterior distribution for the thermal conductivity.

Risk Taking and Fiscal Smoothing with Sovereign Wealth Funds in Advanced Economies

In an economy with a sovereign wealth fund (SWF), the government may draw on the fund to supplement other government revenues. If the fund is invested in risky assets, this introduces a new stochastic element into the government’s budget. We analyze the interaction between the draw from and risk taking in the SWF. Using non-expected utility preferences, we distinguish between intended changes and stochastic changes in the SWF draws over time. We show that the desire for smoothness in taxes and public services translates into smoothing of SWF draws and lower risk taking. It can even lead to procyclical rebalancing of the SWF portfolio. Future interest rates are associated with interest-rate risk. We show that this risk may lead to a higher optimal equity share in the SWF portfolio. Policy makers can use the draws from the SWF to smooth over time variation in risk-free rates.

NUMERICAL APPROXIMATION OF ELLIPTIC PROBLEMS WITH LOG-NORMAL RANDOM COEFFICIENTS

In this work, we consider a non-standard preconditioning strategy for the numerical approximation of the classical elliptic equations with log-normal random coefficients. In \cite{Wan_model}, a Wick-type elliptic model was proposed by modeling the random flux through the Wick product. Due to the lower-triangular structure of the uncertainty propagator, this model can be approximated efficiently using the Wiener chaos expansion in the probability space. Such a Wick-type model provides, in general, a second-order approximation of the classical one in terms of the standard deviation of the underlying Gaussian process. Furthermore, when the correlation length of the underlying Gaussian process goes to infinity, the Wick-type model yields the same solution as the classical one. These observations imply that the Wick-type elliptic equation can provide an effective preconditioner for the classical random elliptic equation under appropriate conditions. We use the Wick-type elliptic model to accelerate the Monte Carlo method and the stochastic Galerkin finite element method. Numerical results are presented and discussed.

A Bramble--Pasciak Conjugate Gradient Method for Discrete Stokes Equations with Random Viscosity

We study the iterative solution of linear systems of equations arising from stochastic Galerkin finite element discretizations of saddle point problems. We focus on the Stokes model with random dat...

Convergence of Adaptive Stochastic Galerkin FEM

We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite element approximations. The algorithms are driven by the energy error reduction estimates derived from two-level a posteriori error indicators for spatial approximations and hierarchical a posteriori error indicators for parametric approximations. The focus of this work is on the mathematical foundation of the adaptive algorithms in the sense of rigorous convergence analysis. In particular, we prove that the proposed algorithms drive the underlying energy error estimates to zero.